Self-normalized central limit theorem for sums of weakly dependent random variables

Let {X _n,n1} be a strictly stationary sequence of weakly dependent random variables satisfyingEX _n=,EX _n ² <,Var S _n /n² and the central limit theorem. This paper presents two estimators of ². Their weak and strong consistence as well as their rate of convergence are obtained for -mixing, -mixing and associated sequences.Supported by a NSF grant and a Taft travel grant. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025.Supported by a Taft Post-doctoral Fellowship at the University of Cincinnati and by the Fok Yingtung Education Foundation of China. Hangzhou University, Hangzhou, Zhejiang, P.R. China and Department of Mathematics, National University of Singapore, Singapore 0511.     

A random CLT for dependent random variables

We introduce a new condition for {Y_τn} to have the same asymptotic distribution that {Y_n} has, where {Y_n} is a sequence of random elements of a metric space (S, d) and {τ_n} is a sequence of random indices. The condition on {Y_n} is that max_iDnd(Y_i, Y_an)→^p0 as n → ∞, where D_n = {i: |k_i−k_an| ≤ δ_ank_an} and {δ_n} is a nonincreasing sequence of positive numbers. The condition on {τ_n} is that P(|(k_τn/k_an)−1| > δ_an) → 0 as n → ∞. Under these conditions, we will show that d(Y_τn, Y_an) → ^P0 and apply this result to the CLT for a general class of sequences of dependent random variables.     

Maximal Inequalities and an Invariance Principle for a Class of Weakly Dependent Random Variables

The aim of this paper is to investigate the properties of the maximum of partial sums for a class of weakly dependent random variables which includes the instantaneous filters of a Gaussian sequence having a positive continuous spectral density. The results are used to obtain an invariance principle for strongly mixing sequences of random variables in the absence of stationarity or strong mixing rates. An additional condition is imposed to the coefficients of interlaced mixing. The results are applied to linear processes of strongly mixing sequences.     

Refinement of the almost sure central limit theorem for associated processes

In this paper, for the partial sumsS _n of a stationary associated random process it is proved that the logarithmic averages converge almost surely. The asymptotic normality of the normalized difference between the logarithmic averages and their limiting value is established. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 513–522, October, 2000.     

相伴随机变量序列的泛函型几乎处处中心极限定理

对于均值为零的平稳相伴随机变量序列,首先证明了在L(n)=EX_1~2 2 sum from n to j=2 Cov(X_1,X_j)是一个缓变函数的条件下的泛函型几乎处处中心极限定理.另外还给出了正则化部分和函数的对数平均几乎处处收敛性.     

The almost sure central limit theorems for the maxima of sums under some new weak dependence assumptions

We prove the almost sure central limit theorems for the maxima of partial sums of r.v.’s under a general condition of dependence due to Doukhan and Louhichi. We will separately consider the centered sequences and the sequences with positive expected values.     

Mann-Whitney test for associated sequences

Let {X ₁, ...,X _m } and {Y ₁, ...,Y _n } be two samples independent of each other, but the random variables within each sample are stationary associated with one dimensional marginal distribution functionsF andG, respectively. We study the properties of the classical Wilcoxon-Mann-Whitney statistic for testing for stochastic dominance in the above set up.     

An Asymptotic Behavior of the Remainder in the Central Limit Theorem for Moments of Sums of Independent Random Variables

The objective of the paper is to study the asymptotic behavior of the reminder in the central limit theorem for moments of sums of independent random variables.