正相协列乘积和的重对数律与强大数律

证明了强平稳正相协列乘积和的重对数律与不同分布正相协列乘积和的强大数律,指出了部分和服从强大数律但乘积和未必服从强大数律这一事实,并讨论了定理2中一个条件的必要性．     

The Law of the Logarithm for Weighted Sums of Independent Random Variables

Let X,X _n ;n1 be a sequence of real-valued i.i.d. random variables with E(X)=0. Assume B(u) is positive, strictly increasing and regularly-varying at infinity with index 1/2<1. Set b _n =B(n),n1. If

线性过程的强逼近和重对数律

本文讨论由独立同分布随机变量列产生的线性过程的泛函型重对数律和强逼近, 同时又给出由NA随机变量列产生的线性过程的重对数律.     

NA序列重对数律的几个极限定理

设{X_n;n≥1}均值为零、方差有限的NA平稳序列。记S_n=∑_(k=1)~n X_k,M_n=maxk≤n|S_k|,n≥1.假设σ~2=EX_1~2+2∑_(k=2)~∞EX_1X_k>0。本文讨论了:当ε 0时,P{M_n≥εσ(2nloglogn)~(1/2)的一类加权级数的精确渐近性质,以及当ε∞时,P{M_n≤εσ(π~2n/(8loglogn))~(1/2)}的一类加权级数的精确渐近性质。这些性质与重对数律和Chung重对数律的速度有关。     

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.     

Extreme Values and the Multivariate Compact Law of the Iterated Logarithm

For a sequence of independent identically distributed Euclidean random vectors, we prove a compact Law of the iterated logarithm when finitely many maximal terms are omitted from the partial sum. With probability one, the limiting cluster set of the appropriately operator normed partial sums is the closed unit Euclidean ball. The result is proved under the hypotheses that the random vectors belong to the Generalized Domain of Attraction of the multivariate Gaussian law and satisfy a mild integrability condition. The integrability condition characterizes how many maximal terms must be omitted from the partial sum sequence.     

The Functional Law of the Iterated Logarithm for the Empirical Process Based on Sample Means

Consider a double array of i.i.d. random variables with mean and variance and set . Let denote the empirical distribution function of Z_{1, n} ,..., Z _{N, n} and let be the standard normal distribution function. The main result establishes a functional law of the iterated logarithm for , where n=n(N) as N. For the proof, some lemmas are derived which may be of independent interest. Some corollaries of the main result are also presented.     

A Berry-Esseen Theorem and a Law of the Iterated Logarithm for Asymptotically Negatively Associated Sequences

Negatively associated sequences have been studied extensively in recent years, Asymptotically negative association is a generalization of negative association, In this paper a Berry Esseen theorem and a law of the iterated logarithm are obtained for asymptotically negatively associated sequences.     

A Compact Law of the Iterated Logarithm for Random Vectors in the Generalized Domain of Attraction of the Multivariate Gaussian Law

For a sequence of independent identically distributed random vectors, we prove that the limiting cluster set of the appropriately operator normed partial sums is, with probability one, the closed unit euclidean ball. The result is proved under the hypotheses that the law of the random vectors belongs to the Generalized Domain of Attraction of the multivariate Gaussian law and satisfy a mild integrability condition. The two conditions together are still weaker than finite second normed moment and are necessary and sufficient.     

迭代Brown运动的一个Chung型重对数律

X及Y分别为Rd1及Rd2中的相互独立的标准Brown运动,满足X（0）=Y（0）=0．定义,称为一个迭代Brown运动．本文给出了关于Zd1,d2的一个Chung型重对数律．     

两参数OU过程的钟重对数律（英）

在本文中，我们证明了两参数OU过程的钟重对数律。     

加权U—统计量的一个重对数律

证明了关于独立同分布随机变量序列的加权Ｕ－统计量的一个重对数律,类似于献「３」证明了一个加权Ｕ－统计量的解耦不等式。     

A Glivenko-Cantelli Theorem and Strong Laws for L-Statistics

We prove a weighted Glivenko-Cantelli theorem and apply it to study the rate of convergence in the strong law for L-statistics.     

关于可交换随机变量序列的重对数律

本文讨论了可交换随机变量序列{Xn:n≥1}的重对数律。     

On the Law of The Iterated Logarithm for Rapidly Increasing Subsequences

The usual law of the iterated logarithm states that the partial sums S_n of independent and identically distributed random variables can be normalized by the sequence a_n = √nlog log n, such that limsup_n→∞ S_n/a_n = √2 a.s. As has been pointed out by Gut (1986) the law fails if one considers the limsup along subsequences which increase faster than exponentially. In particular, for very rapidly increasing subsequences {n_k≥1} one has limsup_k→∞ S_nk/a_nk = 0 a.s. In these cases the normalizing constants a_nk have to be replaced by √nk log k to obtain a non-trivial limiting behaviour: limsup_k→∞ S_nk/ √nk log k = √2 a.s. We will present an intelligible argument for this structural change and apply it to related results.     

巴氏空间中随机元的极限理论及其应用

本文综述了巴拿赫空间中随机元的极限理论及其应用的一些结果，主要内容为：极限理论的某些新结果；等周方法的优化技巧；巴氏空间几何的概率方法；在经验过程研究中的应用。重点是近十年来有关问题研究的一些进展。     

自正则Chung型重对数律的精确渐近性

设X,X_1,X_2,…为零均值、非退化、吸引域为正态吸引场的独立同分布随机变量序列,记S_n=■X_j,M_n=■|S_k|,V_n~2=■X_j~2,n≥1.证明了当b>-1时,■δ~(-2(b 1))■(log log n)~P/(n log n)P(Mn/V_n≤ε~(π~2)/(8lgo log n)~(1/2)) =4/πГ(b 1)■~(-1)~k/(2k 1)~(2b 3).     

The Compact Law of the Iterated Logarithm for Multivariate Stochastic Approximation Algorithms

Abstract

We consider a sequence (Z _n ) _n≥1 defined by a general multivariate stochastic approximation algorithm and assume that (Z _n ) converges to a solution z* almost surely. We establish the compact law of the iterated logarithm for Z _n by proving that, with probability one, the limit set of the sequence (Z _n  ? z*) suitably normalized is an ellipsoid. We also give the law of the iterated logarithm for the l ^p norms, p ∈ [1, ∞], of (Z _n  ? z*).