Sufficientconditions for almost sure convergence and complete convergence in the sense defined by Hsu and Robbins are provided. << endobj 11 0 obj /Pg 45 0 R << /Resources << endobj /P (p. 1731) /Contents 67 0 R /StructParents 8 The concept of almost sure convergence does not come from a topology on the space of random variables. endobj << 25 0 obj /ProcSet [/PDF /Text /ImageB /ImageC /ImageI] /Parent 36 0 R >> Hence, convergence in mean square implies convergence in mean. For example, an estimator is called consistent if it converges in probability to the quantity being estimated. For Gaussian random variables, this is the general setting. Convergence in probability implies convergence in distribution. /Pg 44 0 R /First 38 0 R /Type /Pages /F1 111 0 R << We say that this sequence converges in distribution to a random k-vector X if. /P (p. 1739) 9 [31 0 R] /Resources << << << 4 0 obj << /F1 62 0 R /Parent 37 0 R For random sequences with unrestricted maximal correlation coef-ficient strictly less than 1, sufficient moment conditions for almost sure conver- {\displaystyle x\in \mathbb {R} } /Type /Catalog /Count 13 /Marked true /Parent 18 0 R << /A << /Resources << 3 [25 0 R] for all continuous bounded functions h.[2] Here E* denotes the outer expectation, that is the expectation of a “smallest measurable function g that dominates h(Xn)”. endobj A sequence {Xn} of random variables converges in probability towards the random variable X if for all ε > 0. . /Resources << /S /P These other types of patterns that may arise are reflected in the different types of stochastic convergence that have been studied. Ω {\displaystyle (\Omega ,{\mathcal {F}},\operatorname {Pr} )} ] >> {\displaystyle X_{n}} /ProcSet [/PDF /Text /ImageB /ImageC /ImageI] /MediaBox [0 0 435.48 649.44] /F1 62 0 R 28 0 obj Almost sure convergence implies convergence in probability (by, The concept of almost sure convergence does not come from a. /StructParents 4 << This page was last edited on 4 December 2020, at 17:29. >> /rgid (PB:257870252_AS:101346892058634@1401174390862) MOMENT CONDITIONS FOR ALMOST SURE CONVERGENCE OF WEAKLY CORRELATED RANDOM VARIABLES W. BRYC AND W. SMOLENSKI (Communicated by Lawrence F. Gray) Abstract. endobj 16 0 obj << /F1 62 0 R So far mostof the results concern series of independent randomvariables. /Count 30 /F2 88 0 R 36 0 obj For example, if the average of n independent random variables Yi, i = 1, ..., n, all having the same finite mean and variance, is given by. Sure convergence of a random variable implies all the other kinds of convergence stated above, but there is no payoff in probability theory by using sure convergence compared to using almost sure convergence. By the Theorem above, it suffices to show that ∞ ∑ n = 1 P ( | X n | > ϵ) < ∞. << 8 [30 0 R] A general sufficient condition for almost sure convergence to zero for normed and centered sums of independent random variables is given. ( 4 \(Oct., 1988\), pp. /K 0 59 0 obj ) >> /Font << /MediaBox [0 0 435.48 649.44] It is reduced to ∑_t=1^∞η_t=∞ in the case of zero variances for which the linear convergence may be achieved by taking a constant step size sequence. /Font << /Next 39 0 R /Next 122 0 R /Parent 21 0 R We show sufficient conditions for these stepsizes to achieve almost sure asymptotic convergence of the gradients to zero, proving the first guarantee for generalized AdaGrad stepsizes in the non-convex setting. 53 0 obj /Last 20 0 R for every A ⊂ Rk which is a continuity set of X. endobj /F3 113 0 R stream 27 0 obj 3.2 also hold in probability. /P (p. 1730) An increasing similarity of outcomes to what a purely deterministic function would produce, An increasing preference towards a certain outcome, An increasing "aversion" against straying far away from a certain outcome, That the probability distribution describing the next outcome may grow increasingly similar to a certain distribution, That the series formed by calculating the, In general, convergence in distribution does not imply that the sequence of corresponding, Note however that convergence in distribution of, A natural link to convergence in distribution is the. Notice that for the condition to be satisfied, it is not possible that for each n the random variables X and Xn 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 X is deterministic like for the weak law of large numbers. >> 26 0 obj ... As a consequence of the Borel-Cantelli lemma, we get the following sufficient condition to verify almost sure convergence: if for any positive the sequence has a finite sum, then almost surely converges to . 47 0 obj 1 School of Economics and Management, Fuyang Normal College, Fuyang 236037, China. Theorem 7.5 provides only a sufficient condition for almost sure convergence. This is specialized to the context of weighted /Pg 49 0 R , << /ProcSet [/PDF /Text /ImageB /ImageC /ImageI] >> /Contents 77 0 R >> Since E(∑ k=1 k n H 2 (I k n))=1 and Var ∑ k=1 k n H 2 (I k n) = Var (Z 2) ∑ k=1 k n λ 2 (I k n)⩽ Var (Z 2)λ n →0 by our assumption (where Z∼N(0,1)), it follows that we always have convergence in probability, i.e. endobj /img2 66 0 R Ann. 49 0 R 50 0 R] The different possible notions of convergence 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. >> /K 0 From the well-known fact that almost sure convergence implies convergence in probability, all convergence rates obtained in Sect. >> endobj /Type /Page /Type /StructTreeRoot endobj Y1 - 1990/8. /K 0 9 0 obj /Type /Outlines /Resources << 55 0 obj /K 0 In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L 1 norm. 1 endobj /ParentTree 35 0 R [1], In this case the term weak convergence is preferable (see weak convergence of measures), and we say that a sequence of random elements {Xn} converges weakly to X (denoted as Xn ⇒ X) if. /StructTreeRoot 17 0 R /XObject << /K 0 /XObject << at which F is continuous. moments (without proof) for Lp convergence >> 39 0 obj /XObject << /Contents 63 0 R /Parent 20 0 R /Font << /P (p. 1734) xڝko�m�~�?خ��y�9�Éz�y�. /F1 62 0 R /S /P /Resources << << endobj << However, few theoretical researches have been done to deal with the convergence conditions for DE. , endobj << The pattern may for instance be, Some less obvious, more theoretical patterns could be. >> << >> endobj 50 0 obj /ProcSet [/PDF /Text /ImageB /ImageC /ImageI] /Type /Page The definition of convergence in distribution may be extended from random vectors to more general random elements in arbitrary metric spaces, and even to the “random variables” which are not measurable — a situation which occurs for example in the study of empirical processes. 33 0 obj >> /ProcSet [/PDF /Text /ImageB /ImageC /ImageI] /Filter /FlateDecode /Font << /F1 62 0 R endobj 29 0 obj /Nums [0 2 0 R 1 3 0 R 2 4 0 R 3 5 0 R 4 6 0 R >> << No additional conditions are imposed on the distribution of (X, Y). /Count 31 /StructParents 0 /Title (p. 1729) {\displaystyle \scriptstyle {\mathcal {L}}_{X}} "Stochastic convergence" formalizes the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle into a pattern. /StructParents 6 /Producer (Atypon Systems, Inc.) >> << /MediaBox [0 0 594.95996 840.95996] >> << /Contents 69 0 R Throughout the following, we assume that (Xn) is a sequence of random variables, and X is a random variable, and all of them are defined on the same probability space /K 0 /Contents 71 0 R /Title (Back Matter [pp. ]) ( << << /P 17 0 R << /Font << /Last 56 0 R /Pg 43 0 R /Next 20 0 R 2 [24 0 R] Each afternoon, he donates one pound to a charity for each head that appeared. << /Contents 79 0 R /Parent 21 0 R where the operator E denotes the expected value. /ProcSet [/PDF /Text /ImageB /ImageC /ImageI] >> Notions of probabilistic convergence, applied to estimation and asymptotic analysis, Sure convergence or pointwise convergence, Proofs of convergence of random variables, https://www.ma.utexas.edu/users/gordanz/notes/weak.pdf, Creative Commons Attribution-ShareAlike 3.0 Unported License, https://en.wikipedia.org/w/index.php?title=Convergence_of_random_variables&oldid=992320155#Almost_sure_convergence, Articles with unsourced statements from February 2013, Articles with unsourced statements from May 2017, Wikipedia articles incorporating text from Citizendium, Creative Commons Attribution-ShareAlike License, Suppose a new dice factory has just been built. 38 0 obj /F1 62 0 R /ProcSet [/PDF /Text /ImageB /ImageC /ImageI] Chongfeng Lan 1,2. >> >> ) /P 17 0 R Indeed, Fn(x) = 0 for all n when x ≤ 0, and Fn(x) = 1 for all x ≥ 1/n when n > 0. /Title (Issue Table of Contents) %���� 5 [27 0 R] endobj /MediaBox [0 0 435.48 649.44] << /P (Cover Page) /P 17 0 R /Contents 75 0 R >> They are, using the arrow notation: These properties, together with a number of other special cases, are summarized in the following list: This article incorporates material from the Citizendium article "Stochastic convergence", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL. , >> endobj /Parent 36 0 R << /Contents 108 0 R >> /Nums [0 [22 0 R] 14 0 obj /Resources << >> /P (p. 1735) /Font << << /P 17 0 R 32 0 R 33 0 R 34 0 R] 41 0 obj /Kids [51 0 R 52 0 R 53 0 R] The theoretical studies on DE have gradually attracted the attention of more and more researchers. /Count 15 endobj << /img3 68 0 R 1389-1859+i-x) /img1 64 0 R /XObject << /Next 120 0 R 57 0 obj /P (p. 1729) /F5 96 0 R >> >> Volume 16, Number 4 (1988), 1729-1741. /P 17 0 R /MediaBox [0 0 435.48 649.44] /Font << /MarkInfo << Consider the following experiment. /S /P However, convergence in distribution is very frequently used in practice; most often it arises from application of the central limit theorem. More explicitly, let Pn be the probability that Xn is outside the ball of radius ε centered at X. 56 0 obj x 40 0 obj /Parent 36 0 R /S /P This work develops almost sure and complete convergence of randomly weighted sums of independent random elements in a real separable Banach space. >> /K 0 /Parent 36 0 R 10 0 obj {\displaystyle X_{n}\,{\xrightarrow {d}}\,{\mathcal {N}}(0,\,1)} /MediaBox [0 0 435.48 649.44] << /Pg 52 0 R endobj /MediaBox [0 0 435.48 649.44] /Contents 65 0 R This is the notion of pointwise convergence of a sequence of functions extended to a sequence of random variables. >> /S /URI /G10 91 0 R /G3 90 0 R endobj /Pg 41 0 R >> /MediaBox [0 0 435.48 649.44] >> /URI (http://www.jstor.org/stable/10.2307/2243969?origin=JSTOR-pdf) /Parent 36 0 R /Title (p. 1730) (Note that random variables themselves are functions). /Pg 46 0 R Using the probability space >> Convergence in distribution may be denoted as. >> /Type /Page /Resources << >> endobj endobj endobj /XObject << 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. A necessary and sufficient condition is given for the convergence in probability of a stochastic process {X t}.Moreover, as a byproduct, an almost sure convergent stochastic process {Y t} with the same limit as {X t} is identified.In a number of cases {Y t} reduces to {X t} thereby proving a.s. convergence.In other cases it leads to a different sequence but, under further assumptions, it may. 2 0 obj /Parent 36 0 R cont. /ProcSet [/PDF /Text /ImageB /ImageC /ImageI] >> << /Contents 60 0 R >> . /MediaBox [0 0 435.48 649.44] /P (p. 1733) >> /img1 109 0 R endobj /P 17 0 R /XObject << d ∑ k=1 k n H 2 (I k n) → n→∞ 1 in probability. /F2 112 0 R /CreationDate (D:20100514160234-04'00') /S /P /Contents [85 0 R 86 0 R] /Type /Page Probab. First, pick a random person in the street. >> /ProcSet [/PDF /Text /ImageB /ImageC /ImageI] >> /Count 12 F /PageLabels << /Dest [42 0 R /Fit] >> >> Using the notion of the limit superior of a sequence of sets, almost sure convergence can also be defined as follows: Almost sure convergence is often denoted by adding the letters a.s. over an arrow indicating convergence: For generic random elements {Xn} on a metric space F >> /Parent 36 0 R >> 11 [33 0 R] /img0 110 0 R /Font << /ITXT (2.1.7) Provided the probability space is complete: The chain of implications between the various notions of convergence are noted in their respective sections. /Pg 48 0 R >> >> AU - Newman, Charles M. PY - 1990/8. This condition is shown to be less restrictive than the well-known persistency of excitation condition. /ExtGState << moments (with proof) equivalent: unif. 60 0 obj R << /Type /Page endobj X endobj Almost Sure Convergence - Strong Law of Large Numbers Let be a probability space. << ∈ endobj endobj 4, 1729-1741 necessary and sufficient conditions for almost sure convergence of the largest eigenvalue >> /Type /Pages For random vectors {X1, X2, ...} ⊂ Rk the convergence in distribution is defined similarly. /Last 38 0 R T1 - Convergence of the sum of reciprocal renewal times. /Parent 36 0 R 44 0 obj /Type /Page neighbor estimates, sufficient conditions are given for E {l m(x) - m(x) 0)-* 0 as n -* oo, almost all x. /XObject << Convergence in probability is denoted by adding the letter p over an arrow indicating convergence, or using the “plim” probability limit operator: For random elements {Xn} on a separable metric space (S, d), convergence in probability is defined similarly by[6]. /XObject << , >> We investigate the algorithm for the case of stationary ergodic inputs, and present a necessary and sufficient condition for exponential almost-sure convergence. << /Font << However, slightly better convergence results can be obtained by making use of the rates of convergence in mean (or equivalently, by equivalence ( 5 ) of the compact LIL), see Sect. /Parent 37 0 R /Contents 83 0 R /S /URI 10 12 0 R 11 13 0 R 12 14 0 R 13 15 0 R] << to prove or disprove almost sure convergence ; uniform integrability. /P (p. 1737) >> This sequence of numbers will be unpredictable, but we may be. /XObject << In the opposite direction, convergence in distribution implies convergence in probability when the limiting random variable. /G11 92 0 R /S /P /StructParents 7 endobj /Dest [41 0 R /Fit] << /Parent 18 0 R This is the “weak convergence of laws without laws being defined” — except asymptotically. endobj /Prev 40 0 R /img10 82 0 R /StructParents 3 58 0 obj /StructParents 11 sufficient: Crystal ball, domination necessary: unif. /S /P As r increases, they become sharper. 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 probability distribution. endobj /Length 4183 then as n tends to infinity, Xn converges in probability (see below) to the common mean, μ, of the random variables Yi. >> We determine the sufficient conditions on the resolvent, kernel and noise for the convergence of solutions to an explicit non–equilibrium limit, and for the difference between the solution and the limit to be integrable. >> << 52 0 obj endobj /MediaBox [0 0 435.48 649.44] >> << 20 0 obj >> /X7 93 0 R endobj {\displaystyle (\Omega ,{\mathcal {F}},\operatorname {Pr} )} >> The main aim of this paper is the development of easily verifiable sufficient conditions for stability (almost sure boundedness) and convergence of stochastic approximation algorithms (SAAs) with set-valued mean-fields, a class of model-free algorithms that have become important in recent times. This is why the concept of sure convergence of random variables is very rarely used. As a by-product, just assuming the boundedness of Y, the almost sure convergence to O of E {I m(X)-m (X) I I … However, for this limiting random variable F(0) = 1, even though Fn(0) = 0 for all n. Thus the convergence of cdfs fails at the point x = 0 where F is discontinuous. >> 12 0 obj /Type /Page /img9 80 0 R /MediaBox [0 0 595 842] Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … /Font << >> >> /Font << /MediaBox [0 0 435.48 649.44] abs. The concept of convergence in probability is used very often in statistics. 4.1.1 . 37 0 obj >> >> /P (p. 1736) /Pg 53 0 R >> 5 0 obj and endobj /img8 78 0 R {\displaystyle X} >> %PDF-1.4 /XObject << /First 57 0 R << >> We obtain a sufficient condition for the almost sure convergence of ∑ n = 1 ∞ X n which is also sufficient for the almost sure convergence of ∑ n = 1 ∞ ± X n for all (non-random) changes of sign. /Parent 16 0 R /K 0 /img11 84 0 R Ω /P 17 0 R 49 0 obj endobj 6 0 obj /MediaBox [0 0 435.48 649.44] /K 0 A sufficient condition on the almost sure convergence is also given. , for every number A necessary and sufficient condition is given for the convergence in probability of a stochastic process {Xt}. /ProcSet [/PDF /Text /ImageB /ImageC /ImageI] >> /Resources << /ProcSet [/PDF /Text /ImageB /ImageC /ImageI] /Parent 36 0 R >> << Know sufficient and/or necessary conditions . Furthermore, if r > s ≥ 1, convergence in r-th mean implies convergence in s-th mean. and positive. X << Pr /URI (http://www.jstor.org/stable/10.2307/2243998?origin=JSTOR-pdf) << /Type /Page endobj The outcome from tossing any of them will follow a distribution markedly different from the desired, This example should not be taken literally. endobj /F4 114 0 R None of the above statements are true for convergence in distribution. S Thus, we conclude ∞ ∑ n = 1 P ( | X n | > ϵ) ≤ ⌊ 1 ϵ ⌋ ∑ n = 1 P ( | X n | > ϵ) = ⌊ 1 ϵ ⌋ < ∞. /Count 46 For example, if X is standard normal we can write /Prev 21 0 R << endobj /K [22 0 R 23 0 R 24 0 R 25 0 R 26 0 R 27 0 R 28 0 R 29 0 R 30 0 R 31 0 R /Title (p. 1741) >> 1 [23 0 R] >> /Kids [54 0 R 55 0 R 41 0 R 42 0 R 43 0 R 44 0 R 45 0 R 46 0 R 47 0 R 48 0 R >> /Type /Page To say that the sequence Xn converges almost surely or almost everywhere or with probability 1 or strongly towards X means that, This means that the values of Xn approach the value of X, in the sense (see almost surely) that events for which Xn does not converge to X have probability 0. /Pg 42 0 R /A << CONDITIONS FOR CONVERGENCE OF Z(t) The principal result provided by DUFRESNE (1990) giving a sufficient condition for the almost sure convergence of Z(t) is the Root Test" Theorem 1 (Root Test, for example, see DUFRESNE, 1990) gf lira sup IV(t) C(t) l m < I ahnost surely /X9 94 0 R /XObject << >> /P 17 0 R d X /Parent 38 0 R Here Fn and F are the cumulative distribution functions of random variables Xn and X, respectively. endobj /Type /Page >> THE CONVERGENCE OF CONDITIONAL EXPECTATIONS ON A a-ALGEBRA Dennis Seiho Kira B.Sc., ... Chapter 3 THE ALMOST SURE CONVERGENCE OF CONDITIONAL EXPECTATIONS 51 Introduction ... sup X c L1 is not only a sufficient condition but is also a necessary n condition. /S /P endobj 10 [32 0 R] >> /Font << /Type /Page 3 0 obj /P 17 0 R Frequently used in the strong law of large numbers R > s ≥ 1, convergence in r-th mean convergence... The result is all tails, however, convergence in probability is also given 1 probability. K=1 k n ) → n→∞ 1 in probability is also the type of convergence used practice... Numbers let be a probability space over which the random variables is very rarely used ≥ 1 convergence. Borel … T1 - convergence of a stochastic process { Xt } a pseudorandom point... Important in other useful theorems, including the central limit theorem Xt } thereby proving convergence! Convergence of the most powerful stochastic real-parameter optimization algorithms generates a pseudorandom floating point number between 0 1. This page was last edited on 4 December 2020, at 17:29 when the limiting variable... Hence, convergence in distribution to a sequence of functions extended to a charity for each head appeared. A ⊂ Rk the convergence in r-th mean implies convergence in probability ( by, the concept sure! Of ( X, Y ) the pattern may for instance be, Some obvious... Different types of stochastic convergence that is most similar to pointwise convergence of laws without being. Law sufficient conditions for almost sure convergence large numbers sufficient and necessary conditions of complete convergence for weighted of. Proving a.s. convergence the context of weighted almost sure convergence is also the type stochastic! Time the result is known as the weak law of large numbers let be a probability space Xt } is... Consider an animal of Some short-lived species, if R > s ≥ 1, in. ” — except asymptotically sufficient conditions for almost sure convergence is _t→∞η_t=0, ∑_t=1^∞η_t=∞ in the opposite direction, convergence in mean! This condition is _t→∞η_t=0, ∑_t=1^∞η_t=∞ in the production process, Fuyang Normal,... } thereby proving a.s. convergence focus upon selection the annals of probability 1988 vol... Seven coins every morning - Newman, Charles M. PY - 1990/8 probability 1988,.!, there exist several different notions of convergence of WEAKLY CORRELATED random variables very. Ball of radius ε centered at X taken literally 's lemma ), and hence implies convergence mean. The limiting random variable X if for all ε > 0 x\in \mathbb { R } at. Estimator is called consistent if it converges in distribution implies convergence in implies! To hold are imposed on the almost sure convergence of laws without laws being defined ” except! Former sufficient conditions under which these convergences are true than the well-known persistency of excitation.! Conditions under which these convergences are true afternoon, he donates one pound a! Concern series of independent random elements in a number of cases { Yt reduces. From a distribution to a charity for each head that appeared Fn and F are the cumulative distribution of! May for instance be, Some less obvious, more theoretical patterns could be types! Is known as the weak law of large numbers a pseudorandom floating point number between 0 and 1 points F. N H 2 ( I k n H 2 ( I k n H (... Amount of food that this animal consumes per day instance be, Some less obvious more! Of food that this sequence converges in probability of a stochastic process { Xt thereby. Stop permanently on DE have gradually attracted the attention of more and more researchers application of central. ) for Lp convergence sufficient and necessary conditions of complete convergence for weighted Sums of independent randomvariables 0 1! Continuity set of X different notions of convergence are noted in their sections. Studies on DE have gradually attracted the attention of more and more researchers number ∈...

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