Convergence results for a normalized triangular array of symmetric random variables

For a triangular array of symmetric random variables (without any integrability condition) we replace the classical assumption of row-wise independence by that of row-wise joint symmetry. Under this weaker assumption we prove some results concerning the convergence in distribution of a suitable sequence of randomly normalized sums to the standard normal distribution. Then we exhibit a class of row-wise independent triangular arrays for which the ordinary sums fail to converge in distribution, while our results enable us to affirm the convergence in distribution of the normalized sums.     

On Some Conditions for Complete Convergence for Arrays of Rowwise Negatively Dependent Random Variables

Abstract

This note contains some sufficient conditions for the complete convergence in the strong law of large numbers for arrays of rowwise negatively dependent random variables. Moreover, the rowwise sums are randomly indexed.     

行为混合阵列加权和的收敛性

本文研究了行为混合阵列加权和的收敛性.利用混合序列的Rosenthal型最大值不等式,讨论了混合阵列加权和的L1收敛性,依概率收敛性,几乎处处收敛性,及完全收敛性之间的等价关系,推广了行独立随机变量阵列相应的结果.     

Upper and lower classes for triangular arrays

Summary We derive characterizations of upper and lower classes in the law of the iterated logarithm for row sums of triangular arrays of Gaussian random variables. We also derive strong approximations for more general triangular arrays by Gaussian arrays. By combining these results we deduce characterizations of upper and lower classes for row sums from general arrays.Work done while the first named author was a Visiting Fellow of the Australian National University in the Department of Statistics, The Faculties. He expresses his deep gratitude to Professor C.R. Heathcote for his invitation     

行为NA的随机变量阵列加权和的完全收敛性(Ⅱ)

本文研究了行为NA的随机变量阵列加权和的完全收敛性,推广了行独立随机变量阵列相应的结果.且得到了任意随机变量阵列加权和完全收敛的一个定理.     

Asymptotic Expansion of the Distribution Density Function for the Sum of Random Variables in the Series Scheme in Large Deviation Zones

The work is designated for obtaining asymptotic expansions and determination of structures of the remainder terms that take into consideration large deviations both in the Cramer zone and Linnik power zones for the distribution density function of sums of independent random variables in a triangular array scheme. The result was obtained using general Lemma 6.1 of Saulis and Statuleviius in Limit Theorems for Large Deviations (Kluwer, 1991) and joining the methods of characteristic functions and cumulants. The work extends the theory of sums of random variables and in a special case, improves S. A.Book's results on sums of random variables with weights.     

Precise asymptotics of weighted sequences and their applications

We study an asymptotic property of weighted sequences of nonnegative functions which extends and unifies previous results concerned with precise asymptotics. As applications, we prove precise asymptotics for partial sums of independent identically distributed classical, noncommutative and free random variables.

     

Moment bounds for mixing random variables useful in nonparametric function estimation

Bounds for even moments of sums of strong mixing random variables are given which extend existing bounds. The method of proof uses simple facts about strong mixing random variables and combinatorial methods. The bound is particularly useful for triangular arrays with entries decreasing in size. To illustrate this, applications are being discussed to nonparametric kernel estimation with dependent observations.     

行为ND随机变量阵列加权和的完全收敛性

在非同分布的情况下,给出了行为ND随机变量阵列加权和的完全收敛性的充分条件,所得结果部分地推广了独立随机变量和NA随机变量的相应结果.作为其应用,获得了ND随机变量序列加权和的Marcinkiewicz-Zygmund型强大数定律.     

&rho|混合随机变量加权和的收敛性

该文研究了ρ 混合随机变量加权和的强大数律及完全收敛性, 获得了一些新的结果. 该文的结果推广和改进了Bai 等^[1]及Baum 等^[18] 在 i.i.d. 情形时相应的结果, 也推广和改进了Volodin 等^[4]在实值独立时相应的结果. 该文还得到了一关于任意随机变量阵列加权和的完全收敛性定理.     

Complete convergence of weighted sums for arrays of rowwise φ-mixing random variables

In this paper, we establish the complete convergence and complete moment convergence of weighted sums for arrays of rowwise φ-mixing random variables, and the Baum-Katz-type result for arrays of rowwise φ-mixing random variables. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of φ-mixing random variables is obtained. We extend and complement the corresponding results of X. J. Wang, S. H. Hu (2012).     

行为负相依随机变量阵列加权和的完全收敛性

本文研究了行为NOD随机变量阵列加权和的完全收敛性.运用NOD随机变量列的矩不等式以及截尾的方法,得到了关于行为NOD随机变量阵列加权和的完全收敛性的充分条件.利用获得的充分条件,推广了Baek(2008)关于行为NA随机变量阵列加权和的完全收敛性的结论,得到了比吴群英(2012)更为一般的结果.     

Trimmed sums of i.i.d. Banach space valued random variables

Limit laws for trimmed sums of triangular arrays of i.i.d. Banach space valued random variables are studied. It is shown that if the array belongs to the domain of attraction of an infinitely divisible law without Gaussian component on a separable Banach space of type 2, then the trimmed sum converges weakly to a nondegenerate Banach space valued random variable.     

A new look at Bergström’s theorem on convergence in distribution for sums of dependent random variables

In his 1972Periodica Mathematica Hungarica paper, H. Bergström stated a theorem on convergence in distribution for triangular arrays of dependent random variables satisfying, a ?-mixing condition. A gap in his proof of this theorem is explained and a more general version is proved under weakened hypotheses. The method used consists of comparisons between the given array and associated arrays which are parameterized by a truncation variable. In addition to the main theorem, this method yields a proof of equality of limiting finite-dimensional distributions for processes generated by the given associated arrays as well as the result that if a limit distribution for the centered row sums does exist, it must be infinitely divisible. Several corollaries to the main theorem specialize this result for convergence to distributions within certain subclasses of the infinitely divisible laws.     

Asymptotic Expansions in the Compound Poisson Limit Theorem

A triangular array of independent infinitesimal integer-valued random variables is considered. Asymptotic expansions for the probability distributions of sums of these variables are investigated in the case of the limiting compound Poisson laws.     

Limiting behavior for arrays of rowwise ρ *-mixing random variables

We study the limiting behavior of maximal partial sums for arrays of rowwise ?? ^*-mixing random variables and obtain some new results that improve the corresponding theorem of Zhu [M.H. Zhu, Strong laws of large numbers for arrays of rowwise ?? ^*-mixing random variables, Discrete Dyn. Nat. Soc., 2007, Article ID 74296, 6 pp., 2007].     

On complete moment convergence of weighted sums for arrays of row-wise negatively associated random variables

Negatively associated (NA) random variables are a more general class of random variables which include a set of independent random variables and have been applied to many practical fields. In this paper, the complete moment convergence of weighted sums for arrays of row-wise NA random variables is investigated. Some sufficient conditions for complete moment convergence of weighted sums for arrays of row-wise NA random variables are established. Moreover, under the weaker conditions, we extend the results of Baek et al. [J. Korean Stat. Soc. 37 (2008), pp. 73–80] and Sung [Abstr. Appl. Anal. 2011 (2011)]. As an application, the complete moment convergence of moving average processes based on an NA random sequence is obtained, which improves the result of Li and Zhang [Stat. Probab. Lett. 70 (2004), pp. 191–197 ].     

On the limiting behavior of randomly weighted partial sums

We study the almost sure limiting behavior and convergence in probability of weighted partial sums of the form where {W_nj, 1jn, n1} and {X_nj, 1jn, n1} are triangular arrays of random variables. The results obtain irrespective of the joint distributions of the random variables within each array. Applications concerning the Efron bootstrap and queueing theory are discussed.     

The Asymptotic Distribution of Self-Normalized Triangular Arrays

We develop a general approach to the asymptotic distribution of self-normalized triangular arrays of independent infinitesimal random variables     

The Bergström–Grigelionis asymptotic expansions

We consider a triangular array of independent identically distributed discrete random variables. We assume that the probability distribution of sums satisfies the necessary and sufficient conditions for the weak convergence to the compound Poisson distribution. The first known result (the case where random variables take only integer values) is due to B. Grigelionis, who estimated the convergence rate to the compound Poisson distribution. We extend the summation of random variables by including the variables taking discrete values and by using the Grigelionis ideas to obtain “lengthy” asymptotic expansions. These expansions are based on the well-known Bergstrom identity [H. Bergstrom, On asymptotic expansions of probability functions, Scand. Actuarial J., 34(1):1–33, 1951].