1876:
8643:
1591:
8484:
5493:
10696:
10093:
9785:
9532:
1871:{\displaystyle {\begin{aligned}{}&X_{n}\ \xrightarrow {d} \ X,\ \ X_{n}\ \xrightarrow {\mathcal {D}} \ X,\ \ X_{n}\ \xrightarrow {\mathcal {L}} \ X,\ \ X_{n}\ \xrightarrow {d} \ {\mathcal {L}}_{X},\\&X_{n}\rightsquigarrow X,\ \ X_{n}\Rightarrow X,\ \ {\mathcal {L}}(X_{n})\to {\mathcal {L}}(X),\\\end{aligned}}}
8638:{\displaystyle {\begin{matrix}{\xrightarrow {\overset {}{L^{s}}}}&{\underset {s>r\geq 1}{\Rightarrow }}&{\xrightarrow {\overset {}{L^{r}}}}&&\\&&\Downarrow &&\\{\xrightarrow {\text{a.s.}}}&\Rightarrow &{\xrightarrow {p}}&\Rightarrow &{\xrightarrow {d}}\end{matrix}}}
5760:
73:
and they formalize the idea that certain properties of a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence
127:
While the above discussion has related to the convergence of a single series to a limiting value, the notion of the convergence of two series towards each other is also important, but this is easily handled by studying the sequence defined as either the difference or the ratio of the two series.
3692:
represent the distribution of possible outputs by the algorithm. Because the pseudorandom number is generated deterministically, its next value is not truly random. Suppose that as you observe a sequence of randomly generated numbers, you can deduce a pattern and make increasingly accurate
5348:
856:
Convergence in distribution is the weakest form of convergence typically discussed, since it is implied by all other types of convergence mentioned in this article. However, convergence in distribution is very frequently used in practice; most often it arises from application of the
4126:
4542:
9188:
7262:
9071:
9324:
8964:
5785:
on the space of random variables. This means there is no topology on the space of random variables such that the almost surely convergent sequences are exactly the converging sequences with respect to that topology. In particular, there is no metric of almost sure
10554:
8868:
8739:
74:
relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution.
5337:
9931:
9623:
5636:
9378:
6623:
4257:
796:
6502:
513:
Suppose a new dice factory has just been built. The first few dice come out quite biased, due to imperfections in the production process. The outcome from tossing any of them will follow a distribution markedly different from the desired
3863:
4948:
10355:
6631:
by using sure convergence compared to using almost sure convergence. The difference between the two only exists on sets with probability zero. This is why the concept of sure convergence of random variables is very rarely used.
4642:
Not every sequence of random variables which converges to another random variable in distribution also converges in probability to that random variable. As an example, consider a sequence of standard normal random variables
2172:
8184:
6840:
4426:
5155:
3099:
3001:
839:
674:
5569:
8461:
2074:
50:. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit
8253:
7928:
8102:
8051:
7570:
7519:
2906:
4000:
2592:
3390:
3300:
11677:
6975:
3191:
1997:
10982:
5005:
Consider a man who tosses seven coins every morning. Each afternoon, he donates one pound to a charity for each head that appeared. The first time the result is all tails, however, he will stop permanently.
2344:
10885:
8372:
8314:
7997:
7715:
7657:
276:
9086:
7314:
6047:
7846:
7795:
7436:
7385:
10825:
5488:{\displaystyle \mathbb {P} {\Bigl (}\limsup _{n\to \infty }{\bigl \{}\omega \in \Omega :|X_{n}(\omega )-X(\omega )|>\varepsilon {\bigr \}}{\Bigr )}=0\quad {\text{for all}}\quad \varepsilon >0.}
5238:
2814:
2698:
7117:
2510:
provides several equivalent definitions of convergence in distribution. Although these definitions are less intuitive, they are used to prove a number of statistical theorems. The lemma states that
10762:
8979:
5229:
492:
6167:
10691:{\displaystyle \left.{\begin{matrix}X_{n}\xrightarrow {\overset {}{\text{a.s.}}} X\\|X_{n}|<Y\\\mathbb {E} <\infty \end{matrix}}\right\}\quad \Rightarrow \quad X_{n}\xrightarrow {L^{1}} X}
1596:
9236:
1062:
8879:
8783:
82:"Stochastic convergence" formalizes the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle into a pattern. The pattern may for instance be
6530:
1925:
8657:
2635:
1214:
6274:
54:
of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the value a random variable will take, rather than just the distribution.
10505:
5942:
4402:
3977:
are independent (and thus convergence in probability is a condition on the joint cdf's, as opposed to convergence in distribution, which is a condition on the individual cdf's), unless
2434:
5881:
4632:
3985:
cannot, whenever the deterministic value is a discontinuity point (not isolated), be handled by convergence in distribution, where discontinuity points have to be explicitly excluded.
2210:
6419:
5982:
10088:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{p}}}\ X,\ \ Y_{n}\ {\xrightarrow {\overset {}{p}}}\ Y\ \quad \Rightarrow \quad (X_{n},Y_{n})\ {\xrightarrow {\overset {}{p}}}\ (X,Y)}
9780:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{d}}}\ X,\ \ Y_{n}\ {\xrightarrow {\overset {}{d}}}\ c\ \quad \Rightarrow \quad (X_{n},Y_{n})\ {\xrightarrow {\overset {}{d}}}\ (X,c)}
4343:
1409:
190:
9527:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{d}}}\ X,\ \ |X_{n}-Y_{n}|\ {\xrightarrow {\overset {}{p}}}\ 0\ \quad \Rightarrow \quad Y_{n}\ {\xrightarrow {\overset {}{d}}}\ X}
4727:
966:
913:
4806:
1093:
521:
As the factory is improved, the dice become less and less loaded, and the outcomes from tossing a newly produced die will follow the uniform distribution more and more closely.
11034:
4991:
Consider an animal of some short-lived species. We record the amount of food that this animal consumes per day. This sequence of numbers will be unpredictable, but we may be
4174:
6732:
6703:
2747:
11902:
6206:
849:
Loosely, with this mode of convergence, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by a given
6100:
1512:
1376:
1286:
6307:
5826:
722:
8776:
5755:{\displaystyle \mathbb {P} {\Bigl (}\omega \in \Omega \colon \,d{\big (}X_{n}(\omega ),X(\omega ){\big )}\,{\underset {n\to \infty }{\longrightarrow }}\,0{\Bigr )}=1}
1334:
425:
1470:
1435:
9861:
9614:
6882:
5631:
4781:
4754:
4668:
1167:
381:
326:
354:
7741:
7596:
7462:
3778:
1558:
1244:
5088:
6902:
6367:
6347:
6327:
6067:
4970:
4801:
4303:
3414:
3324:
3234:
3214:
3122:
3024:
2926:
2837:
2725:
1578:
1532:
1308:
1140:
1113:
445:
299:
10277:
11681:
12109:
3732:
The basic idea behind this type of convergence is that the probability of an âunusualâ outcome becomes smaller and smaller as the sequence progresses.
2094:
12126:
11346:
8107:
6755:
5342:
853:. More precisely, the distribution of the associated random variable in the sequence becomes arbitrarily close to a specified fixed distribution.
8478:
The chain of implications between the various notions of convergence are noted in their respective sections. They are, using the arrow notation:
3031:
2933:
613:
5509:
8377:
2008:
3739:
if it converges in probability to the quantity being estimated. Convergence in probability is also the type of convergence established by the
11546:
8197:
7851:
4121:{\displaystyle X_{n}\ \xrightarrow {p} \ X,\ \ X_{n}\ \xrightarrow {P} \ X,\ \ {\underset {n\to \infty }{\operatorname {plim} }}\,X_{n}=X.}
3578:
8056:
8005:
7524:
7473:
2844:
2530:
3331:
3241:
11074:
is essentially a question of convergence, and many of the same concepts and relationships used above apply to the continuity question.
6919:
4537:{\displaystyle d(X,Y)=\inf \!{\big \{}\varepsilon >0:\ \mathbb {P} {\big (}|X-Y|>\varepsilon {\big )}\leq \varepsilon {\big \}}}
3129:
1938:
11052:
10892:
10532:
However, for a sequence of mutually independent random variables, convergence in probability does not imply almost sure convergence.
2280:
124:
These other types of patterns that may arise are reflected in the different types of stochastic convergence that have been studied.
11926:
10832:
9183:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ X\quad \Rightarrow \quad X_{n}\ {\xrightarrow {\overset {}{L^{s}}}}\ X,}
8319:
8261:
7941:
7662:
7604:
706:
206:
17:
7270:
7257:{\displaystyle {\overset {}{X_{n}\xrightarrow {L^{r}} X}}\quad \Rightarrow \quad \lim _{n\to \infty }\mathbb {E} =\mathbb {E} .}
5987:
7800:
7749:
7390:
7339:
4547:
10781:
9066:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ X\quad \Rightarrow \quad X_{n}\ {\xrightarrow {\overset {}{p}}}\ X}
2758:
2642:
11635:
11616:
11597:
11575:
11556:
11527:
11504:
11485:
11466:
11439:
11420:
11384:
11241:
11087:
3617:
4271:
In the opposite direction, convergence in distribution implies convergence in probability when the limiting random variable
10722:
9319:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{d}}}\ c\quad \Rightarrow \quad X_{n}\ {\xrightarrow {\overset {}{p}}}\ c,}
5190:
3688:
Suppose that a random number generator generates a pseudorandom floating point number between 0 and 1. Let random variable
453:
8959:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{p}}}\ X\quad \Rightarrow \quad X_{n}\ {\xrightarrow {\overset {}{d}}}\ X}
6108:
12058:
11709:
8863:{\displaystyle X_{n}\ \xrightarrow {\overset {}{p}} \ X\quad \Rightarrow \quad X_{n_{k}}\ \xrightarrow {\text{a.s.}} \ X}
6627:
Sure convergence of a random variable implies all the other kinds of convergence stated above, but there is no payoff in
997:
12219:
11984:
8734:{\displaystyle X_{n}\ {\xrightarrow {\text{a.s.}}}\ X\quad \Rightarrow \quad X_{n}\ {\xrightarrow {\overset {}{p}}}\ X}
12121:
12040:
2231:, and even to the ârandom variablesâ which are not measurable â a situation which occurs for example in the study of
1897:
1170:
710:
135:
101:
That the probability distribution describing the next outcome may grow increasingly similar to a certain distribution
9081:-th order mean implies convergence in lower order mean, assuming that both orders are greater than or equal to one:
2597:
6209:
1176:
985:
920:
6215:
3704:
random numbers. As you learn the pattern and your guesses become more accurate, not only will the distribution of
3559:
10450:
5886:
5332:{\displaystyle \mathbb {P} {\Bigl (}\omega \in \Omega :\lim _{n\to \infty }X_{n}(\omega )=X(\omega ){\Bigr )}=1.}
515:
11673:
10199:. Usually, convergence in distribution does not imply convergence almost surely. However, for a given sequence {
3735:
The concept of convergence in probability is used very often in statistics. For example, an estimator is called
2387:
12020:
12000:
5831:
4349:
2180:
5947:
12214:
12116:
12050:
11954:
11837:
11082:
11067:
10538:
11257:
2498:
2459:
2243:
3610:
1381:
836:
grows larger, the shape of the probability density function gets closer and closer to the
Gaussian curve.
141:
12104:
11974:
4673:
4308:
925:
872:
6618:{\displaystyle \left\{\omega \in \Omega :\lim _{n\to \infty }X_{n}(\omega )=X(\omega )\right\}=\Omega .}
12068:
12010:
11867:
11102:
8648:
These properties, together with a number of other special cases, are summarized in the following list:
4282:
4252:{\displaystyle \forall \varepsilon >0,\mathbb {P} {\big (}d(X_{n},X)\geq \varepsilon {\big )}\to 0.}
3981:
is deterministic like for the weak law of large numbers. At the same time, the case of a deterministic
3422:
1288:
1070:
5774:), and hence implies convergence in distribution. It is the notion of convergence used in the strong
12073:
12005:
10989:
3740:
384:
12025:
12099:
12078:
12015:
10424:
850:
838:
51:
6715:
6686:
2730:
2223:
The definition of convergence in distribution may be extended from random vectors to more general
791:{\displaystyle \scriptstyle Z_{n}={\scriptscriptstyle {\frac {1}{\sqrt {n}}}}\sum _{i=1}^{n}X_{i}}
11880:
11702:
11077:
11057:
6497:{\displaystyle \forall \omega \in \Omega \colon \ \lim _{n\to \infty }X_{n}(\omega )=X(\omega ),}
6172:
1217:
6072:
1475:
1339:
1249:
597:
grows larger, this distribution will gradually start to take shape more and more similar to the
12030:
11959:
11852:
11832:
11062:
7089:
6279:
5798:
4154:
552:
11412:
11857:
11749:
11092:
8748:
6518:
5043:
3591:
3393:
3303:
2235:. This is the âweak convergence of laws without laws being definedâ â except asymptotically.
1313:
858:
799:
681:
587:
397:
388:
10525:
converges in probability. The proof can be found in Page 126 (Theorem 5.3.4) of the book by
7317:
3858:{\displaystyle \lim _{n\to \infty }\mathbb {P} {\big (}|X_{n}-X|>\varepsilon {\big )}=0.}
1440:
1414:
12035:
11921:
11806:
11537:
11097:
11037:
10772:
9834:
9587:
6860:
6681:
5775:
5604:
4943:{\displaystyle P(|X_{n}-Y_{n}|\geq \epsilon )=P(|X_{n}|\cdot |(1-(-1)^{n})|\geq \epsilon )}
4759:
4732:
4646:
3736:
3664:
1145:
359:
304:
2458:
In general, convergence in distribution does not imply that the sequence of corresponding
339:
92:
An increasing similarity of outcomes to what a purely deterministic function would produce
8:
12191:
11979:
11862:
11801:
11770:
11655:
7720:
7575:
7441:
2705:
1537:
1223:
1116:
598:
86:
10405:
sufficiently quickly (i.e. the above sequence of tail probabilities is summable for all
12161:
12063:
11969:
11916:
11842:
11775:
11695:
11405:
11251:
11071:
10350:{\displaystyle \sum _{n}\mathbb {P} \left(|X_{n}-X|>\varepsilon \right)<\infty ,}
6887:
6628:
6352:
6332:
6312:
6052:
4955:
4786:
4288:
3399:
3309:
3219:
3199:
3107:
3009:
2911:
2822:
2817:
2710:
2452:
1563:
1517:
1293:
1125:
1098:
430:
284:
62:
31:
3992:
over an arrow indicating convergence, or using the "plim" probability limit operator:
11949:
11631:
11612:
11593:
11585:
11571:
11552:
11523:
11500:
11481:
11462:
11435:
11416:
11380:
11237:
6511:
6391:
2507:
2232:
448:
11347:"real analysis - Generalizing Scheffe's Lemma using only Convergence in Probability"
387:. Other forms of convergence are important in other useful theorems, including the
12151:
12090:
11911:
11811:
11766:
11754:
11515:
11454:
11322:
7329:
6522:
6378:
5771:
3660:
2701:
2462:
will also converge. As an example one may consider random variables with densities
2443:
to be in a given range is approximately equal to the probability that the value of
89:
in the classical sense to a fixed value, perhaps itself coming from a random event
27:
Notions of probabilistic convergence, applied to estimation and asymptotic analysis
9803:
converges to a constant is important, if it were to converge to a random variable
5036:
of days, there is a nonzero probability the terminating condition will not occur.
4412:
on the space of random variables over a fixed probability space. This topology is
3640:
Consider the following experiment. First, pick a random person in the street. Let
11533:
2167:{\displaystyle \lim _{n\to \infty }\mathbb {P} (X_{n}\in A)=\mathbb {P} (X\in A)}
916:
117:
11398:. Wiley Series in Probability and Mathematical Statistics (2nd ed.). Wiley.
11375:
Bickel, Peter J.; Klaassen, Chris A.J.; Ritov, Yaâacov; Wellner, Jon A. (1998).
3960:
Notice that for the condition to be satisfied, it is not possible that for each
2076:
the convergence in distribution is defined similarly. We say that this sequence
11964:
11780:
8179:{\displaystyle aX_{n}+bY_{n}\ {\xrightarrow {\overset {}{\text{a.s.}}}}\ aX+bY}
6846:
6395:
6329:
almost everywhere (in fact the set on which this sequence does not converge to
5588:
3700:
be your guess of the value of the next random number after observing the first
3194:
2750:
2224:
2213:
110:
11458:
12208:
12176:
12171:
12156:
12146:
11847:
11761:
11736:
10526:
7464:
6835:{\displaystyle \lim _{n\to \infty }\mathbb {E} \left(|X_{n}-X|^{r}\right)=0,}
5183:
5047:
3648:
a random variable. Then ask other people to estimate this height by eye. Let
2003:
11933:
11732:
6507:
5599:
5150:{\displaystyle \mathbb {P} \!\left(\lim _{n\to \infty }\!X_{n}=X\right)=1.}
4995:
that one day the number will become zero, and will stay zero forever after.
3094:{\displaystyle \lim \sup \mathbb {P} (X_{n}\in F)\leq \mathbb {P} (X\in F)}
2996:{\displaystyle \lim \inf \mathbb {P} (X_{n}\in G)\geq \mathbb {P} (X\in G)}
2228:
669:{\displaystyle \scriptstyle Z_{n}={\frac {\sqrt {n}}{\sigma }}(X_{n}-\mu )}
7108:-th mean. Hence, convergence in mean square implies convergence in mean.
5564:{\displaystyle {\overset {}{X_{n}\,{\xrightarrow {\mathrm {a.s.} }}\,X.}}}
11669:
8456:{\displaystyle aX_{n}+bY_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ aX+bY}
2069:{\displaystyle \left\{X_{1},X_{2},\dots \right\}\subset \mathbb {R} ^{k}}
98:
An increasing "aversion" against straying far away from a certain outcome
66:
5029:
that one day this amount will be zero, and stay zero forever after that.
57:
The concept is important in probability theory, and its applications to
12186:
11877:
8248:{\displaystyle X_{n}Y_{n}{\xrightarrow {\overset {}{\text{a.s.}}}}\ XY}
4413:
3693:
predictions as to what the next randomly generated number will be. Let
3606:
3102:
58:
8474:
None of the above statements are true for convergence in distribution.
12181:
12166:
7923:{\displaystyle aX_{n}+bY_{n}\ {\xrightarrow {\overset {}{p}}}\ aX+bY}
10854:
10802:
10741:
10672:
10579:
10057:
9996:
9953:
9749:
9688:
9645:
9508:
9466:
9400:
9297:
9258:
9154:
9108:
9047:
9001:
8940:
8901:
8848:
8804:
8715:
8679:
8624:
8605:
8586:
8547:
8497:
8418:
8341:
8283:
8226:
8148:
8078:
8027:
7970:
7892:
7822:
7771:
7684:
7626:
7546:
7495:
7412:
7361:
7292:
7138:
6942:
5532:
4376:
4326:
4059:
4021:
1958:
1738:
1698:
1658:
1620:
680:
to the standard normal, the result that follows from the celebrated
11821:
11790:
11741:
11718:
8097:{\displaystyle Y_{n}\ {\xrightarrow {\overset {}{\text{a.s.}}}}\ Y}
8046:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{\text{a.s.}}}}\ X}
7565:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{\text{a.s.}}}}\ Y}
7514:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{\text{a.s.}}}}\ X}
6665:
5782:
5042:
This is the type of stochastic convergence that is most similar to
4409:
3004:
2901:{\displaystyle \lim \inf \mathbb {E} f(X_{n})\geq \mathbb {E} f(X)}
197:
113:
of the outcome's distance from a particular value may converge to 0
4278:
Convergence in probability does not imply almost sure convergence.
2587:{\displaystyle \mathbb {P} (X_{n}\leq x)\to \mathbb {P} (X\leq x)}
10541:
gives sufficient conditions for almost sure convergence to imply
3385:{\displaystyle \liminf \mathbb {E} f(X_{n})\geq \mathbb {E} f(X)}
3295:{\displaystyle \limsup \mathbb {E} f(X_{n})\leq \mathbb {E} f(X)}
2436:, the convergence in distribution means that the probability for
11497:
8874:
Convergence in probability implies convergence in distribution:
6970:{\displaystyle {\overset {}{X_{n}\,{\xrightarrow {L^{r}}}\,X.}}}
3186:{\displaystyle \mathbb {P} (X_{n}\in B)\to \mathbb {P} (X\in B)}
1992:{\displaystyle X_{n}\,{\xrightarrow {d}}\,{\mathcal {N}}(0,\,1)}
10977:{\displaystyle \mathbb {E} \rightarrow \mathbb {E} <\infty }
8745:
Convergence in probability implies there exists a sub-sequence
6907:
This type of convergence is often denoted by adding the letter
5770:
Almost sure convergence implies convergence in probability (by
5497:
Almost sure convergence is often denoted by adding the letters
5231:
and the concept of the random variable as a function from Ω to
4418:
4268:
Convergence in probability implies convergence in distribution.
2486:. These random variables converge in distribution to a uniform
2490:(0, 1), whereas their densities do not converge at all.
2339:{\displaystyle \mathbb {E} ^{*}h(X_{n})\to \mathbb {E} \,h(X)}
447:
is a random variable, and all of them are defined on the same
10880:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ X}
8367:{\displaystyle Y_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ Y}
8309:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ X}
7992:{\displaystyle X_{n}Y_{n}{\xrightarrow {\overset {}{p}}}\ XY}
7710:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ Y}
7652:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ X}
2357:, that is the expectation of a âsmallest measurable function
11687:
11678:
Creative
Commons Attribution-ShareAlike 3.0 Unported License
8652:
Almost sure convergence implies convergence in probability:
7267:
The converse is not necessarily true, however it is true if
6372:
5781:
The concept of almost sure convergence does not come from a
11480:(2nd ed.). Clarendon Press, Oxford. pp. 271â285.
11377:
Efficient and adaptive estimation for semiparametric models
10559:
3988:
Convergence in probability is denoted by adding the letter
832:
distribution. This convergence is shown in the picture: as
538:
be the fraction of heads after tossing up an unbiased coin
333:
271:{\displaystyle X_{n}={\frac {1}{n}}\sum _{i=1}^{n}Y_{i}\,,}
193:
11374:
11121:
10213:
it is always possible to find a new probability space (Ω,
7309:{\displaystyle {\overset {}{X_{n}\,\xrightarrow {p} \,X}}}
6042:{\displaystyle P(|X_{n}|\geq \varepsilon )={\frac {1}{n}}}
5345:, almost sure convergence can also be defined as follows:
7841:{\displaystyle Y_{n}\ {\xrightarrow {\overset {}{p}}}\ Y}
7790:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{p}}}\ X}
7431:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{p}}}\ Y}
7380:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{p}}}\ X}
6525:. (Note that random variables themselves are functions).
2525:
if and only if any of the following statements are true:
10820:{\displaystyle X_{n}\ \xrightarrow {\overset {}{p}} \ X}
2809:{\displaystyle \mathbb {E} f(X_{n})\to \mathbb {E} f(X)}
2693:{\displaystyle \mathbb {E} f(X_{n})\to \mathbb {E} f(X)}
5022:, ⊠be the daily amounts the charity received from him.
11499:(3rd ed.). HCĂ-tryk, Copenhagen. pp. 18â20.
10562:
8489:
4352:
4311:
1901:
1216:, then this sequence converges in distribution to the
741:
726:
617:
105:
Some less obvious, more theoretical patterns could be
11883:
10992:
10895:
10835:
10784:
10757:{\displaystyle X_{n}{\xrightarrow {\overset {}{P}}}X}
10725:
10557:
10453:
10280:
9934:
9837:
9626:
9590:
9381:
9239:
9089:
8982:
8882:
8786:
8751:
8660:
8487:
8380:
8322:
8264:
8200:
8110:
8059:
8008:
7944:
7854:
7803:
7752:
7723:
7665:
7607:
7578:
7527:
7476:
7444:
7393:
7342:
7273:
7120:
6922:
6890:
6863:
6758:
6718:
6689:
6533:
6521:
of a sequence of functions extended to a sequence of
6422:
6355:
6335:
6315:
6282:
6218:
6175:
6111:
6075:
6055:
5990:
5950:
5889:
5834:
5801:
5639:
5607:
5512:
5351:
5241:
5224:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
5193:
5091:
4958:
4809:
4789:
4762:
4735:
4676:
4649:
4550:
4429:
4291:
4177:
4168:, convergence in probability is defined similarly by
4003:
3781:
3616:
A natural link to convergence in distribution is the
3402:
3334:
3312:
3244:
3222:
3202:
3132:
3110:
3034:
3012:
2936:
2914:
2847:
2825:
2761:
2733:
2713:
2645:
2600:
2533:
2390:
2283:
2183:
2097:
2011:
1941:
1900:
1594:
1566:
1540:
1520:
1478:
1443:
1417:
1384:
1342:
1316:
1296:
1252:
1226:
1179:
1148:
1128:
1101:
1073:
1000:
928:
875:
725:
616:
487:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
456:
433:
400:
362:
342:
307:
287:
209:
144:
11584:
11133:
6162:{\displaystyle \sum _{n\geq 1}P(X_{n}=1)\to \infty }
3674:
will converge in probability to the random variable
601:
of the normal distribution. If we shift and rescale
120:
describing the next event grows smaller and smaller.
11566:Grimmett, Geoffrey R.; Stirzaker, David R. (2020).
8974:-th order mean implies convergence in probability:
1142:should be considered is essential. For example, if
1122:The requirement that only the continuity points of
11896:
11404:
11028:
10976:
10879:
10819:
10756:
10690:
10499:
10349:
10087:
9855:
9779:
9608:
9526:
9318:
9182:
9065:
8958:
8862:
8770:
8733:
8637:
8455:
8366:
8308:
8247:
8178:
8096:
8045:
7991:
7922:
7840:
7789:
7735:
7709:
7651:
7590:
7564:
7513:
7456:
7430:
7379:
7308:
7256:
6969:
6896:
6876:
6834:
6726:
6697:
6617:
6496:
6361:
6341:
6321:
6301:
6268:
6200:
6161:
6094:
6061:
6041:
5976:
5936:
5875:
5820:
5754:
5633:, convergence almost surely is defined similarly:
5625:
5563:
5487:
5331:
5223:
5149:
4964:
4942:
4795:
4775:
4748:
4721:
4662:
4626:
4536:
4396:
4337:
4297:
4251:
4120:
3857:
3408:
3384:
3318:
3294:
3228:
3208:
3185:
3116:
3093:
3018:
2995:
2920:
2900:
2831:
2808:
2741:
2719:
2692:
2629:
2586:
2428:
2338:
2204:
2166:
2068:
1991:
1919:
1870:
1572:
1552:
1534:. Thus the convergence of cdfs fails at the point
1526:
1506:
1464:
1429:
1403:
1370:
1328:
1302:
1280:
1238:
1208:
1161:
1134:
1107:
1087:
1057:{\displaystyle \lim _{n\to \infty }F_{n}(x)=F(x),}
1056:
960:
907:
798:be their (normalized) sums. Then according to the
790:
668:
486:
439:
419:
375:
348:
320:
293:
270:
184:
95:An increasing preference towards a certain outcome
11565:
11475:
11270:
5741:
5647:
5458:
5359:
5318:
5249:
5119:
5097:
4454:
3473:Note however that convergence in distribution of
2246:), and we say that a sequence of random elements
12206:
11411:(2nd ed.). John Wiley & Sons. pp.
10178:are almost surely bounded above and below, then
7164:
6760:
6552:
6442:
6226:
5365:
5267:
5104:
4972:. So we do not have convergence in probability.
4582:
4451:
3783:
3335:
3245:
3038:
3035:
2940:
2937:
2851:
2848:
2099:
1002:
11513:
1920:{\displaystyle \scriptstyle {\mathcal {L}}_{X}}
6853:-th mean tells us that the expectation of the
2630:{\displaystyle x\mapsto \mathbb {P} (X\leq x)}
1583:Convergence in distribution may be denoted as
497:
65:. The same concepts are known in more general
11703:
11606:
11548:Counterexamples in Probability and Statistics
11545:Romano, Joseph P.; Siegel, Andrew F. (1985).
11522:. Berlin: Springer-Verlag. pp. xii+480.
11453:. New York: Springer Science+Business Media.
11434:. Cambridge, UK: Cambridge University Press.
11282:
11219:
11172:
6514:over which the random variables are defined.
5711:
5670:
5451:
5382:
4529:
4516:
4484:
4457:
4238:
4200:
3844:
3805:
3624:
3577:if and only if the sequence of corresponding
1437:. However, for this limiting random variable
1209:{\displaystyle \left(0,{\frac {1}{n}}\right)}
11668:This article incorporates material from the
11544:
11448:
11294:
11145:
11023:
10993:
7088:â„ 1, implies convergence in probability (by
6296:
6283:
6269:{\displaystyle P(\limsup _{n}\{X_{n}=1\})=1}
6254:
6235:
6195:
6176:
5815:
5802:
36:convergence of sequences of random variables
11647:Stochastic Processes in Engineering Systems
11402:
11393:
11323:"Proofs of convergence of random variables"
10500:{\displaystyle S_{n}=X_{1}+\cdots +X_{n}\,}
6993:The most important cases of convergence in
5937:{\displaystyle P(X_{n}=0)=1-{\frac {1}{n}}}
34:, there exist several different notions of
12127:Vitale's random BrunnâMinkowski inequality
11710:
11696:
10715:A necessary and sufficient condition for
9807:then we wouldn't be able to conclude that
5828:of independent random variables such that
4975:
4397:{\textstyle g(X_{n})\xrightarrow {p} g(X)}
4285:states that for every continuous function
2908:for all nonnegative, continuous functions
2429:{\displaystyle F(a)=\mathbb {P} (X\leq a)}
109:That the series formed by calculating the
11644:
11570:(4th ed.). Oxford University Press.
11053:Proofs of convergence of random variables
10938:
10897:
10626:
10496:
10292:
7298:
7287:
7221:
7180:
6956:
6936:
6776:
6720:
6691:
6373:Sure convergence or pointwise convergence
5876:{\displaystyle P(X_{n}=1)={\frac {1}{n}}}
5735:
5716:
5664:
5641:
5550:
5526:
5353:
5243:
5214:
5093:
4627:{\displaystyle d(X,Y)=\mathbb {E} \left.}
4573:
4478:
4194:
4098:
3799:
3366:
3339:
3276:
3249:
3164:
3134:
3072:
3042:
2974:
2944:
2882:
2855:
2790:
2763:
2735:
2674:
2647:
2608:
2565:
2535:
2407:
2323:
2319:
2286:
2205:{\displaystyle A\subset \mathbb {R} ^{k}}
2192:
2145:
2115:
2056:
1982:
1965:
1952:
1927:is the law (probability distribution) of
1081:
477:
394:Throughout the following, we assume that
264:
11653:
11625:
11611:. New York: Cambridge University Press.
11590:Weak convergence and empirical processes
11494:
11476:Grimmett, G.R.; Stirzaker, D.R. (1992).
10423:. This is a direct implication from the
10390:then it also converges almost surely to
9556:converges in distribution to a constant
9214:converges in distribution to a constant
5977:{\displaystyle 0<\varepsilon <1/2}
3920: > 0 there exists a number
11451:A Modern Approach to Probability Theory
11449:Fristedt, Bert; Gray, Lawrence (1997).
11157:
10518:converges almost surely if and only if
9362:converges in probability to zero, then
503:Examples of convergence in distribution
427:is a sequence of random variables, and
14:
12207:
11654:Zitkovic, Gordan (November 17, 2013).
11429:
11215:
11213:
11211:
11209:
11207:
11205:
11196:
11184:
11151:
6911:over an arrow indicating convergence:
6635:
5501:over an arrow indicating convergence:
5235:, this is equivalent to the statement
3905:is said to converge in probability to
3659:responses. Then (provided there is no
3630:Examples of convergence in probability
3425:states that for a continuous function
11691:
11551:. Great Britain: Chapman & Hall.
11306:
11300:
10228:= 0, 1, ...} defined on it such that
10206:} which converges in distribution to
5170:, in the sense that events for which
4408:Convergence in probability defines a
3507:imply convergence in distribution of
2349:for all continuous bounded functions
12140:Applications & related
10548:
10386:converges almost completely towards
6913:
6857:-th power of the difference between
5790:
5503:
5343:limit superior of a sequence of sets
4338:{\textstyle X_{n}\xrightarrow {p} X}
3994:
2501:implies convergence in distribution.
1585:
1404:{\displaystyle x\geq {\frac {1}{n}}}
185:{\displaystyle Y_{i},\ i=1,\dots ,n}
12059:Marcinkiewicz interpolation theorem
11407:Convergence of probability measures
11231:
11202:
10443:real independent random variables:
4981:Examples of almost sure convergence
4722:{\displaystyle Y_{n}=(-1)^{n}X_{n}}
3683:Predicting random number generation
961:{\displaystyle F_{1},F_{2},\ldots }
908:{\displaystyle X_{1},X_{2},\ldots }
24:
11985:Symmetric decreasing rearrangement
11889:
11088:Skorokhod's representation theorem
11070:: the question of continuity of a
10971:
10642:
10341:
9369:also converges in distribution to
7328:Provided the probability space is
7174:
6770:
6609:
6562:
6545:
6452:
6432:
6423:
6156:
5729:
5658:
5540:
5534:
5393:
5375:
5277:
5260:
5205:
5197:
5114:
4729:. Notice that the distribution of
4637:
4178:
4092:
3793:
3618:Skorokhod's representation theorem
2109:
1968:
1905:
1847:
1821:
1750:
1700:
1660:
1012:
837:
569:. The subsequent random variables
468:
460:
25:
12231:
10827:, the followings are equivalent
6845:where the operator E denotes the
3722:will converge to the outcomes of
1088:{\displaystyle x\in \mathbb {R} }
921:cumulative distribution functions
11568:Probability and Random Processes
11478:Probability and random processes
11160:Probability: Theory and Examples
11134:van der Vaart & Wellner 1996
10419:also converges almost surely to
7104:-th mean implies convergence in
4756:is equal to the distribution of
3711:converge to the distribution of
1935:is standard normal we can write
986:cumulative distribution function
11676:", which is licensed under the
11339:
11315:
11288:
11276:
11264:
11093:The Tweedie convergence theorem
11029:{\displaystyle \{|X_{n}|^{r}\}}
10657:
10653:
10019:
10015:
9711:
9707:
9489:
9485:
9278:
9274:
9135:
9131:
9028:
9024:
8921:
8917:
8823:
8819:
8778:which almost surely converges:
8696:
8692:
7323:
7162:
7158:
5764:
5475:
5469:
5187:). Using the probability space
5053:
4262:
3746:
3605:Convergence in distribution is
3594:to the characteristic function
383:. This result is known as the
131:For example, if the average of
11630:. Cambridge University Press.
11607:van der Vaart, Aad W. (1998).
11309:A Course in Probability Theory
11234:Probability: A graduate course
11225:
11190:
11178:
11166:
11139:
11127:
11115:
11013:
10997:
10965:
10955:
10946:
10942:
10934:
10931:
10921:
10905:
10901:
10654:
10636:
10630:
10611:
10596:
10323:
10302:
10082:
10070:
10046:
10020:
10016:
9850:
9838:
9774:
9762:
9738:
9712:
9708:
9603:
9591:
9486:
9454:
9426:
9275:
9132:
9025:
8918:
8820:
8765:
8752:
8693:
8614:
8595:
8572:
8518:
7316:(by a more general version of
7248:
7238:
7229:
7225:
7214:
7204:
7188:
7184:
7171:
7159:
6808:
6786:
6767:
6672:) towards the random variable
6598:
6592:
6583:
6577:
6559:
6488:
6482:
6473:
6467:
6449:
6257:
6222:
6153:
6150:
6131:
6086:
6023:
6013:
5998:
5994:
5912:
5893:
5857:
5838:
5726:
5719:
5706:
5700:
5691:
5685:
5620:
5608:
5439:
5435:
5429:
5420:
5414:
5400:
5372:
5313:
5307:
5298:
5292:
5274:
5218:
5194:
5159:This means that the values of
5111:
4937:
4927:
4923:
4914:
4904:
4895:
4891:
4883:
4868:
4864:
4855:
4845:
4817:
4813:
4700:
4690:
4613:
4603:
4589:
4585:
4566:
4554:
4544:or alternately by this metric
4504:
4490:
4445:
4433:
4391:
4385:
4369:
4356:
4243:
4227:
4208:
4089:
3890:is outside the ball of radius
3832:
3811:
3790:
3379:
3373:
3359:
3346:
3289:
3283:
3269:
3256:
3180:
3168:
3160:
3157:
3138:
3088:
3076:
3065:
3046:
2990:
2978:
2967:
2948:
2895:
2889:
2875:
2862:
2803:
2797:
2786:
2783:
2770:
2687:
2681:
2670:
2667:
2654:
2624:
2612:
2604:
2581:
2569:
2561:
2558:
2539:
2423:
2411:
2400:
2394:
2333:
2327:
2315:
2312:
2299:
2161:
2149:
2138:
2119:
2106:
1986:
1973:
1858:
1852:
1842:
1839:
1826:
1804:
1779:
1495:
1489:
1453:
1447:
1359:
1353:
1269:
1263:
1048:
1042:
1033:
1027:
1009:
662:
643:
481:
457:
414:
401:
13:
1:
11955:Convergence almost everywhere
11717:
11656:"Lecture 7: Weak Convergence"
11592:. New York: Springer-Verlag.
11432:Real analysis and probability
11403:Billingsley, Patrick (1999).
11394:Billingsley, Patrick (1986).
11379:. New York: Springer-Verlag.
11367:
11271:Grimmett & Stirzaker 2020
11083:Big O in probability notation
11068:Continuous stochastic process
10539:dominated convergence theorem
9796:Note that the condition that
9582:converges in distribution to
9545:converges in distribution to
9344:converges in distribution to
3573:converges in distribution to
3459:converges in distribution to
3440:converges in distribution to
2594:for all continuity points of
2521:converges in distribution to
2499:probability density functions
2460:probability density functions
2378:
864:
192:, all having the same finite
77:
11649:. New York: SpringerâVerlag.
11645:Wong, E.; HĂĄjek, B. (1985).
11628:Probability with Martingales
11520:Probability in Banach spaces
10401:converges in probability to
10235:is equal in distribution to
10167:converges in probability to
10106:converges in probability to
9914:converges in probability to
9888:converges in probability to
9877:converges in probability to
9227:converges in probability to
6727:{\displaystyle \mathbb {E} }
6698:{\displaystyle \mathbb {E} }
6377:To say that the sequence of
3764:towards the random variable
3655:be the average of the first
2742:{\displaystyle \mathbb {E} }
2244:weak convergence of measures
7:
12122:PrĂ©kopaâLeindler inequality
11975:Locally integrable function
11897:{\displaystyle L^{\infty }}
11046:
10704:
10369:converges almost completely
10217:, P) and random variables {
9348:and the difference between
6983:
6647:, we say that the sequence
6210:second Borel Cantelli Lemma
6201:{\displaystyle \{X_{n}=1\}}
5577:
4952:which does not converge to
4134:
3957:(the definition of limit).
2447:is in that range, provided
1884:
498:Convergence in distribution
44:convergence in distribution
10:
12236:
11868:Square-integrable function
11588:; Wellner, Jon A. (1996).
11351:Mathematics Stack Exchange
11103:Continuous mapping theorem
10197:Almost sure representation
6095:{\displaystyle X_{n}\to 0}
5032:However, when we consider
4283:continuous mapping theorem
3881:) be the probability that
3644:be their height, which is
3625:Convergence in probability
3423:continuous mapping theorem
1507:{\displaystyle F_{n}(0)=0}
1371:{\displaystyle F_{n}(x)=1}
1281:{\displaystyle F_{n}(x)=0}
678:converging in distribution
356:, of the random variables
332:(see below) to the common
40:convergence in probability
12220:Convergence (mathematics)
12139:
12117:MinkowskiâSteiner formula
12087:
12049:
11993:
11942:
11876:
11820:
11789:
11725:
11459:10.1007/978-1-4899-2837-5
11236:. Theorem 3.4: Springer.
10171:and all random variables
6302:{\displaystyle \{X_{n}\}}
5821:{\displaystyle \{X_{n}\}}
5058:To say that the sequence
5004:
4999:
4990:
4985:
3741:weak law of large numbers
3687:
3682:
3639:
3634:
3560:LĂ©vyâs continuity theorem
2078:converges in distribution
693:
688:
530:
525:
512:
507:
385:weak law of large numbers
116:That the variance of the
12100:Isoperimetric inequality
11295:Fristedt & Gray 1997
11256:: CS1 maint: location (
11146:Romano & Siegel 1985
11108:
9892:, then the joint vector
9560:, then the joint vector
6390:) defined over the same
5341:Using the notion of the
5181:have probability 0 (see
3762:converges in probability
3579:characteristic functions
970:converge in distribution
851:probability distribution
586:will all be distributed
12105:BrunnâMinkowski theorem
11396:Probability and Measure
11307:Chung, Kai-lai (2001).
11187:, Chapter 9.2, page 287
11078:Asymptotic distribution
11058:Convergence of measures
8771:{\displaystyle (n_{k})}
4976:Almost sure convergence
1329:{\displaystyle x\leq 0}
420:{\displaystyle (X_{n})}
48:almost sure convergence
18:Almost sure convergence
11960:Convergence in measure
11898:
11674:Stochastic convergence
11158:Durrett, Rick (2010).
11063:Convergence in measure
11030:
10978:
10881:
10821:
10758:
10692:
10501:
10394:. In other words, if
10351:
10089:
9857:
9781:
9610:
9528:
9320:
9184:
9067:
8960:
8864:
8772:
8735:
8639:
8463:(for any real numbers
8457:
8368:
8310:
8249:
8186:(for any real numbers
8180:
8098:
8047:
7993:
7930:(for any real numbers
7924:
7842:
7791:
7737:
7711:
7653:
7592:
7566:
7515:
7458:
7432:
7381:
7310:
7258:
6971:
6898:
6878:
6836:
6728:
6699:
6619:
6517:This is the notion of
6498:
6363:
6343:
6323:
6303:
6270:
6202:
6163:
6096:
6063:
6043:
5978:
5938:
5877:
5822:
5756:
5627:
5565:
5489:
5333:
5225:
5166:approach the value of
5151:
5046:known from elementary
4966:
4944:
4797:
4777:
4750:
4723:
4670:and a second sequence
4664:
4628:
4538:
4398:
4339:
4299:
4253:
4155:separable metric space
4122:
3859:
3760:} of random variables
3715:, but the outcomes of
3410:
3386:
3320:
3296:
3230:
3210:
3187:
3118:
3095:
3020:
2997:
2922:
2902:
2833:
2810:
2743:
2721:
2694:
2631:
2588:
2493:However, according to
2430:
2353:. Here E* denotes the
2340:
2238:In this case the term
2206:
2168:
2070:
1993:
1921:
1872:
1574:
1554:
1528:
1508:
1466:
1465:{\displaystyle F(0)=1}
1431:
1430:{\displaystyle n>0}
1405:
1372:
1330:
1304:
1282:
1240:
1210:
1163:
1136:
1109:
1089:
1058:
962:
909:
842:
809:approaches the normal
802:, the distribution of
792:
776:
719:random variables. Let
670:
553:Bernoulli distribution
488:
441:
421:
377:
350:
322:
295:
272:
253:
186:
71:stochastic convergence
12074:RieszâFischer theorem
11899:
11858:Polarization identity
11626:Williams, D. (1991).
11609:Asymptotic statistics
11586:van der Vaart, Aad W.
11495:Jacobsen, M. (1992).
11430:Dudley, R.M. (2002).
11031:
10979:
10882:
10822:
10759:
10693:
10502:
10373:almost in probability
10352:
10160:. In other words, if
10090:
9858:
9856:{\displaystyle (X,Y)}
9782:
9611:
9609:{\displaystyle (X,c)}
9529:
9321:
9185:
9068:
8961:
8865:
8773:
8736:
8640:
8458:
8369:
8311:
8250:
8181:
8099:
8048:
7994:
7925:
7843:
7792:
7738:
7712:
7654:
7593:
7567:
7516:
7459:
7433:
7382:
7311:
7259:
6972:
6899:
6879:
6877:{\displaystyle X_{n}}
6837:
6729:
6700:
6620:
6519:pointwise convergence
6499:
6364:
6344:
6324:
6309:does not converge to
6304:
6271:
6203:
6164:
6097:
6064:
6044:
5979:
5939:
5878:
5823:
5757:
5628:
5626:{\displaystyle (S,d)}
5566:
5490:
5334:
5226:
5177:does not converge to
5152:
5044:pointwise convergence
4967:
4945:
4798:
4778:
4776:{\displaystyle X_{n}}
4751:
4749:{\displaystyle Y_{n}}
4724:
4665:
4663:{\displaystyle X_{n}}
4629:
4539:
4399:
4340:
4300:
4254:
4144:For random elements {
4123:
3964:the random variables
3924:(which may depend on
3868:More explicitly, let
3860:
3611:LĂ©vyâProkhorov metric
3411:
3394:lower semi-continuous
3387:
3321:
3304:upper semi-continuous
3297:
3231:
3211:
3188:
3119:
3096:
3021:
2998:
2923:
2903:
2834:
2811:
2744:
2722:
2695:
2632:
2589:
2497:, convergence of the
2431:
2341:
2207:
2169:
2071:
1994:
1922:
1873:
1575:
1555:
1529:
1509:
1467:
1432:
1406:
1373:
1331:
1305:
1283:
1241:
1211:
1164:
1162:{\displaystyle X_{n}}
1137:
1110:
1090:
1059:
980:to a random variable
963:
910:
859:central limit theorem
841:
800:central limit theorem
793:
756:
682:central limit theorem
671:
489:
442:
422:
389:central limit theorem
378:
376:{\displaystyle Y_{i}}
351:
323:
321:{\displaystyle X_{n}}
296:
273:
233:
187:
12215:Stochastic processes
12079:RieszâThorin theorem
11922:Infimum and supremum
11881:
11807:Lebesgue integration
11038:uniformly integrable
10990:
10893:
10833:
10782:
10773:uniformly integrable
10723:
10555:
10451:
10425:BorelâCantelli lemma
10278:
9932:
9835:
9624:
9588:
9379:
9237:
9087:
8980:
8880:
8784:
8749:
8658:
8485:
8378:
8320:
8262:
8198:
8108:
8057:
8006:
7942:
7852:
7801:
7750:
7721:
7663:
7605:
7576:
7525:
7474:
7442:
7391:
7340:
7271:
7118:
7100:â„ 1, convergence in
6920:
6888:
6861:
6756:
6716:
6687:
6640:Given a real number
6531:
6420:
6353:
6333:
6313:
6280:
6216:
6173:
6109:
6073:
6053:
5988:
5948:
5887:
5832:
5799:
5795:Consider a sequence
5776:law of large numbers
5637:
5605:
5510:
5349:
5239:
5191:
5089:
4956:
4807:
4787:
4760:
4733:
4674:
4647:
4548:
4427:
4350:
4309:
4289:
4175:
4001:
3932:) such that for all
3779:
3665:law of large numbers
3400:
3332:
3310:
3242:
3220:
3200:
3130:
3108:
3032:
3010:
2934:
2912:
2845:
2823:
2759:
2731:
2711:
2706:continuous functions
2643:
2598:
2531:
2388:
2281:
2257:converges weakly to
2181:
2095:
2009:
1939:
1898:
1592:
1564:
1538:
1518:
1476:
1441:
1415:
1382:
1340:
1314:
1294:
1250:
1224:
1177:
1146:
1126:
1099:
1071:
998:
926:
873:
723:
614:
610:appropriately, then
555:with expected value
516:uniform distribution
454:
431:
398:
360:
349:{\displaystyle \mu }
340:
305:
285:
207:
142:
63:stochastic processes
12041:Young's convolution
11980:Measurable function
11863:Pythagorean theorem
11853:Parseval's identity
11802:Integrable function
11232:Gut, Allan (2005).
10868:
10809:
10748:
10683:
10586:
10064:
10003:
9960:
9756:
9695:
9652:
9515:
9473:
9407:
9304:
9265:
9168:
9122:
9054:
9015:
8947:
8908:
8852:
8811:
8722:
8683:
8628:
8609:
8590:
8561:
8511:
8432:
8355:
8297:
8233:
8155:
8085:
8034:
7977:
7899:
7829:
7778:
7736:{\displaystyle X=Y}
7698:
7640:
7591:{\displaystyle X=Y}
7553:
7502:
7457:{\displaystyle X=Y}
7419:
7368:
7296:
7149:
7092:). Furthermore, if
7090:Markov's inequality
7080:Convergence in the
6953:
6904:converges to zero.
6636:Convergence in mean
6276:hence the sequence
6049:which converges to
5547:
4982:
4380:
4330:
4063:
4025:
3631:
3592:converges pointwise
3216:of random variable
2818:Lipschitz functions
2242:is preferable (see
2233:empirical processes
1962:
1742:
1704:
1664:
1624:
1553:{\displaystyle x=0}
1239:{\displaystyle X=0}
504:
301:tends to infinity,
12162:Probability theory
12064:Plancherel theorem
11970:Integral transform
11917:Chebyshev distance
11894:
11843:Euclidean distance
11776:Minkowski distance
11680:but not under the
11283:van der Vaart 1998
11220:van der Vaart 1998
11173:van der Vaart 1998
11122:Bickel et al. 1998
11072:stochastic process
11026:
10974:
10877:
10817:
10764:and the sequence (
10754:
10688:
10647:
10497:
10347:
10290:
10085:
9853:
9777:
9606:
9524:
9316:
9180:
9063:
8956:
8860:
8768:
8731:
8635:
8633:
8538:
8453:
8364:
8306:
8245:
8176:
8094:
8043:
7989:
7920:
7838:
7787:
7733:
7707:
7649:
7588:
7562:
7511:
7454:
7428:
7377:
7306:
7254:
7178:
6967:
6894:
6874:
6832:
6774:
6724:
6695:
6629:probability theory
6615:
6566:
6510:of the underlying
6494:
6456:
6359:
6339:
6319:
6299:
6266:
6234:
6198:
6159:
6127:
6092:
6059:
6039:
5974:
5934:
5873:
5818:
5752:
5733:
5623:
5561:
5485:
5379:
5329:
5281:
5221:
5147:
5118:
5075:with probability 1
4980:
4962:
4940:
4793:
4773:
4746:
4719:
4660:
4624:
4534:
4394:
4345:, then also
4335:
4295:
4249:
4118:
4096:
3855:
3797:
3635:Height of a person
3629:
3429:, if the sequence
3406:
3382:
3316:
3292:
3226:
3206:
3183:
3114:
3091:
3016:
2993:
2918:
2898:
2829:
2806:
2739:
2717:
2690:
2627:
2584:
2453:sufficiently large
2426:
2336:
2202:
2164:
2113:
2066:
1989:
1931:. For example, if
1917:
1916:
1868:
1866:
1580:is discontinuous.
1570:
1550:
1524:
1504:
1462:
1427:
1401:
1368:
1326:
1300:
1278:
1236:
1206:
1159:
1132:
1105:
1085:
1054:
1016:
958:
905:
843:
788:
787:
754:
666:
665:
502:
484:
437:
417:
373:
346:
318:
291:
268:
182:
32:probability theory
12202:
12201:
12135:
12134:
11950:Almost everywhere
11735: &
11637:978-0-521-40605-5
11618:978-0-521-49603-2
11599:978-0-387-94640-5
11577:978-0-198-84760-1
11558:978-0-412-98901-8
11529:978-3-540-52013-9
11516:Talagrand, Michel
11506:978-87-91180-71-2
11487:978-0-19-853665-9
11468:978-1-4899-2837-5
11441:978-0-521-80972-6
11422:978-0-471-19745-4
11386:978-0-387-98473-5
11243:978-0-387-22833-4
11098:Slutsky's theorem
10873:
10869:
10867:
10866:
10848:
10813:
10810:
10808:
10807:
10797:
10749:
10747:
10746:
10712:
10711:
10684:
10587:
10585:
10584:
10583:
10361:then we say that
10281:
10069:
10065:
10063:
10062:
10051:
10014:
10008:
10004:
10002:
10001:
9990:
9977:
9974:
9965:
9961:
9959:
9958:
9947:
9761:
9757:
9755:
9754:
9743:
9706:
9700:
9696:
9694:
9693:
9682:
9669:
9666:
9657:
9653:
9651:
9650:
9639:
9520:
9516:
9514:
9513:
9502:
9484:
9478:
9474:
9472:
9471:
9460:
9424:
9421:
9412:
9408:
9406:
9405:
9394:
9309:
9305:
9303:
9302:
9291:
9270:
9266:
9264:
9263:
9252:
9173:
9169:
9167:
9166:
9148:
9127:
9123:
9121:
9120:
9102:
9059:
9055:
9053:
9052:
9041:
9020:
9016:
9014:
9013:
8995:
8952:
8948:
8946:
8945:
8934:
8913:
8909:
8907:
8906:
8895:
8856:
8853:
8851:
8843:
8815:
8812:
8810:
8809:
8799:
8727:
8723:
8721:
8720:
8709:
8688:
8684:
8682:
8673:
8629:
8610:
8591:
8589:
8562:
8560:
8559:
8517:
8512:
8510:
8509:
8437:
8433:
8431:
8430:
8412:
8360:
8356:
8354:
8353:
8335:
8302:
8298:
8296:
8295:
8277:
8238:
8234:
8232:
8231:
8230:
8160:
8156:
8154:
8153:
8152:
8142:
8090:
8086:
8084:
8083:
8082:
8072:
8039:
8035:
8033:
8032:
8031:
8021:
7982:
7978:
7976:
7975:
7904:
7900:
7898:
7897:
7886:
7834:
7830:
7828:
7827:
7816:
7783:
7779:
7777:
7776:
7765:
7703:
7699:
7697:
7696:
7678:
7645:
7641:
7639:
7638:
7620:
7558:
7554:
7552:
7551:
7550:
7540:
7507:
7503:
7501:
7500:
7499:
7489:
7424:
7420:
7418:
7417:
7406:
7373:
7369:
7367:
7366:
7355:
7304:
7303:
7297:
7163:
7156:
7155:
7150:
7070:in quadratic mean
7057:= 2, we say that
7020:= 1, we say that
6991:
6990:
6965:
6964:
6954:
6897:{\displaystyle X}
6849:. Convergence in
6759:
6551:
6512:probability space
6441:
6440:
6392:probability space
6362:{\displaystyle 1}
6342:{\displaystyle 0}
6322:{\displaystyle 0}
6225:
6208:are independent,
6112:
6062:{\displaystyle 0}
6037:
5932:
5871:
5718:
5585:
5584:
5559:
5558:
5548:
5473:
5364:
5266:
5103:
5071:almost everywhere
5040:
5039:
5034:any finite number
4965:{\displaystyle 0}
4796:{\displaystyle n}
4476:
4381:
4331:
4298:{\displaystyle g}
4142:
4141:
4081:
4079:
4076:
4067:
4064:
4054:
4041:
4038:
4029:
4026:
4016:
3953:) <
3894:centered at
3782:
3730:
3729:
3409:{\displaystyle f}
3319:{\displaystyle f}
3229:{\displaystyle X}
3209:{\displaystyle B}
3117:{\displaystyle F}
3019:{\displaystyle G}
2921:{\displaystyle f}
2832:{\displaystyle f}
2816:for all bounded,
2720:{\displaystyle f}
2508:portmanteau lemma
2495:ScheffĂ©âs theorem
2355:outer expectation
2098:
1963:
1892:
1891:
1818:
1815:
1793:
1790:
1746:
1743:
1733:
1720:
1717:
1708:
1705:
1693:
1680:
1677:
1668:
1665:
1653:
1640:
1637:
1628:
1625:
1615:
1573:{\displaystyle F}
1527:{\displaystyle n}
1399:
1303:{\displaystyle n}
1199:
1135:{\displaystyle F}
1108:{\displaystyle F}
1067:for every number
1001:
847:
846:
752:
751:
641:
637:
449:probability space
440:{\displaystyle X}
294:{\displaystyle n}
231:
160:
138:random variables
16:(Redirected from
12227:
12152:Fourier analysis
12110:Milman's reverse
12093:
12091:Lebesgue measure
12085:
12084:
12069:RiemannâLebesgue
11912:Bounded function
11903:
11901:
11900:
11895:
11893:
11892:
11812:Taxicab geometry
11767:Measurable space
11712:
11705:
11698:
11689:
11688:
11662:
11660:
11650:
11641:
11622:
11603:
11581:
11562:
11541:
11514:Ledoux, Michel;
11510:
11491:
11472:
11445:
11426:
11410:
11399:
11390:
11361:
11360:
11358:
11357:
11343:
11337:
11336:
11334:
11333:
11319:
11313:
11312:
11304:
11298:
11292:
11286:
11280:
11274:
11268:
11262:
11261:
11255:
11247:
11229:
11223:
11217:
11200:
11194:
11188:
11182:
11176:
11170:
11164:
11163:
11155:
11149:
11143:
11137:
11131:
11125:
11119:
11035:
11033:
11032:
11027:
11022:
11021:
11016:
11010:
11009:
11000:
10983:
10981:
10980:
10975:
10964:
10963:
10958:
10949:
10941:
10930:
10929:
10924:
10918:
10917:
10908:
10900:
10886:
10884:
10883:
10878:
10871:
10870:
10865:
10864:
10855:
10850:
10846:
10845:
10844:
10826:
10824:
10823:
10818:
10811:
10803:
10798:
10795:
10794:
10793:
10763:
10761:
10760:
10755:
10750:
10742:
10737:
10735:
10734:
10706:
10697:
10695:
10694:
10689:
10682:
10681:
10668:
10667:
10666:
10652:
10648:
10629:
10614:
10609:
10608:
10599:
10581:
10580:
10575:
10574:
10573:
10549:
10524:
10517:
10506:
10504:
10503:
10498:
10495:
10494:
10476:
10475:
10463:
10462:
10438:
10418:
10411:
10400:
10385:
10367:
10356:
10354:
10353:
10348:
10337:
10333:
10326:
10315:
10314:
10305:
10295:
10289:
10248:
10241:
10184:
10177:
10166:
10159:
10144:
10129:
10123:
10105:
10094:
10092:
10091:
10086:
10067:
10066:
10058:
10053:
10049:
10045:
10044:
10032:
10031:
10012:
10006:
10005:
9997:
9992:
9988:
9987:
9986:
9975:
9972:
9963:
9962:
9954:
9949:
9945:
9944:
9943:
9925:
9913:
9864:
9862:
9860:
9859:
9854:
9828:
9802:
9793:
9786:
9784:
9783:
9778:
9759:
9758:
9750:
9745:
9741:
9737:
9736:
9724:
9723:
9704:
9698:
9697:
9689:
9684:
9680:
9679:
9678:
9667:
9664:
9655:
9654:
9646:
9641:
9637:
9636:
9635:
9617:
9615:
9613:
9612:
9607:
9581:
9544:
9533:
9531:
9530:
9525:
9518:
9517:
9509:
9504:
9500:
9499:
9498:
9482:
9476:
9475:
9467:
9462:
9458:
9457:
9452:
9451:
9439:
9438:
9429:
9422:
9419:
9410:
9409:
9401:
9396:
9392:
9391:
9390:
9343:
9332:
9325:
9323:
9322:
9317:
9307:
9306:
9298:
9293:
9289:
9288:
9287:
9268:
9267:
9259:
9254:
9250:
9249:
9248:
9200:
9189:
9187:
9186:
9181:
9171:
9170:
9165:
9164:
9155:
9150:
9146:
9145:
9144:
9125:
9124:
9119:
9118:
9109:
9104:
9100:
9099:
9098:
9072:
9070:
9069:
9064:
9057:
9056:
9048:
9043:
9039:
9038:
9037:
9018:
9017:
9012:
9011:
9002:
8997:
8993:
8992:
8991:
8965:
8963:
8962:
8957:
8950:
8949:
8941:
8936:
8932:
8931:
8930:
8911:
8910:
8902:
8897:
8893:
8892:
8891:
8869:
8867:
8866:
8861:
8854:
8849:
8844:
8841:
8840:
8839:
8838:
8837:
8813:
8805:
8800:
8797:
8796:
8795:
8777:
8775:
8774:
8769:
8764:
8763:
8740:
8738:
8737:
8732:
8725:
8724:
8716:
8711:
8707:
8706:
8705:
8686:
8685:
8680:
8675:
8671:
8670:
8669:
8644:
8642:
8641:
8636:
8634:
8630:
8620:
8611:
8601:
8592:
8587:
8582:
8577:
8576:
8570:
8569:
8566:
8565:
8563:
8558:
8557:
8548:
8543:
8539:
8537:
8513:
8508:
8507:
8498:
8493:
8470:
8466:
8462:
8460:
8459:
8454:
8435:
8434:
8429:
8428:
8419:
8414:
8410:
8409:
8408:
8393:
8392:
8373:
8371:
8370:
8365:
8358:
8357:
8352:
8351:
8342:
8337:
8333:
8332:
8331:
8315:
8313:
8312:
8307:
8300:
8299:
8294:
8293:
8284:
8279:
8275:
8274:
8273:
8254:
8252:
8251:
8246:
8236:
8235:
8228:
8227:
8222:
8220:
8219:
8210:
8209:
8193:
8189:
8185:
8183:
8182:
8177:
8158:
8157:
8150:
8149:
8144:
8140:
8139:
8138:
8123:
8122:
8103:
8101:
8100:
8095:
8088:
8087:
8080:
8079:
8074:
8070:
8069:
8068:
8052:
8050:
8049:
8044:
8037:
8036:
8029:
8028:
8023:
8019:
8018:
8017:
7998:
7996:
7995:
7990:
7980:
7979:
7971:
7966:
7964:
7963:
7954:
7953:
7937:
7933:
7929:
7927:
7926:
7921:
7902:
7901:
7893:
7888:
7884:
7883:
7882:
7867:
7866:
7847:
7845:
7844:
7839:
7832:
7831:
7823:
7818:
7814:
7813:
7812:
7796:
7794:
7793:
7788:
7781:
7780:
7772:
7767:
7763:
7762:
7761:
7742:
7740:
7739:
7734:
7716:
7714:
7713:
7708:
7701:
7700:
7695:
7694:
7685:
7680:
7676:
7675:
7674:
7658:
7656:
7655:
7650:
7643:
7642:
7637:
7636:
7627:
7622:
7618:
7617:
7616:
7597:
7595:
7594:
7589:
7571:
7569:
7568:
7563:
7556:
7555:
7548:
7547:
7542:
7538:
7537:
7536:
7520:
7518:
7517:
7512:
7505:
7504:
7497:
7496:
7491:
7487:
7486:
7485:
7463:
7461:
7460:
7455:
7437:
7435:
7434:
7429:
7422:
7421:
7413:
7408:
7404:
7403:
7402:
7386:
7384:
7383:
7378:
7371:
7370:
7362:
7357:
7353:
7352:
7351:
7315:
7313:
7312:
7307:
7305:
7302:
7288:
7286:
7285:
7275:
7263:
7261:
7260:
7255:
7247:
7246:
7241:
7232:
7224:
7213:
7212:
7207:
7201:
7200:
7191:
7183:
7177:
7157:
7154:
7148:
7147:
7134:
7133:
7132:
7122:
7063:
7044:
7026:
7007:
6985:
6976:
6974:
6973:
6968:
6966:
6963:
6955:
6952:
6951:
6938:
6935:
6934:
6924:
6914:
6903:
6901:
6900:
6895:
6883:
6881:
6880:
6875:
6873:
6872:
6856:
6852:
6841:
6839:
6838:
6833:
6822:
6818:
6817:
6816:
6811:
6799:
6798:
6789:
6779:
6773:
6744:
6733:
6731:
6730:
6725:
6723:
6704:
6702:
6701:
6696:
6694:
6682:absolute moments
6679:
6653:
6646:
6624:
6622:
6621:
6616:
6605:
6601:
6576:
6575:
6565:
6523:random variables
6503:
6501:
6500:
6495:
6466:
6465:
6455:
6438:
6379:random variables
6368:
6366:
6365:
6360:
6349:has probability
6348:
6346:
6345:
6340:
6328:
6326:
6325:
6320:
6308:
6306:
6305:
6300:
6295:
6294:
6275:
6273:
6272:
6267:
6247:
6246:
6233:
6207:
6205:
6204:
6199:
6188:
6187:
6168:
6166:
6165:
6160:
6143:
6142:
6126:
6102:in probability.
6101:
6099:
6098:
6093:
6085:
6084:
6068:
6066:
6065:
6060:
6048:
6046:
6045:
6040:
6038:
6030:
6016:
6011:
6010:
6001:
5983:
5981:
5980:
5975:
5970:
5943:
5941:
5940:
5935:
5933:
5925:
5905:
5904:
5882:
5880:
5879:
5874:
5872:
5864:
5850:
5849:
5827:
5825:
5824:
5819:
5814:
5813:
5761:
5759:
5758:
5753:
5745:
5744:
5734:
5732:
5715:
5714:
5684:
5683:
5674:
5673:
5651:
5650:
5644:
5632:
5630:
5629:
5624:
5579:
5570:
5568:
5567:
5562:
5560:
5557:
5549:
5546:
5528:
5525:
5524:
5514:
5504:
5494:
5492:
5491:
5486:
5474:
5471:
5462:
5461:
5455:
5454:
5442:
5413:
5412:
5403:
5386:
5385:
5378:
5363:
5362:
5356:
5338:
5336:
5335:
5330:
5322:
5321:
5291:
5290:
5280:
5253:
5252:
5246:
5230:
5228:
5227:
5222:
5217:
5209:
5208:
5176:
5165:
5156:
5154:
5153:
5148:
5140:
5136:
5129:
5128:
5117:
5096:
5064:
4983:
4979:
4971:
4969:
4968:
4963:
4949:
4947:
4946:
4941:
4930:
4922:
4921:
4894:
4886:
4881:
4880:
4871:
4848:
4843:
4842:
4830:
4829:
4820:
4802:
4800:
4799:
4794:
4782:
4780:
4779:
4774:
4772:
4771:
4755:
4753:
4752:
4747:
4745:
4744:
4728:
4726:
4725:
4720:
4718:
4717:
4708:
4707:
4686:
4685:
4669:
4667:
4666:
4661:
4659:
4658:
4633:
4631:
4630:
4625:
4620:
4616:
4606:
4592:
4576:
4543:
4541:
4540:
4535:
4533:
4532:
4520:
4519:
4507:
4493:
4488:
4487:
4481:
4474:
4461:
4460:
4405:
4403:
4401:
4400:
4395:
4372:
4368:
4367:
4344:
4342:
4341:
4336:
4322:
4321:
4320:
4304:
4302:
4301:
4296:
4258:
4256:
4255:
4250:
4242:
4241:
4220:
4219:
4204:
4203:
4197:
4167:
4136:
4127:
4125:
4124:
4119:
4108:
4107:
4097:
4095:
4077:
4074:
4065:
4055:
4052:
4051:
4050:
4039:
4036:
4027:
4017:
4014:
4013:
4012:
3995:
3915:
3904:
3864:
3862:
3861:
3856:
3848:
3847:
3835:
3824:
3823:
3814:
3809:
3808:
3802:
3796:
3725:
3721:
3714:
3710:
3703:
3699:
3691:
3677:
3673:
3661:systematic error
3658:
3654:
3643:
3632:
3628:
3601:
3597:
3590:
3576:
3572:
3553:
3549:
3534:
3524:
3503:does in general
3502:
3498:
3487:
3483:
3469:
3458:
3443:
3439:
3428:
3415:
3413:
3412:
3407:
3391:
3389:
3388:
3383:
3369:
3358:
3357:
3342:
3325:
3323:
3322:
3317:
3301:
3299:
3298:
3293:
3279:
3268:
3267:
3252:
3235:
3233:
3232:
3227:
3215:
3213:
3212:
3207:
3192:
3190:
3189:
3184:
3167:
3150:
3149:
3137:
3123:
3121:
3120:
3115:
3100:
3098:
3097:
3092:
3075:
3058:
3057:
3045:
3025:
3023:
3022:
3017:
3002:
3000:
2999:
2994:
2977:
2960:
2959:
2947:
2927:
2925:
2924:
2919:
2907:
2905:
2904:
2899:
2885:
2874:
2873:
2858:
2838:
2836:
2835:
2830:
2815:
2813:
2812:
2807:
2793:
2782:
2781:
2766:
2748:
2746:
2745:
2740:
2738:
2726:
2724:
2723:
2718:
2699:
2697:
2696:
2691:
2677:
2666:
2665:
2650:
2636:
2634:
2633:
2628:
2611:
2593:
2591:
2590:
2585:
2568:
2551:
2550:
2538:
2524:
2520:
2485:
2450:
2446:
2442:
2435:
2433:
2432:
2427:
2410:
2374:
2360:
2352:
2345:
2343:
2342:
2337:
2322:
2311:
2310:
2295:
2294:
2289:
2273:
2260:
2256:
2240:weak convergence
2219:
2211:
2209:
2208:
2203:
2201:
2200:
2195:
2173:
2171:
2170:
2165:
2148:
2131:
2130:
2118:
2112:
2087:
2083:
2075:
2073:
2072:
2067:
2065:
2064:
2059:
2050:
2046:
2039:
2038:
2026:
2025:
1998:
1996:
1995:
1990:
1972:
1971:
1964:
1954:
1951:
1950:
1934:
1930:
1926:
1924:
1923:
1918:
1915:
1914:
1909:
1908:
1886:
1877:
1875:
1874:
1869:
1867:
1851:
1850:
1838:
1837:
1825:
1824:
1816:
1813:
1803:
1802:
1791:
1788:
1778:
1777:
1767:
1760:
1759:
1754:
1753:
1744:
1734:
1731:
1730:
1729:
1718:
1715:
1706:
1703:
1694:
1691:
1690:
1689:
1678:
1675:
1666:
1663:
1654:
1651:
1650:
1649:
1638:
1635:
1626:
1616:
1613:
1612:
1611:
1600:
1586:
1579:
1577:
1576:
1571:
1559:
1557:
1556:
1551:
1533:
1531:
1530:
1525:
1513:
1511:
1510:
1505:
1488:
1487:
1471:
1469:
1468:
1463:
1436:
1434:
1433:
1428:
1410:
1408:
1407:
1402:
1400:
1392:
1377:
1375:
1374:
1369:
1352:
1351:
1335:
1333:
1332:
1327:
1309:
1307:
1306:
1301:
1287:
1285:
1284:
1279:
1262:
1261:
1245:
1243:
1242:
1237:
1220:random variable
1215:
1213:
1212:
1207:
1205:
1201:
1200:
1192:
1169:are distributed
1168:
1166:
1165:
1160:
1158:
1157:
1141:
1139:
1138:
1133:
1114:
1112:
1111:
1106:
1094:
1092:
1091:
1086:
1084:
1063:
1061:
1060:
1055:
1026:
1025:
1015:
990:
983:
967:
965:
964:
959:
951:
950:
938:
937:
917:random variables
914:
912:
911:
906:
898:
897:
885:
884:
835:
831:
829:
827:
826:
823:
820:
808:
797:
795:
794:
789:
786:
785:
775:
770:
755:
753:
747:
743:
736:
735:
718:
704:
675:
673:
672:
667:
655:
654:
642:
633:
632:
627:
626:
609:
596:
585:
568:
561:
550:
541:
537:
505:
501:
493:
491:
490:
485:
480:
472:
471:
446:
444:
443:
438:
426:
424:
423:
418:
413:
412:
382:
380:
379:
374:
372:
371:
355:
353:
352:
347:
327:
325:
324:
319:
317:
316:
300:
298:
297:
292:
277:
275:
274:
269:
263:
262:
252:
247:
232:
224:
219:
218:
191:
189:
188:
183:
158:
154:
153:
21:
12235:
12234:
12230:
12229:
12228:
12226:
12225:
12224:
12205:
12204:
12203:
12198:
12131:
12088:
12083:
12045:
12021:HausdorffâYoung
12001:BabenkoâBeckner
11989:
11938:
11888:
11884:
11882:
11879:
11878:
11872:
11816:
11785:
11781:Sequence spaces
11721:
11716:
11665:
11658:
11638:
11619:
11600:
11578:
11559:
11530:
11507:
11488:
11469:
11442:
11423:
11387:
11370:
11365:
11364:
11355:
11353:
11345:
11344:
11340:
11331:
11329:
11321:
11320:
11316:
11305:
11301:
11293:
11289:
11281:
11277:
11269:
11265:
11249:
11248:
11244:
11230:
11226:
11218:
11203:
11195:
11191:
11183:
11179:
11171:
11167:
11156:
11152:
11144:
11140:
11132:
11128:
11124:, A.8, page 475
11120:
11116:
11111:
11049:
11017:
11012:
11011:
11005:
11001:
10996:
10991:
10988:
10987:
10959:
10954:
10953:
10945:
10937:
10925:
10920:
10919:
10913:
10909:
10904:
10896:
10894:
10891:
10890:
10860:
10856:
10849:
10840:
10836:
10834:
10831:
10830:
10789:
10785:
10783:
10780:
10779:
10769:
10736:
10730:
10726:
10724:
10721:
10720:
10719:convergence is
10677:
10673:
10662:
10658:
10646:
10645:
10625:
10622:
10621:
10610:
10604:
10600:
10595:
10592:
10591:
10569:
10565:
10561:
10558:
10556:
10553:
10552:
10523:
10519:
10516:
10512:
10490:
10486:
10471:
10467:
10458:
10454:
10452:
10449:
10448:
10437:
10433:
10417:
10413:
10406:
10399:
10395:
10384:
10380:
10366:
10362:
10322:
10310:
10306:
10301:
10300:
10296:
10291:
10285:
10279:
10276:
10275:
10262:
10254:
10243:
10240:
10236:
10233:
10222:
10212:
10204:
10183:
10179:
10176:
10172:
10165:
10161:
10154:
10143:
10139:
10121:
10116:
10111:
10104:
10100:
10052:
10040:
10036:
10027:
10023:
9991:
9982:
9978:
9948:
9939:
9935:
9933:
9930:
9929:
9915:
9911:
9902:
9893:
9886:
9875:
9836:
9833:
9832:
9830:
9826:
9817:
9808:
9801:
9797:
9787:
9744:
9732:
9728:
9719:
9715:
9683:
9674:
9670:
9640:
9631:
9627:
9625:
9622:
9621:
9589:
9586:
9585:
9583:
9579:
9570:
9561:
9554:
9543:
9539:
9503:
9494:
9490:
9461:
9453:
9447:
9443:
9434:
9430:
9425:
9395:
9386:
9382:
9380:
9377:
9376:
9367:
9360:
9353:
9342:
9338:
9326:
9292:
9283:
9279:
9253:
9244:
9240:
9238:
9235:
9234:
9226:
9213:
9190:
9160:
9156:
9149:
9140:
9136:
9114:
9110:
9103:
9094:
9090:
9088:
9085:
9084:
9077:Convergence in
9042:
9033:
9029:
9007:
9003:
8996:
8987:
8983:
8981:
8978:
8977:
8970:Convergence in
8935:
8926:
8922:
8896:
8887:
8883:
8881:
8878:
8877:
8833:
8829:
8828:
8824:
8791:
8787:
8785:
8782:
8781:
8759:
8755:
8750:
8747:
8746:
8710:
8701:
8697:
8674:
8665:
8661:
8659:
8656:
8655:
8632:
8631:
8619:
8617:
8612:
8600:
8598:
8593:
8581:
8578:
8575:
8567:
8564:
8553:
8549:
8542:
8540:
8521:
8516:
8514:
8503:
8499:
8492:
8488:
8486:
8483:
8482:
8468:
8464:
8424:
8420:
8413:
8404:
8400:
8388:
8384:
8379:
8376:
8375:
8347:
8343:
8336:
8327:
8323:
8321:
8318:
8317:
8289:
8285:
8278:
8269:
8265:
8263:
8260:
8259:
8221:
8215:
8211:
8205:
8201:
8199:
8196:
8195:
8191:
8187:
8143:
8134:
8130:
8118:
8114:
8109:
8106:
8105:
8073:
8064:
8060:
8058:
8055:
8054:
8022:
8013:
8009:
8007:
8004:
8003:
7965:
7959:
7955:
7949:
7945:
7943:
7940:
7939:
7935:
7931:
7887:
7878:
7874:
7862:
7858:
7853:
7850:
7849:
7817:
7808:
7804:
7802:
7799:
7798:
7766:
7757:
7753:
7751:
7748:
7747:
7722:
7719:
7718:
7690:
7686:
7679:
7670:
7666:
7664:
7661:
7660:
7632:
7628:
7621:
7612:
7608:
7606:
7603:
7602:
7577:
7574:
7573:
7541:
7532:
7528:
7526:
7523:
7522:
7490:
7481:
7477:
7475:
7472:
7471:
7443:
7440:
7439:
7407:
7398:
7394:
7392:
7389:
7388:
7356:
7347:
7343:
7341:
7338:
7337:
7326:
7318:Scheffé's lemma
7281:
7277:
7276:
7274:
7272:
7269:
7268:
7242:
7237:
7236:
7228:
7220:
7208:
7203:
7202:
7196:
7192:
7187:
7179:
7167:
7143:
7139:
7128:
7124:
7123:
7121:
7119:
7116:
7115:
7111:Additionally,
7066:in mean square
7062:
7058:
7043:
7039:
7025:
7021:
7006:
7002:
6947:
6943:
6937:
6930:
6926:
6925:
6923:
6921:
6918:
6917:
6889:
6886:
6885:
6868:
6864:
6862:
6859:
6858:
6854:
6850:
6812:
6807:
6806:
6794:
6790:
6785:
6784:
6780:
6775:
6763:
6757:
6754:
6753:
6743:
6739:
6719:
6717:
6714:
6713:
6710:
6690:
6688:
6685:
6684:
6677:
6652:
6648:
6641:
6638:
6571:
6567:
6555:
6538:
6534:
6532:
6529:
6528:
6506:where Ω is the
6461:
6457:
6445:
6421:
6418:
6417:
6389:
6375:
6354:
6351:
6350:
6334:
6331:
6330:
6314:
6311:
6310:
6290:
6286:
6281:
6278:
6277:
6242:
6238:
6229:
6217:
6214:
6213:
6183:
6179:
6174:
6171:
6170:
6169:and the events
6138:
6134:
6116:
6110:
6107:
6106:
6080:
6076:
6074:
6071:
6070:
6054:
6051:
6050:
6029:
6012:
6006:
6002:
5997:
5989:
5986:
5985:
5966:
5949:
5946:
5945:
5924:
5900:
5896:
5888:
5885:
5884:
5863:
5845:
5841:
5833:
5830:
5829:
5809:
5805:
5800:
5797:
5796:
5793:
5791:Counterexamples
5767:
5740:
5739:
5722:
5717:
5710:
5709:
5679:
5675:
5669:
5668:
5646:
5645:
5640:
5638:
5635:
5634:
5606:
5603:
5602:
5596:
5589:random elements
5533:
5527:
5520:
5516:
5515:
5513:
5511:
5508:
5507:
5470:
5457:
5456:
5450:
5449:
5438:
5408:
5404:
5399:
5381:
5380:
5368:
5358:
5357:
5352:
5350:
5347:
5346:
5317:
5316:
5286:
5282:
5270:
5248:
5247:
5242:
5240:
5237:
5236:
5213:
5204:
5203:
5192:
5189:
5188:
5175:
5171:
5164:
5160:
5124:
5120:
5107:
5102:
5098:
5092:
5090:
5087:
5086:
5063:
5059:
5056:
5031:
5030:
5024:
5023:
5021:
5014:
5007:
5006:
4978:
4957:
4954:
4953:
4926:
4917:
4913:
4890:
4882:
4876:
4872:
4867:
4844:
4838:
4834:
4825:
4821:
4816:
4808:
4805:
4804:
4788:
4785:
4784:
4767:
4763:
4761:
4758:
4757:
4740:
4736:
4734:
4731:
4730:
4713:
4709:
4703:
4699:
4681:
4677:
4675:
4672:
4671:
4654:
4650:
4648:
4645:
4644:
4640:
4638:Counterexamples
4602:
4588:
4581:
4577:
4572:
4549:
4546:
4545:
4528:
4527:
4515:
4514:
4503:
4489:
4483:
4482:
4477:
4456:
4455:
4428:
4425:
4424:
4363:
4359:
4351:
4348:
4347:
4346:
4316:
4312:
4310:
4307:
4306:
4290:
4287:
4286:
4265:
4237:
4236:
4215:
4211:
4199:
4198:
4193:
4176:
4173:
4172:
4157:
4152:
4103:
4099:
4085:
4080:
4046:
4042:
4008:
4004:
4002:
3999:
3998:
3976:
3948:
3910:
3903:
3899:
3889:
3876:
3843:
3842:
3831:
3819:
3815:
3810:
3804:
3803:
3798:
3786:
3780:
3777:
3776:
3759:
3749:
3723:
3720:
3716:
3712:
3709:
3705:
3701:
3698:
3694:
3689:
3675:
3672:
3668:
3667:, the sequence
3656:
3653:
3649:
3641:
3627:
3599:
3595:
3587:
3581:
3574:
3569:
3563:
3562:: The sequence
3551:
3546:
3542:
3536:
3526:
3521:
3514:
3508:
3500:
3495:
3489:
3485:
3480:
3474:
3460:
3455:
3445:
3441:
3436:
3430:
3426:
3401:
3398:
3397:
3365:
3353:
3349:
3338:
3333:
3330:
3329:
3311:
3308:
3307:
3275:
3263:
3259:
3248:
3243:
3240:
3239:
3221:
3218:
3217:
3201:
3198:
3197:
3195:continuity sets
3163:
3145:
3141:
3133:
3131:
3128:
3127:
3109:
3106:
3105:
3071:
3053:
3049:
3041:
3033:
3030:
3029:
3011:
3008:
3007:
2973:
2955:
2951:
2943:
2935:
2932:
2931:
2913:
2910:
2909:
2881:
2869:
2865:
2854:
2846:
2843:
2842:
2824:
2821:
2820:
2789:
2777:
2773:
2762:
2760:
2757:
2756:
2734:
2732:
2729:
2728:
2712:
2709:
2708:
2673:
2661:
2657:
2646:
2644:
2641:
2640:
2607:
2599:
2596:
2595:
2564:
2546:
2542:
2534:
2532:
2529:
2528:
2522:
2517:
2511:
2484:
2468:
2463:
2448:
2444:
2441:
2437:
2406:
2389:
2386:
2385:
2381:
2371:
2362:
2361:that dominates
2358:
2350:
2318:
2306:
2302:
2290:
2285:
2284:
2282:
2279:
2278:
2267:
2262:
2258:
2253:
2247:
2225:random elements
2217:
2196:
2191:
2190:
2182:
2179:
2178:
2144:
2126:
2122:
2114:
2102:
2096:
2093:
2092:
2085:
2081:
2060:
2055:
2054:
2034:
2030:
2021:
2017:
2016:
2012:
2010:
2007:
2006:
1967:
1966:
1953:
1946:
1942:
1940:
1937:
1936:
1932:
1928:
1910:
1904:
1903:
1902:
1899:
1896:
1895:
1865:
1864:
1846:
1845:
1833:
1829:
1820:
1819:
1798:
1794:
1773:
1769:
1765:
1764:
1755:
1749:
1748:
1747:
1725:
1721:
1699:
1685:
1681:
1659:
1645:
1641:
1607:
1603:
1601:
1599:
1595:
1593:
1590:
1589:
1565:
1562:
1561:
1539:
1536:
1535:
1519:
1516:
1515:
1483:
1479:
1477:
1474:
1473:
1442:
1439:
1438:
1416:
1413:
1412:
1391:
1383:
1380:
1379:
1347:
1343:
1341:
1338:
1337:
1315:
1312:
1311:
1295:
1292:
1291:
1257:
1253:
1251:
1248:
1247:
1225:
1222:
1221:
1191:
1184:
1180:
1178:
1175:
1174:
1153:
1149:
1147:
1144:
1143:
1127:
1124:
1123:
1100:
1097:
1096:
1080:
1072:
1069:
1068:
1021:
1017:
1005:
999:
996:
995:
988:
981:
978:converge in law
974:converge weakly
946:
942:
933:
929:
927:
924:
923:
915:of real-valued
893:
889:
880:
876:
874:
871:
870:
867:
833:
824:
821:
818:
817:
815:
810:
807:
803:
781:
777:
771:
760:
742:
740:
731:
727:
724:
721:
720:
713:
701:
695:
689:Graphic example
650:
646:
631:
622:
618:
615:
612:
611:
607:
602:
594:
592:
591:
583:
576:
570:
563:
556:
549:
543:
539:
536:
532:
520:
519:
500:
476:
467:
466:
455:
452:
451:
432:
429:
428:
408:
404:
399:
396:
395:
367:
363:
361:
358:
357:
341:
338:
337:
312:
308:
306:
303:
302:
286:
283:
282:
258:
254:
248:
237:
223:
214:
210:
208:
205:
204:
149:
145:
143:
140:
139:
118:random variable
80:
28:
23:
22:
15:
12:
11:
5:
12233:
12223:
12222:
12217:
12200:
12199:
12197:
12196:
12195:
12194:
12189:
12179:
12174:
12169:
12164:
12159:
12154:
12149:
12143:
12141:
12137:
12136:
12133:
12132:
12130:
12129:
12124:
12119:
12114:
12113:
12112:
12102:
12096:
12094:
12082:
12081:
12076:
12071:
12066:
12061:
12055:
12053:
12047:
12046:
12044:
12043:
12038:
12033:
12028:
12023:
12018:
12013:
12008:
12003:
11997:
11995:
11991:
11990:
11988:
11987:
11982:
11977:
11972:
11967:
11965:Function space
11962:
11957:
11952:
11946:
11944:
11940:
11939:
11937:
11936:
11931:
11930:
11929:
11919:
11914:
11908:
11906:
11891:
11887:
11874:
11873:
11871:
11870:
11865:
11860:
11855:
11850:
11845:
11840:
11838:CauchyâSchwarz
11835:
11829:
11827:
11818:
11817:
11815:
11814:
11809:
11804:
11798:
11796:
11787:
11786:
11784:
11783:
11778:
11773:
11764:
11759:
11758:
11757:
11747:
11739:
11737:Hilbert spaces
11729:
11727:
11726:Basic concepts
11723:
11722:
11715:
11714:
11707:
11700:
11692:
11664:
11663:
11651:
11642:
11636:
11623:
11617:
11604:
11598:
11582:
11576:
11563:
11557:
11542:
11528:
11511:
11505:
11492:
11486:
11473:
11467:
11446:
11440:
11427:
11421:
11400:
11391:
11385:
11371:
11369:
11366:
11363:
11362:
11338:
11314:
11311:. p. 126.
11299:
11297:, Theorem 14.5
11287:
11275:
11263:
11242:
11224:
11201:
11189:
11177:
11165:
11150:
11148:, Example 5.26
11138:
11126:
11113:
11112:
11110:
11107:
11106:
11105:
11100:
11095:
11090:
11085:
11080:
11075:
11065:
11060:
11055:
11048:
11045:
11044:
11043:
11042:
11041:
11025:
11020:
11015:
11008:
11004:
10999:
10995:
10985:
10973:
10970:
10967:
10962:
10957:
10952:
10948:
10944:
10940:
10936:
10933:
10928:
10923:
10916:
10912:
10907:
10903:
10899:
10888:
10876:
10863:
10859:
10853:
10843:
10839:
10816:
10806:
10801:
10792:
10788:
10776:
10767:
10753:
10745:
10740:
10733:
10729:
10710:
10709:
10700:
10698:
10687:
10680:
10676:
10671:
10665:
10661:
10656:
10651:
10644:
10641:
10638:
10635:
10632:
10628:
10624:
10623:
10620:
10617:
10613:
10607:
10603:
10598:
10594:
10593:
10590:
10578:
10572:
10568:
10564:
10563:
10560:
10547:
10546:
10535:
10534:
10533:
10530:
10521:
10514:
10509:
10508:
10507:
10493:
10489:
10485:
10482:
10479:
10474:
10470:
10466:
10461:
10457:
10435:
10430:
10429:
10428:
10415:
10397:
10382:
10364:
10359:
10358:
10357:
10346:
10343:
10340:
10336:
10332:
10329:
10325:
10321:
10318:
10313:
10309:
10304:
10299:
10294:
10288:
10284:
10264:
10263:almost surely.
10260:
10252:
10238:
10231:
10220:
10210:
10202:
10194:
10181:
10174:
10163:
10141:
10119:
10102:
10097:
10096:
10095:
10084:
10081:
10078:
10075:
10072:
10061:
10056:
10048:
10043:
10039:
10035:
10030:
10026:
10022:
10018:
10011:
10000:
9995:
9985:
9981:
9971:
9968:
9957:
9952:
9942:
9938:
9907:
9898:
9884:
9873:
9868:
9867:
9866:
9852:
9849:
9846:
9843:
9840:
9822:
9813:
9799:
9794:
9792:is a constant.
9776:
9773:
9770:
9767:
9764:
9753:
9748:
9740:
9735:
9731:
9727:
9722:
9718:
9714:
9710:
9703:
9692:
9687:
9677:
9673:
9663:
9660:
9649:
9644:
9634:
9630:
9605:
9602:
9599:
9596:
9593:
9575:
9566:
9552:
9541:
9536:
9535:
9534:
9523:
9512:
9507:
9497:
9493:
9488:
9481:
9470:
9465:
9456:
9450:
9446:
9442:
9437:
9433:
9428:
9418:
9415:
9404:
9399:
9389:
9385:
9365:
9358:
9351:
9340:
9335:
9334:
9333:
9331:is a constant.
9315:
9312:
9301:
9296:
9286:
9282:
9277:
9273:
9262:
9257:
9247:
9243:
9222:
9209:
9203:
9202:
9201:
9179:
9176:
9163:
9159:
9153:
9143:
9139:
9134:
9130:
9117:
9113:
9107:
9097:
9093:
9075:
9074:
9073:
9062:
9051:
9046:
9036:
9032:
9027:
9023:
9010:
9006:
9000:
8990:
8986:
8968:
8967:
8966:
8955:
8944:
8939:
8929:
8925:
8920:
8916:
8905:
8900:
8890:
8886:
8872:
8871:
8870:
8859:
8847:
8836:
8832:
8827:
8822:
8818:
8808:
8803:
8794:
8790:
8767:
8762:
8758:
8754:
8743:
8742:
8741:
8730:
8719:
8714:
8704:
8700:
8695:
8691:
8678:
8668:
8664:
8646:
8645:
8627:
8623:
8618:
8616:
8613:
8608:
8604:
8599:
8597:
8594:
8585:
8580:
8579:
8574:
8571:
8568:
8556:
8552:
8546:
8541:
8536:
8533:
8530:
8527:
8524:
8520:
8515:
8506:
8502:
8496:
8491:
8490:
8476:
8475:
8472:
8452:
8449:
8446:
8443:
8440:
8427:
8423:
8417:
8407:
8403:
8399:
8396:
8391:
8387:
8383:
8363:
8350:
8346:
8340:
8330:
8326:
8305:
8292:
8288:
8282:
8272:
8268:
8256:
8244:
8241:
8225:
8218:
8214:
8208:
8204:
8175:
8172:
8169:
8166:
8163:
8147:
8137:
8133:
8129:
8126:
8121:
8117:
8113:
8093:
8077:
8067:
8063:
8042:
8026:
8016:
8012:
8000:
7988:
7985:
7974:
7969:
7962:
7958:
7952:
7948:
7919:
7916:
7913:
7910:
7907:
7896:
7891:
7881:
7877:
7873:
7870:
7865:
7861:
7857:
7837:
7826:
7821:
7811:
7807:
7786:
7775:
7770:
7760:
7756:
7744:
7743:almost surely.
7732:
7729:
7726:
7706:
7693:
7689:
7683:
7673:
7669:
7648:
7635:
7631:
7625:
7615:
7611:
7599:
7598:almost surely.
7587:
7584:
7581:
7561:
7545:
7535:
7531:
7510:
7494:
7484:
7480:
7468:
7453:
7450:
7447:
7427:
7416:
7411:
7401:
7397:
7376:
7365:
7360:
7350:
7346:
7325:
7322:
7301:
7295:
7291:
7284:
7280:
7265:
7264:
7253:
7250:
7245:
7240:
7235:
7231:
7227:
7223:
7219:
7216:
7211:
7206:
7199:
7195:
7190:
7186:
7182:
7176:
7173:
7170:
7166:
7161:
7153:
7146:
7142:
7137:
7131:
7127:
7084:-th mean, for
7078:
7077:
7060:
7041:
7036:
7023:
7004:
6997:-th mean are:
6989:
6988:
6979:
6977:
6962:
6959:
6950:
6946:
6941:
6933:
6929:
6893:
6871:
6867:
6847:expected value
6843:
6842:
6831:
6828:
6825:
6821:
6815:
6810:
6805:
6802:
6797:
6793:
6788:
6783:
6778:
6772:
6769:
6766:
6762:
6741:
6722:
6708:
6693:
6650:
6637:
6634:
6614:
6611:
6608:
6604:
6600:
6597:
6594:
6591:
6588:
6585:
6582:
6579:
6574:
6570:
6564:
6561:
6558:
6554:
6550:
6547:
6544:
6541:
6537:
6493:
6490:
6487:
6484:
6481:
6478:
6475:
6472:
6469:
6464:
6460:
6454:
6451:
6448:
6444:
6437:
6434:
6431:
6428:
6425:
6396:random process
6385:
6374:
6371:
6358:
6338:
6318:
6298:
6293:
6289:
6285:
6265:
6262:
6259:
6256:
6253:
6250:
6245:
6241:
6237:
6232:
6228:
6227:lim sup
6224:
6221:
6197:
6194:
6191:
6186:
6182:
6178:
6158:
6155:
6152:
6149:
6146:
6141:
6137:
6133:
6130:
6125:
6122:
6119:
6115:
6091:
6088:
6083:
6079:
6058:
6036:
6033:
6028:
6025:
6022:
6019:
6015:
6009:
6005:
6000:
5996:
5993:
5973:
5969:
5965:
5962:
5959:
5956:
5953:
5931:
5928:
5923:
5920:
5917:
5914:
5911:
5908:
5903:
5899:
5895:
5892:
5870:
5867:
5862:
5859:
5856:
5853:
5848:
5844:
5840:
5837:
5817:
5812:
5808:
5804:
5792:
5789:
5788:
5787:
5779:
5766:
5763:
5751:
5748:
5743:
5738:
5731:
5728:
5725:
5721:
5713:
5708:
5705:
5702:
5699:
5696:
5693:
5690:
5687:
5682:
5678:
5672:
5667:
5663:
5660:
5657:
5654:
5649:
5643:
5622:
5619:
5616:
5613:
5610:
5594:
5583:
5582:
5573:
5571:
5556:
5553:
5545:
5542:
5539:
5536:
5531:
5523:
5519:
5484:
5481:
5478:
5468:
5465:
5460:
5453:
5448:
5445:
5441:
5437:
5434:
5431:
5428:
5425:
5422:
5419:
5416:
5411:
5407:
5402:
5398:
5395:
5392:
5389:
5384:
5377:
5374:
5371:
5367:
5366:lim sup
5361:
5355:
5328:
5325:
5320:
5315:
5312:
5309:
5306:
5303:
5300:
5297:
5294:
5289:
5285:
5279:
5276:
5273:
5269:
5265:
5262:
5259:
5256:
5251:
5245:
5220:
5216:
5212:
5207:
5202:
5199:
5196:
5173:
5162:
5146:
5143:
5139:
5135:
5132:
5127:
5123:
5116:
5113:
5110:
5106:
5101:
5095:
5061:
5055:
5052:
5038:
5037:
5019:
5012:
5002:
5001:
4997:
4996:
4988:
4987:
4977:
4974:
4961:
4939:
4936:
4933:
4929:
4925:
4920:
4916:
4912:
4909:
4906:
4903:
4900:
4897:
4893:
4889:
4885:
4879:
4875:
4870:
4866:
4863:
4860:
4857:
4854:
4851:
4847:
4841:
4837:
4833:
4828:
4824:
4819:
4815:
4812:
4792:
4770:
4766:
4743:
4739:
4716:
4712:
4706:
4702:
4698:
4695:
4692:
4689:
4684:
4680:
4657:
4653:
4639:
4636:
4635:
4634:
4623:
4619:
4615:
4612:
4609:
4605:
4601:
4598:
4595:
4591:
4587:
4584:
4580:
4575:
4571:
4568:
4565:
4562:
4559:
4556:
4553:
4531:
4526:
4523:
4518:
4513:
4510:
4506:
4502:
4499:
4496:
4492:
4486:
4480:
4473:
4470:
4467:
4464:
4459:
4453:
4450:
4447:
4444:
4441:
4438:
4435:
4432:
4406:
4393:
4390:
4387:
4384:
4379:
4375:
4371:
4366:
4362:
4358:
4355:
4334:
4329:
4325:
4319:
4315:
4294:
4279:
4276:
4275:is a constant.
4269:
4264:
4261:
4260:
4259:
4248:
4245:
4240:
4235:
4232:
4229:
4226:
4223:
4218:
4214:
4210:
4207:
4202:
4196:
4192:
4189:
4186:
4183:
4180:
4148:
4140:
4139:
4130:
4128:
4117:
4114:
4111:
4106:
4102:
4094:
4091:
4088:
4084:
4073:
4070:
4062:
4058:
4049:
4045:
4035:
4032:
4024:
4020:
4011:
4007:
3972:
3944:
3901:
3885:
3872:
3866:
3865:
3854:
3851:
3846:
3841:
3838:
3834:
3830:
3827:
3822:
3818:
3813:
3807:
3801:
3795:
3792:
3789:
3785:
3755:
3748:
3745:
3728:
3727:
3718:
3707:
3696:
3685:
3684:
3680:
3679:
3670:
3651:
3637:
3636:
3626:
3623:
3622:
3621:
3614:
3603:
3585:
3567:
3557:
3556:
3555:
3544:
3540:
3519:
3512:
3493:
3478:
3453:
3434:
3419:
3418:
3417:
3416:bounded below.
3405:
3381:
3378:
3375:
3372:
3368:
3364:
3361:
3356:
3352:
3348:
3345:
3341:
3337:
3336:lim inf
3327:
3326:bounded above;
3315:
3291:
3288:
3285:
3282:
3278:
3274:
3271:
3266:
3262:
3258:
3255:
3251:
3247:
3246:lim sup
3237:
3225:
3205:
3182:
3179:
3176:
3173:
3170:
3166:
3162:
3159:
3156:
3153:
3148:
3144:
3140:
3136:
3125:
3113:
3090:
3087:
3084:
3081:
3078:
3074:
3070:
3067:
3064:
3061:
3056:
3052:
3048:
3044:
3040:
3037:
3027:
3015:
2992:
2989:
2986:
2983:
2980:
2976:
2972:
2969:
2966:
2963:
2958:
2954:
2950:
2946:
2942:
2939:
2929:
2917:
2897:
2894:
2891:
2888:
2884:
2880:
2877:
2872:
2868:
2864:
2861:
2857:
2853:
2850:
2840:
2828:
2805:
2802:
2799:
2796:
2792:
2788:
2785:
2780:
2776:
2772:
2769:
2765:
2754:
2751:expected value
2737:
2716:
2689:
2686:
2683:
2680:
2676:
2672:
2669:
2664:
2660:
2656:
2653:
2649:
2638:
2626:
2623:
2620:
2617:
2614:
2610:
2606:
2603:
2583:
2580:
2577:
2574:
2571:
2567:
2563:
2560:
2557:
2554:
2549:
2545:
2541:
2537:
2515:
2504:
2503:
2502:
2482:
2474:) = (1 + cos(2
2466:
2456:
2439:
2425:
2422:
2419:
2416:
2413:
2409:
2405:
2402:
2399:
2396:
2393:
2380:
2377:
2369:
2347:
2346:
2335:
2332:
2329:
2326:
2321:
2317:
2314:
2309:
2305:
2301:
2298:
2293:
2288:
2265:
2251:
2214:continuity set
2199:
2194:
2189:
2186:
2175:
2174:
2163:
2160:
2157:
2154:
2151:
2147:
2143:
2140:
2137:
2134:
2129:
2125:
2121:
2117:
2111:
2108:
2105:
2101:
2063:
2058:
2053:
2049:
2045:
2042:
2037:
2033:
2029:
2024:
2020:
2015:
2004:random vectors
1988:
1985:
1981:
1978:
1975:
1970:
1961:
1957:
1949:
1945:
1913:
1907:
1890:
1889:
1880:
1878:
1863:
1860:
1857:
1854:
1849:
1844:
1841:
1836:
1832:
1828:
1823:
1812:
1809:
1806:
1801:
1797:
1787:
1784:
1781:
1776:
1772:
1768:
1766:
1763:
1758:
1752:
1741:
1737:
1728:
1724:
1714:
1711:
1702:
1697:
1688:
1684:
1674:
1671:
1662:
1657:
1648:
1644:
1634:
1631:
1623:
1619:
1610:
1606:
1602:
1598:
1597:
1569:
1549:
1546:
1543:
1523:
1503:
1500:
1497:
1494:
1491:
1486:
1482:
1472:, even though
1461:
1458:
1455:
1452:
1449:
1446:
1426:
1423:
1420:
1398:
1395:
1390:
1387:
1367:
1364:
1361:
1358:
1355:
1350:
1346:
1325:
1322:
1319:
1299:
1277:
1274:
1271:
1268:
1265:
1260:
1256:
1235:
1232:
1229:
1204:
1198:
1195:
1190:
1187:
1183:
1156:
1152:
1131:
1104:
1083:
1079:
1076:
1065:
1064:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1024:
1020:
1014:
1011:
1008:
1004:
957:
954:
949:
945:
941:
936:
932:
904:
901:
896:
892:
888:
883:
879:
866:
863:
845:
844:
805:
784:
780:
774:
769:
766:
763:
759:
750:
746:
739:
734:
730:
699:
691:
690:
686:
685:
664:
661:
658:
653:
649:
645:
640:
636:
630:
625:
621:
605:
581:
574:
547:
534:
528:
527:
523:
522:
510:
509:
499:
496:
483:
479:
475:
470:
465:
462:
459:
436:
416:
411:
407:
403:
370:
366:
345:
330:in probability
315:
311:
290:
279:
278:
267:
261:
257:
251:
246:
243:
240:
236:
230:
227:
222:
217:
213:
200:, is given by
181:
178:
175:
172:
169:
166:
163:
157:
152:
148:
122:
121:
114:
111:expected value
103:
102:
99:
96:
93:
90:
79:
76:
26:
9:
6:
4:
3:
2:
12232:
12221:
12218:
12216:
12213:
12212:
12210:
12193:
12190:
12188:
12185:
12184:
12183:
12180:
12178:
12177:Sobolev space
12175:
12173:
12172:Real analysis
12170:
12168:
12165:
12163:
12160:
12158:
12157:Lorentz space
12155:
12153:
12150:
12148:
12147:Bochner space
12145:
12144:
12142:
12138:
12128:
12125:
12123:
12120:
12118:
12115:
12111:
12108:
12107:
12106:
12103:
12101:
12098:
12097:
12095:
12092:
12086:
12080:
12077:
12075:
12072:
12070:
12067:
12065:
12062:
12060:
12057:
12056:
12054:
12052:
12048:
12042:
12039:
12037:
12034:
12032:
12029:
12027:
12024:
12022:
12019:
12017:
12014:
12012:
12009:
12007:
12004:
12002:
11999:
11998:
11996:
11992:
11986:
11983:
11981:
11978:
11976:
11973:
11971:
11968:
11966:
11963:
11961:
11958:
11956:
11953:
11951:
11948:
11947:
11945:
11941:
11935:
11932:
11928:
11925:
11924:
11923:
11920:
11918:
11915:
11913:
11910:
11909:
11907:
11905:
11885:
11875:
11869:
11866:
11864:
11861:
11859:
11856:
11854:
11851:
11849:
11848:Hilbert space
11846:
11844:
11841:
11839:
11836:
11834:
11831:
11830:
11828:
11826:
11824:
11819:
11813:
11810:
11808:
11805:
11803:
11800:
11799:
11797:
11795:
11793:
11788:
11782:
11779:
11777:
11774:
11772:
11768:
11765:
11763:
11762:Measure space
11760:
11756:
11753:
11752:
11751:
11748:
11746:
11744:
11740:
11738:
11734:
11731:
11730:
11728:
11724:
11720:
11713:
11708:
11706:
11701:
11699:
11694:
11693:
11690:
11686:
11685:
11683:
11679:
11675:
11671:
11657:
11652:
11648:
11643:
11639:
11633:
11629:
11624:
11620:
11614:
11610:
11605:
11601:
11595:
11591:
11587:
11583:
11579:
11573:
11569:
11564:
11560:
11554:
11550:
11549:
11543:
11539:
11535:
11531:
11525:
11521:
11517:
11512:
11508:
11502:
11498:
11493:
11489:
11483:
11479:
11474:
11470:
11464:
11460:
11456:
11452:
11447:
11443:
11437:
11433:
11428:
11424:
11418:
11414:
11409:
11408:
11401:
11397:
11392:
11388:
11382:
11378:
11373:
11372:
11352:
11348:
11342:
11328:
11324:
11318:
11310:
11303:
11296:
11291:
11284:
11279:
11273:, p. 354
11272:
11267:
11259:
11253:
11245:
11239:
11235:
11228:
11222:, Theorem 2.7
11221:
11216:
11214:
11212:
11210:
11208:
11206:
11199:, p. 289
11198:
11193:
11186:
11181:
11174:
11169:
11162:. p. 84.
11161:
11154:
11147:
11142:
11135:
11130:
11123:
11118:
11114:
11104:
11101:
11099:
11096:
11094:
11091:
11089:
11086:
11084:
11081:
11079:
11076:
11073:
11069:
11066:
11064:
11061:
11059:
11056:
11054:
11051:
11050:
11039:
11018:
11006:
11002:
10986:
10968:
10960:
10950:
10926:
10914:
10910:
10889:
10874:
10861:
10857:
10851:
10841:
10837:
10829:
10828:
10814:
10804:
10799:
10790:
10786:
10777:
10774:
10770:
10751:
10743:
10738:
10731:
10727:
10718:
10714:
10713:
10708:
10701:
10699:
10685:
10678:
10674:
10669:
10663:
10659:
10649:
10639:
10633:
10618:
10615:
10605:
10601:
10588:
10576:
10570:
10566:
10551:
10550:
10545:-convergence:
10544:
10540:
10536:
10531:
10528:
10527:Kai Lai Chung
10510:
10491:
10487:
10483:
10480:
10477:
10472:
10468:
10464:
10459:
10455:
10447:
10446:
10445:
10444:
10442:
10431:
10426:
10422:
10409:
10404:
10393:
10389:
10378:
10374:
10370:
10360:
10344:
10338:
10334:
10330:
10327:
10319:
10316:
10311:
10307:
10297:
10286:
10282:
10274:
10273:
10272:
10271:
10269:
10265:
10259:
10256:converges to
10255:
10246:
10234:
10227:
10223:
10216:
10209:
10205:
10198:
10195:
10192:
10188:
10185:converges to
10170:
10157:
10152:
10148:
10145:converges in
10137:
10133:
10127:
10122:
10114:
10109:
10098:
10079:
10076:
10073:
10059:
10054:
10041:
10037:
10033:
10028:
10024:
10009:
9998:
9993:
9983:
9979:
9969:
9966:
9955:
9950:
9940:
9936:
9928:
9927:
9923:
9919:
9910:
9906:
9901:
9897:
9891:
9887:
9880:
9876:
9869:
9847:
9844:
9841:
9829:converges to
9825:
9821:
9816:
9812:
9806:
9795:
9791:
9771:
9768:
9765:
9751:
9746:
9733:
9729:
9725:
9720:
9716:
9701:
9690:
9685:
9675:
9671:
9661:
9658:
9647:
9642:
9632:
9628:
9620:
9619:
9600:
9597:
9594:
9578:
9574:
9569:
9565:
9559:
9555:
9548:
9537:
9521:
9510:
9505:
9495:
9491:
9479:
9468:
9463:
9448:
9444:
9440:
9435:
9431:
9416:
9413:
9402:
9397:
9387:
9383:
9375:
9374:
9372:
9368:
9361:
9354:
9347:
9336:
9330:
9313:
9310:
9299:
9294:
9284:
9280:
9271:
9260:
9255:
9245:
9241:
9233:
9232:
9230:
9225:
9221:
9217:
9212:
9208:
9204:
9198:
9194:
9177:
9174:
9161:
9157:
9151:
9141:
9137:
9128:
9115:
9111:
9105:
9095:
9091:
9083:
9082:
9080:
9076:
9060:
9049:
9044:
9034:
9030:
9021:
9008:
9004:
8998:
8988:
8984:
8976:
8975:
8973:
8969:
8953:
8942:
8937:
8927:
8923:
8914:
8903:
8898:
8888:
8884:
8876:
8875:
8873:
8857:
8845:
8834:
8830:
8825:
8816:
8806:
8801:
8792:
8788:
8780:
8779:
8760:
8756:
8744:
8728:
8717:
8712:
8702:
8698:
8689:
8676:
8666:
8662:
8654:
8653:
8651:
8650:
8649:
8625:
8621:
8606:
8602:
8583:
8554:
8550:
8544:
8534:
8531:
8528:
8525:
8522:
8504:
8500:
8494:
8481:
8480:
8479:
8473:
8450:
8447:
8444:
8441:
8438:
8425:
8421:
8415:
8405:
8401:
8397:
8394:
8389:
8385:
8381:
8361:
8348:
8344:
8338:
8328:
8324:
8303:
8290:
8286:
8280:
8270:
8266:
8257:
8242:
8239:
8223:
8216:
8212:
8206:
8202:
8173:
8170:
8167:
8164:
8161:
8145:
8135:
8131:
8127:
8124:
8119:
8115:
8111:
8091:
8075:
8065:
8061:
8040:
8024:
8014:
8010:
8001:
7986:
7983:
7972:
7967:
7960:
7956:
7950:
7946:
7917:
7914:
7911:
7908:
7905:
7894:
7889:
7879:
7875:
7871:
7868:
7863:
7859:
7855:
7835:
7824:
7819:
7809:
7805:
7784:
7773:
7768:
7758:
7754:
7745:
7730:
7727:
7724:
7704:
7691:
7687:
7681:
7671:
7667:
7646:
7633:
7629:
7623:
7613:
7609:
7600:
7585:
7582:
7579:
7559:
7543:
7533:
7529:
7508:
7492:
7482:
7478:
7469:
7466:
7465:almost surely
7451:
7448:
7445:
7425:
7414:
7409:
7399:
7395:
7374:
7363:
7358:
7348:
7344:
7335:
7334:
7333:
7331:
7321:
7319:
7299:
7293:
7289:
7282:
7278:
7251:
7243:
7233:
7217:
7209:
7197:
7193:
7168:
7151:
7144:
7140:
7135:
7129:
7125:
7114:
7113:
7112:
7109:
7107:
7103:
7099:
7095:
7091:
7087:
7083:
7075:
7071:
7067:
7056:
7052:
7048:
7045:converges in
7037:
7034:
7030:
7019:
7015:
7011:
7008:converges in
7000:
6999:
6998:
6996:
6987:
6980:
6978:
6960:
6957:
6948:
6944:
6939:
6931:
6927:
6916:
6915:
6912:
6910:
6905:
6891:
6869:
6865:
6848:
6829:
6826:
6823:
6819:
6813:
6803:
6800:
6795:
6791:
6781:
6764:
6752:
6751:
6750:
6748:
6737:
6711:
6683:
6675:
6671:
6670:
6668:
6661:
6659:
6644:
6633:
6630:
6625:
6612:
6606:
6602:
6595:
6589:
6586:
6580:
6572:
6568:
6556:
6548:
6542:
6539:
6535:
6526:
6524:
6520:
6515:
6513:
6509:
6504:
6491:
6485:
6479:
6476:
6470:
6462:
6458:
6446:
6435:
6429:
6426:
6415:
6413:
6409:
6405:
6401:
6397:
6393:
6388:
6384:
6380:
6370:
6356:
6336:
6316:
6291:
6287:
6263:
6260:
6251:
6248:
6243:
6239:
6230:
6219:
6212:ensures that
6211:
6192:
6189:
6184:
6180:
6147:
6144:
6139:
6135:
6128:
6123:
6120:
6117:
6113:
6103:
6089:
6081:
6077:
6056:
6034:
6031:
6026:
6020:
6017:
6007:
6003:
5991:
5971:
5967:
5963:
5960:
5957:
5954:
5951:
5929:
5926:
5921:
5918:
5915:
5909:
5906:
5901:
5897:
5890:
5868:
5865:
5860:
5854:
5851:
5846:
5842:
5835:
5810:
5806:
5784:
5780:
5777:
5773:
5772:Fatou's lemma
5769:
5768:
5762:
5749:
5746:
5736:
5723:
5703:
5697:
5694:
5688:
5680:
5676:
5665:
5661:
5655:
5652:
5617:
5614:
5611:
5601:
5597:
5590:
5581:
5574:
5572:
5554:
5551:
5543:
5537:
5529:
5521:
5517:
5506:
5505:
5502:
5500:
5495:
5482:
5479:
5476:
5466:
5463:
5446:
5443:
5432:
5426:
5423:
5417:
5409:
5405:
5396:
5390:
5387:
5369:
5344:
5339:
5326:
5323:
5310:
5304:
5301:
5295:
5287:
5283:
5271:
5263:
5257:
5254:
5234:
5210:
5200:
5186:
5185:
5184:Almost surely
5180:
5169:
5157:
5144:
5141:
5137:
5133:
5130:
5125:
5121:
5108:
5099:
5084:
5080:
5076:
5072:
5068:
5067:almost surely
5051:
5049:
5048:real analysis
5045:
5035:
5028:
5018:
5011:
5003:
4998:
4994:
4993:quite certain
4989:
4984:
4973:
4959:
4950:
4934:
4931:
4918:
4910:
4907:
4901:
4898:
4887:
4877:
4873:
4861:
4858:
4852:
4849:
4839:
4835:
4831:
4826:
4822:
4810:
4790:
4768:
4764:
4741:
4737:
4714:
4710:
4704:
4696:
4693:
4687:
4682:
4678:
4655:
4651:
4621:
4617:
4610:
4607:
4599:
4596:
4593:
4578:
4569:
4563:
4560:
4557:
4551:
4524:
4521:
4511:
4508:
4500:
4497:
4494:
4471:
4468:
4465:
4462:
4448:
4442:
4439:
4436:
4430:
4422:
4420:
4415:
4411:
4407:
4388:
4382:
4377:
4373:
4364:
4360:
4353:
4332:
4327:
4323:
4317:
4313:
4292:
4284:
4280:
4277:
4274:
4270:
4267:
4266:
4246:
4233:
4230:
4224:
4221:
4216:
4212:
4205:
4190:
4187:
4184:
4181:
4171:
4170:
4169:
4165:
4161:
4156:
4151:
4147:
4138:
4131:
4129:
4115:
4112:
4109:
4104:
4100:
4086:
4082:
4071:
4068:
4060:
4056:
4047:
4043:
4033:
4030:
4022:
4018:
4009:
4005:
3997:
3996:
3993:
3991:
3986:
3984:
3980:
3975:
3971:
3967:
3963:
3958:
3956:
3952:
3947:
3943:
3939:
3936: â„
3935:
3931:
3927:
3923:
3919:
3913:
3908:
3897:
3893:
3888:
3884:
3880:
3875:
3871:
3852:
3849:
3839:
3836:
3828:
3825:
3820:
3816:
3787:
3775:
3774:
3773:
3771:
3767:
3763:
3758:
3754:
3744:
3742:
3738:
3733:
3686:
3681:
3666:
3662:
3647:
3638:
3633:
3619:
3615:
3612:
3608:
3604:
3593:
3588:
3580:
3570:
3561:
3558:
3547:
3533:
3529:
3522:
3515:
3506:
3496:
3481:
3472:
3471:
3467:
3463:
3456:
3449:
3437:
3424:
3420:
3403:
3395:
3376:
3370:
3362:
3354:
3350:
3343:
3328:
3313:
3305:
3286:
3280:
3272:
3264:
3260:
3253:
3238:
3223:
3203:
3196:
3177:
3174:
3171:
3154:
3151:
3146:
3142:
3126:
3111:
3104:
3085:
3082:
3079:
3068:
3062:
3059:
3054:
3050:
3028:
3013:
3006:
2987:
2984:
2981:
2970:
2964:
2961:
2956:
2952:
2930:
2915:
2892:
2886:
2878:
2870:
2866:
2859:
2841:
2826:
2819:
2800:
2794:
2778:
2774:
2767:
2755:
2752:
2714:
2707:
2703:
2684:
2678:
2662:
2658:
2651:
2639:
2621:
2618:
2615:
2601:
2578:
2575:
2572:
2555:
2552:
2547:
2543:
2527:
2526:
2518:
2509:
2505:
2500:
2496:
2492:
2491:
2489:
2481:
2477:
2473:
2469:
2461:
2457:
2454:
2420:
2417:
2414:
2403:
2397:
2391:
2383:
2382:
2376:
2372:
2365:
2356:
2330:
2324:
2307:
2303:
2296:
2291:
2277:
2276:
2275:
2272:
2268:
2254:
2245:
2241:
2236:
2234:
2230:
2229:metric spaces
2227:in arbitrary
2226:
2221:
2215:
2197:
2187:
2184:
2158:
2155:
2152:
2141:
2135:
2132:
2127:
2123:
2103:
2091:
2090:
2089:
2079:
2061:
2051:
2047:
2043:
2040:
2035:
2031:
2027:
2022:
2018:
2013:
2005:
2000:
1983:
1979:
1976:
1959:
1955:
1947:
1943:
1911:
1888:
1881:
1879:
1861:
1855:
1834:
1830:
1810:
1807:
1799:
1795:
1785:
1782:
1774:
1770:
1761:
1756:
1739:
1735:
1726:
1722:
1712:
1709:
1695:
1686:
1682:
1672:
1669:
1655:
1646:
1642:
1632:
1629:
1621:
1617:
1608:
1604:
1588:
1587:
1584:
1581:
1567:
1547:
1544:
1541:
1521:
1501:
1498:
1492:
1484:
1480:
1459:
1456:
1450:
1444:
1424:
1421:
1418:
1396:
1393:
1388:
1385:
1365:
1362:
1356:
1348:
1344:
1323:
1320:
1317:
1297:
1290:
1275:
1272:
1266:
1258:
1254:
1233:
1230:
1227:
1219:
1202:
1196:
1193:
1188:
1185:
1181:
1173:on intervals
1172:
1154:
1150:
1129:
1120:
1118:
1102:
1077:
1074:
1051:
1045:
1039:
1036:
1030:
1022:
1018:
1006:
994:
993:
992:
987:
979:
975:
971:
968:, is said to
955:
952:
947:
943:
939:
934:
930:
922:
918:
902:
899:
894:
890:
886:
881:
877:
862:
860:
854:
852:
840:
813:
801:
782:
778:
772:
767:
764:
761:
757:
748:
744:
737:
732:
728:
716:
712:
708:
702:
692:
687:
683:
679:
659:
656:
651:
647:
638:
634:
628:
623:
619:
608:
600:
589:
580:
573:
566:
562:and variance
559:
554:
546:
529:
526:Tossing coins
524:
517:
511:
506:
495:
473:
463:
450:
434:
409:
405:
392:
390:
386:
368:
364:
343:
335:
331:
313:
309:
288:
265:
259:
255:
249:
244:
241:
238:
234:
228:
225:
220:
215:
211:
203:
202:
201:
199:
195:
179:
176:
173:
170:
167:
164:
161:
155:
150:
146:
137:
134:
129:
125:
119:
115:
112:
108:
107:
106:
100:
97:
94:
91:
88:
85:
84:
83:
75:
72:
68:
64:
60:
55:
53:
49:
45:
41:
37:
33:
19:
11994:Inequalities
11934:Uniform norm
11822:
11791:
11742:
11667:
11666:
11646:
11627:
11608:
11589:
11567:
11547:
11519:
11496:
11477:
11450:
11431:
11406:
11395:
11376:
11354:. Retrieved
11350:
11341:
11330:. Retrieved
11326:
11317:
11308:
11302:
11290:
11278:
11266:
11233:
11227:
11192:
11180:
11168:
11159:
11153:
11141:
11129:
11117:
10765:
10716:
10702:
10542:
10440:
10439:is a sum of
10420:
10407:
10402:
10391:
10387:
10376:
10372:
10368:
10267:
10257:
10250:
10244:
10229:
10225:
10218:
10214:
10207:
10200:
10196:
10190:
10189:also in any
10186:
10168:
10155:
10150:
10146:
10135:
10131:
10125:
10117:
10112:
10107:
9921:
9917:
9908:
9904:
9899:
9895:
9889:
9882:
9878:
9871:
9823:
9819:
9814:
9810:
9804:
9789:
9576:
9572:
9567:
9563:
9557:
9550:
9546:
9370:
9363:
9356:
9349:
9345:
9328:
9228:
9223:
9219:
9215:
9210:
9206:
9196:
9192:
9078:
8971:
8647:
8477:
7327:
7266:
7110:
7105:
7101:
7097:
7093:
7085:
7081:
7079:
7073:
7069:
7065:
7054:
7050:
7049:-th mean to
7046:
7032:
7028:
7017:
7013:
7012:-th mean to
7009:
6994:
6992:
6981:
6908:
6906:
6844:
6746:
6735:
6706:
6673:
6666:
6663:
6657:
6655:
6642:
6639:
6626:
6527:
6516:
6508:sample space
6505:
6416:
6411:
6407:
6403:
6399:
6398:) converges
6386:
6382:
6376:
6104:
5794:
5786:convergence.
5600:metric space
5592:
5587:For generic
5586:
5575:
5498:
5496:
5340:
5232:
5182:
5178:
5167:
5158:
5082:
5078:
5074:
5070:
5066:
5057:
5041:
5033:
5026:
5016:
5009:
4992:
4951:
4641:
4417:
4272:
4163:
4159:
4149:
4145:
4143:
4132:
3989:
3987:
3982:
3978:
3973:
3969:
3965:
3961:
3959:
3954:
3950:
3945:
3941:
3937:
3933:
3929:
3925:
3921:
3917:
3911:
3906:
3895:
3891:
3886:
3882:
3878:
3873:
3869:
3867:
3769:
3765:
3761:
3756:
3752:
3751:A sequence {
3750:
3734:
3731:
3645:
3583:
3565:
3538:
3531:
3527:
3517:
3510:
3504:
3491:
3476:
3465:
3461:
3451:
3447:
3432:
2749:denotes the
2513:
2494:
2487:
2479:
2475:
2471:
2464:
2367:
2363:
2354:
2348:
2270:
2263:
2261:(denoted as
2249:
2239:
2237:
2222:
2176:
2080:to a random
2077:
2001:
1893:
1882:
1582:
1121:
1066:
977:
973:
969:
868:
855:
848:
811:
714:
709:sequence of
697:
677:
603:
578:
571:
564:
557:
544:
542:times. Then
508:Dice factory
393:
329:
280:
132:
130:
126:
123:
104:
81:
70:
56:
52:distribution
47:
43:
39:
38:, including
35:
29:
12192:Von Neumann
12006:Chebyshev's
11670:Citizendium
11197:Dudley 2002
11185:Dudley 2002
11175:, Lemma 2.2
11136:, p. 4
10266:If for all
10149:th mean to
6749:exist, and
5085:means that
5027:almost sure
3909:if for any
3768:if for all
2212:which is a
869:A sequence
136:independent
87:Convergence
67:mathematics
12209:Categories
12187:C*-algebra
12011:Clarkson's
11368:References
11356:2022-03-12
11332:2024-09-23
7324:Properties
7064:converges
7027:converges
6654:converges
6404:everywhere
5765:Properties
5065:converges
5054:Definition
5025:We may be
4414:metrizable
4263:Properties
3747:Definition
3737:consistent
3607:metrizable
3392:for every
3302:for every
3103:closed set
3101:for every
3003:for every
2753:operator);
2379:Properties
2177:for every
1246:. Indeed,
1218:degenerate
1117:continuous
865:Definition
599:bell curve
588:binomially
328:converges
78:Background
59:statistics
12182:*-algebra
12167:Quasinorm
12036:Minkowski
11927:Essential
11890:∞
11719:Lp spaces
11672:article "
11327:Knowledge
11285:, Th.2.19
11252:cite book
10972:∞
10935:→
10655:⇒
10643:∞
10481:⋯
10342:∞
10331:ε
10317:−
10283:∑
10242:for each
10134:and some
10124:| â€
10110:, and if
10017:⇒
9788:provided
9709:⇒
9487:⇒
9441:−
9327:provided
9276:⇒
9191:provided
9133:⇒
9026:⇒
8919:⇒
8821:⇒
8694:⇒
8615:⇒
8596:⇒
8573:⇓
8532:≥
8519:⇒
7175:∞
7172:→
7160:⇒
6801:−
6771:∞
6768:→
6676:, if the
6610:Ω
6596:ω
6581:ω
6563:∞
6560:→
6546:Ω
6543:∈
6540:ω
6486:ω
6471:ω
6453:∞
6450:→
6436::
6433:Ω
6430:∈
6427:ω
6424:∀
6408:pointwise
6394:(i.e., a
6157:∞
6154:→
6121:≥
6114:∑
6087:→
6021:ε
6018:≥
5958:ε
5922:−
5730:∞
5727:→
5720:⟶
5704:ω
5689:ω
5662::
5659:Ω
5656:∈
5653:ω
5477:ε
5447:ε
5433:ω
5424:−
5418:ω
5394:Ω
5391:∈
5388:ω
5376:∞
5373:→
5311:ω
5296:ω
5278:∞
5275:→
5261:Ω
5258:∈
5255:ω
5198:Ω
5115:∞
5112:→
5000:Example 2
4986:Example 1
4935:ϵ
4932:≥
4908:−
4902:−
4888:⋅
4853:ϵ
4850:≥
4832:−
4694:−
4597:−
4525:ε
4522:≤
4512:ε
4498:−
4463:ε
4244:→
4234:ε
4231:≥
4182:ε
4179:∀
4093:∞
4090:→
3840:ε
3826:−
3794:∞
3791:→
3663:) by the
3396:function
3363:≥
3306:function
3273:≤
3175:∈
3161:→
3152:∈
3083:∈
3069:≤
3060:∈
2985:∈
2971:≥
2962:∈
2879:≥
2787:→
2671:→
2619:≤
2605:↦
2576:≤
2562:→
2553:≤
2418:≤
2316:→
2292:∗
2188:⊂
2156:∈
2133:∈
2110:∞
2107:→
2052:⊂
2044:…
1843:→
1805:⇒
1780:⇝
1389:≥
1321:≤
1171:uniformly
1095:at which
1078:∈
1013:∞
1010:→
956:…
903:…
758:∑
660:μ
657:−
639:σ
461:Ω
344:μ
235:∑
174:…
12031:Markov's
12026:Hölder's
12016:Hanner's
11833:Bessel's
11771:function
11755:Lebesgue
11518:(1991).
11047:See also
10852:→
10800:→
10739:→
10670:→
10577:→
10412:), then
10375:towards
10270:> 0,
10193:th mean.
10153:for all
10130:for all
10055:→
9994:→
9951:→
9747:→
9686:→
9643:→
9506:→
9464:→
9398:→
9295:→
9256:→
9152:→
9106:→
9045:→
8999:→
8938:→
8899:→
8846:→
8802:→
8713:→
8677:→
8622:→
8603:→
8584:→
8545:→
8495:→
8416:→
8339:→
8281:→
8224:→
8146:→
8076:→
8025:→
7968:→
7890:→
7820:→
7769:→
7682:→
7624:→
7544:→
7493:→
7410:→
7359:→
7330:complete
7290:→
7136:→
7029:in mean
6940:→
6660:-th mean
6410:towards
5984:we have
5783:topology
5530:→
5081:towards
5079:strongly
4783:for all
4410:topology
4374:→
4324:→
4057:→
4019:→
3916:and any
3193:for all
3005:open set
2700:for all
2084:-vector
1956:→
1736:→
1696:→
1656:→
1618:→
1514:for all
1378:for all
694:Suppose
676:will be
551:has the
281:then as
198:variance
12051:Results
11750:Measure
11538:1102015
10379:. When
10138:, then
10115:(|
9920:,
9903:,
9863:
9831:
9818:,
9616:
9584:
9571:,
9218:, then
8374:, then
8104:, then
7848:, then
7717:, then
7572:, then
7438:, then
6712:|) and
6664:in the
6656:in the
5598:} on a
5472:for all
4803:, but:
4416:by the
4153:} on a
3898:. Then
3772:> 0
3646:ex ante
3609:by the
3444:, then
2727:(where
2702:bounded
1289:for all
919:, with
828:
816:
717:(â1, 1)
711:uniform
11904:spaces
11825:spaces
11794:spaces
11745:spaces
11733:Banach
11634:
11615:
11596:
11574:
11555:
11536:
11526:
11503:
11484:
11465:
11438:
11419:
11383:
11240:
10872:
10847:
10812:
10796:
10410:> 0
10249:, and
10068:
10050:
10013:
10007:
9989:
9976:
9973:
9964:
9946:
9760:
9742:
9705:
9699:
9681:
9668:
9665:
9656:
9638:
9519:
9501:
9483:
9477:
9459:
9423:
9420:
9411:
9393:
9308:
9290:
9269:
9251:
9172:
9147:
9126:
9101:
9058:
9040:
9019:
8994:
8951:
8933:
8912:
8894:
8855:
8842:
8814:
8798:
8726:
8708:
8687:
8672:
8436:
8411:
8359:
8334:
8301:
8276:
8237:
8194:) and
8159:
8141:
8089:
8071:
8038:
8020:
7981:
7938:) and
7903:
7885:
7833:
7815:
7782:
7764:
7702:
7677:
7644:
7619:
7557:
7539:
7506:
7488:
7423:
7405:
7372:
7354:
6738:|) of
6439:
6414:means
6400:surely
6105:Since
6069:hence
5944:. For
4475:
4421:metric
4419:Ky Fan
4078:
4075:
4066:
4053:
4040:
4037:
4028:
4015:
3914:> 0
3535:or of
2384:Since
1894:where
1817:
1814:
1792:
1789:
1745:
1732:
1719:
1716:
1707:
1692:
1679:
1676:
1667:
1652:
1639:
1636:
1627:
1614:
1560:where
1336:, and
705:is an
567:= 0.25
159:
46:, and
11659:(PDF)
11109:Notes
10771:) is
10511:then
10371:, or
10128:) = 1
7096:>
7072:) to
7038:When
7001:When
6669:-norm
4305:, if
2483:(0,1)
2274:) if
1411:when
1310:when
984:with
976:, or
972:, or
584:, ...
560:= 0.5
12089:For
11943:Maps
11682:GFDL
11632:ISBN
11613:ISBN
11594:ISBN
11572:ISBN
11553:ISBN
11524:ISBN
11501:ISBN
11482:ISBN
11463:ISBN
11436:ISBN
11417:ISBN
11413:1â28
11381:ISBN
11258:link
11238:ISBN
10969:<
10640:<
10616:<
10582:a.s.
10537:The
10339:<
10328:>
9881:and
9549:and
9355:and
9199:â„ 1.
8850:a.s.
8681:a.s.
8588:a.s.
8526:>
8467:and
8316:and
8229:a.s.
8190:and
8151:a.s.
8081:a.s.
8053:and
8030:a.s.
7934:and
7797:and
7659:and
7549:a.s.
7521:and
7498:a.s.
7387:and
7068:(or
7053:for
7016:for
6884:and
6745:and
6680:-th
6662:(or
5961:<
5955:<
5883:and
5499:a.s.
5480:>
5444:>
5008:Let
4509:>
4466:>
4281:The
4185:>
4083:plim
3968:and
3928:and
3837:>
3488:and
3421:The
2506:The
2002:For
1422:>
814:(0,
531:Let
334:mean
196:and
194:mean
61:and
11455:doi
11036:is
10778:If
10432:If
10247:â„ 0
10158:â„ 1
10099:If
9870:If
9538:If
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7320:).
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7031:to
6761:lim
6645:â„ 1
6553:lim
6443:lim
6406:or
6402:or
6369:).
5268:lim
5105:lim
5077:or
5073:or
5069:or
4583:min
4452:inf
3784:lim
3598:of
3550:to
3525:to
3505:not
3499:to
3484:to
3457:)}
3039:sup
3036:lim
2941:inf
2938:lim
2852:inf
2849:lim
2476:Ïnx
2451:is
2375:â.
2216:of
2100:lim
2088:if
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991:if
707:iid
593:As
69:as
30:In
12211::
11534:MR
11532:.
11461:.
11415:.
11349:.
11325:.
11254:}}
11250:{{
11204:^
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7332::
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5050:.
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3743:.
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1999:.
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861:.
703:}
577:,
518:.
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11823:L
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11769:/
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11711:e
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10637:]
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10631:[
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10543:L
10529:.
10522:n
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10515:n
10513:S
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10469:X
10465:=
10460:n
10456:S
10441:n
10436:n
10434:S
10427:.
10421:X
10416:n
10414:X
10408:Δ
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10113:P
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10083:)
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10071:(
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10047:)
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10025:X
10021:(
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9999:p
9984:n
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9916:(
9912:)
9909:n
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9900:n
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9894:(
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9865:.
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9848:Y
9845:,
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9839:(
9827:)
9824:n
9820:Y
9815:n
9811:X
9809:(
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9800:n
9798:Y
9790:c
9775:)
9772:c
9769:,
9766:X
9763:(
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9739:)
9734:n
9730:Y
9726:,
9721:n
9717:X
9713:(
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9601:c
9598:,
9595:X
9592:(
9580:)
9577:n
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9562:(
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9350:X
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9314:,
9311:c
9300:p
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9224:n
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8761:k
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8729:X
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8535:1
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8465:a
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8445:+
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8439:a
8426:r
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8325:Y
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8255:.
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8217:n
8213:Y
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8112:a
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7880:n
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7836:Y
7825:p
7810:n
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7774:p
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7731:Y
7728:=
7725:X
7705:Y
7692:r
7688:L
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7668:X
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7634:r
7630:L
7614:n
7610:X
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7583:=
7580:X
7560:Y
7534:n
7530:X
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7467:.
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7449:=
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7300:X
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7230:|
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7218:=
7215:]
7210:r
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7185:[
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6984:4
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6820:)
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6782:(
6777:E
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6607:=
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6578:(
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6549::
6536:{
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6477:=
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6468:(
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6264:1
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6244:n
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6236:{
6231:n
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6220:P
6196:}
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6177:{
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5972:2
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5910:0
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5894:(
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