On the Strong Law of Large Numbers for Multivariate Martingales with Continuous Time

We prove the strong law of large numbers for vector martingales with arbitrary operator normalizations. From the theorem proved, we deduce several known results on the strong law of large numbers for martingales with continuous time.     

The strong law of large numbers for random processes

Analogs of the Kolmogorov, Zygmund-Martsenkevich, and Brunk-Prokhorov strong law of large numbers are proved for martingales with continuous parameter. A new generalization of the Brunk—Prokhorov strong law of large numbers is given for martingales with discrete times. Along with convergence almost everywhere, we also prove the average convergence.     

Martingale convergence,series expansion of Gaussian elements,and strong law of large numbers in Fréchet spaces

The main result of this paper is the derivation of a convergence theorem for certain martingales with values in a separable Fréchet space F. It is shown that this result includes a well known theorem due to Chatterji. Moreover, the series expansion of zero-mean Gaussian elements with values in F and the strong law of large numbers for i.i.d. F-valued random elements also follow as applications of the main theorem.     

Maximal inequalities for demimartingales and a strong law of large numbers

Chow's maximal inequality for (sub)martingales is extended to the case of demi(sub)martingales introduced by Newman and Wright (Z. Wahrsch. Verw. Geb. 59 (1982) 361–371). This result serves as a “source” inequality for other inequalities such as the Hajek–Renyi inequality and Doob's maximal inequality and leads to a strong law of large numbers. The partial sum of mean zero associated random variables is a demimartingale. Therefore, maximal inequalities and a strong law of large numbers are obtained for associated random variables as special cases.     

随机变量序列的大数定律及常返性定理

对一类有界独立或相依的随机变量序列|ξn|,获得了它的伯努利大数定律、波雷尔强大数定律及常返性定理.作为应用,得出了Loève专著[1]中的推广的伯努利大数定律、常返性定理,改进了[1]中的推广的波雷尔强大数定律.     

Strong Law of Large Numbers with Operator Normalizations for Martingales and Sums of Orthogonal Random Vectors

We establish the strong law of large numbers with operator normalizations for vector martingales and sums of orthogonal random vectors. We describe its applications to the investigation of the strong consistency of least-squares estimators in a linear regression and the asymptotic behavior of multidimensional autoregression processes.     

Analysis of a generalized Friedman’s urn with multiple drawings

We study a generalized Friedman’s urn model with multiple drawings of white and blue balls. After a drawing, the replacement follows a policy of opposite reinforcement. We give the exact expected value and variance of the number of white balls after a number of draws, and determine the structure of the moments. Moreover, we obtain a strong law of large numbers, and a central limit theorem for the number of white balls. Interestingly, the central limit theorem is obtained combinatorially via the method of moments and probabilistically via martingales. We briefly discuss the merits of each approach. The connection to a few other related urn models is briefly sketched.     

GENERALIZED ROSENTHAL'S INEQUALITY FOR BANACH-SPACE-VALUED MARTINGALES

A generalized Rosenthal's inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and character-izes the p-uniform smoothness and q-uniform convexity of the underlying Banach space. As an application of this inequality, the strong law of large numbers for Banach-space-valued martingales is also given.     

随机序列的变换及其局部强收敛性

本文引入任意随机变量序列停时变换的概念，利用截尾方法构造几乎处处收敛的鞅结合无穷乘积定理，讨论了变换的局部收敛性及强大数定理，作为推论得到了关于赌博系统的若干强极限定理。     

Central limit theorem and weak law of large numbers with rates for martingales in Banach spaces

This paper is concerned with large-$\text{O}$ error estimates concerning convergence in distribution as well as norm convergence for Banach space-valued martingale difference sequences. Indeed, two general limit theorems equipped with rates of convergence for such difference sequences are established. Applications of these lead to the central limit theorem and the weak law of large numbers with rates for Banach space-valued martingales.     

Little's theorem: A stochastic integral approach

In a previous paper we have given a unified approach to the PASTA and the conditional PASTA property that is based upon the observation that the difference between the two limits can be represented as a stochastic integral with respect to a square integrable martingale. The equality of the two limits is then a consequence of a strong law of large numbers for martingales. In this paper we derive a non-standard version of Little's theorem via the same method. The moral of the story is that each of these theorems is but a particular case of a more general theory.     

B值拟鞅不等式及其应用

本文给出B值拟鞅的概率不等式与集合不等式,并用它们刻划了B空间的p可光滑性及q可凸性,作为应用,还证明了B值拟鞅的强大数律,收敛速度及极大值函数的可积性。     

On the rate of convergence in the strong law of large numbers for martingales

The aim of this note is to establish the Baum–Katz type rate of convergence in the Marcinkiewicz–Zygmund strong law of large numbers for martingales, which improves the recent works of Stoica [Series of moderate deviation probabilities for martingales, J. Math. Anal. Appl. 336 (2005), pp. 759–763; Baum–Katz–Nagaev type results for martingales, J. Math. Anal. Appl. 336 (2007), pp. 1489–1492; A note on the rate of convergence in the strong law of large numbers for martingales, J. Math. Anal. Appl. 381 (2011), pp. 910–913]. Furthermore, we also study some relevant limit behaviours for the uniform mixing process. Under some uniform mixing conditions, the sufficient and necessary condition of the convergence of the martingale series is established.     

Laws of large numbers for semimartingales with applications to stochastic regression

Summary Strong laws of large numbers for matrix-normalised vector-valued local martingales are established. The results are derived from strong laws for positive local submartingales and purely discontinuous local martingales and a Borel-Cantelli-type lemma for local martingales of finite variation. The multivariate strong laws are applied to study strong consistency of estimates in stochastic linear regression models.Research supported by the Australian Research Grants Scheme     

Some new results for demimartingales

Harremoёs obtained some new maximal inequalities for non-negative martingales. In this paper, we get some new maximal and minimal inequalities for non-negative demimartingales which generalize the results of Harremoёs. We also obtain an inequality for non-negative demimartingales which generalizes the result of Iksanov and Marynych. Finally we obtain a strong law of large numbers, strong growth rate and integrability of supremum for demimartingales which generalize and improve the result of Chow.     

一类马尔可夫骨架过程的强大数定律和中心极限定理

基于马尔可夫骨架过程极限分布的已有研究结果,本文运用波莱尔-康特立引理、更新理论、科尔莫哥洛夫的强大数定律以及独立同分布情形的中心极限定理等重要理论,分别给出了一类马尔可夫骨架过程对应的累积过程满足强大数定律和中心极限定理的充分条件.     

ON THE STRONG LAW OF LARGE NUMBERS

This paper introduces the concept of BC sequences and investigates some conditions which imply the strong law of large numbers for these sequences. The authors also study the strong law of large numbers for general random variable sequences. As applications of the result the authors characterize p-smoothableness of Banach space. Some generalizations of Petrov theorem, the Marcinkiewicz-Zygmund theorem and Hoffmann-Jorgensen and Pisier theorem are obtained.     

任意随机序列的变换及其a.s.收敛性

本文利用截尾方法构造几乎处处收敛的鞅结合无穷乘积定理，研究随机变量序列变换的局部收敛性及强大数定理，作为推论得到了关于赌博系统的若干强极限定理。     

STRONG LAW OF LARGE NUMBERS AND SHANNON-MCMILLAN THEOREM FOR MARKOV CHAINS FIELD ON CAYLEY TREE

1 lntroductionA tr(t(t is a grapl1 G = {T, E) wllicl1 is co1l11ected a11d colltai1ls no cir(.uits. Give11 a11y twov(irtic'is 't / P E T, let crP bc tl1e ullique patl1 col111ecti11g,v aIld /]. Defille tl1e graph distal1cc(l(rr, p) to hc the Ilu1llber of edges co11tained in the path crP.We discuss ulainly tl1e rooted Cayley tree TC,2(i.e.,binary tree. See Fig.1). In tlle Cayleytree Tc,2,tl1e root (denoted by 0) llas OIlly two 11(tiglll)(irs al1d all otller vertices have threeneighbors. A…     

A Driven Tagged Particle in Symmetric Exclusion Processes with Removals

We consider a driven tagged particle in a symmetric exclusion process on ℤ with a removal rule. In this process, untagged particles are removed once they jump to the left of the tagged particle. We investigate the behavior of the displacement of the tagged particle and prove limit theorems of it: an (annealed) law of large numbers, a central limit theorem, and a large deviation principle. We also characterize a class of ergodic measures for the environment process. Our approach is based on analyzing two auxiliary processes with associated martingales and a regenerative structure. © 2020 Wiley Periodicals LLC