Strong Law of Large Numbers and Complete Convergence for Sequences of -Mixing Random Variables

We give some theorems of strong law of large numbers and complete convergence for sequences of φ-mixing random variables. In particular, Wittmann＇s strong law of large numbers and Teicher＇s strong law of large nnumbers for independent random variables are generalized to the case of φ -minxing random variables.

Complete Convergence Properties of the Sums for -mixing Sequences of Random Variables

This paper discusses complete convergence properties of the sums of -mixing random sequences.As a result,we improve the corresponding results of Wu Qunying(2001). And extended the Baum and Katz complete convergence to the case of -mixing random sequences by moment inequality and truncating without necessarily adding any extra conditions.

Some Convergence Results for Sequences of -mixing Random Variables

In this paper, we will present some strong convergence results for sequences of ψ-mixing random variables. The results for sequences of ψ-mixing random variables generalize the corresponding results for independent random variable sequences without any extra conditions.

Complete Convergence and Weak Law of Large Numbers for <i><span style="text-decoration:overline;">ρ</span></i>-Mixing Sequences of Random Variables

In this paper, the complete convergence and weak law of large numbers are established for ρ-mixing sequences of random variables. Our results extend and improve the Baum and Katz complete convergence theorem and the classical weak law of large numbers, etc. from independent sequences of random variables to ρ-mixing sequences of random variables without necessarily adding any extra conditions.

SOME LIMIT THEOREMS FOR SEQUENCES OF PAIRWISE NQD RANDOM VARIABLES

In this article, the authors study some limit properties for sequences of pairwise NQD random variables, which are not necessarily identically distributed. They obtain Baum and Katz complete convergence and the strong stability of Jamison＇s weighted sums for pairwise NQD random variables, which may have different distributions. Some wellknown results are improved and extended.

Inequalities of Maximum of Partial Sums and Convergence Rates in the Strong Laws for ρ^-mixing Random Variables

In this paper,we establish a Rosenthal-type inequality of partial sums for ρ~mixing random variables.As its applications,we get the complete convergence rates in the strong laws for ρ^-mixing random variables.The result obtained extends the corresponding result.

Hájek-Rényi-type Inequality for a Class of Random Variable Sequences and Its Applications

In this paper, we obtain the Hejek-Renyi-type inequality for a class of random variable sequences and give some applications for associated random variable sequences, strongly positive dependent stochastic sequences and martingale difference sequences which generalize and improve the results of Prakasa Rao and Soo published in Statist. Probab. Lett., 57（2002） and 78（2008）. Using this result, we get the integrability of supremum and the strong law of large numbers for a class of random variable sequences.

Complete Convergence for Arrays of Rowwise Ч-mixing Random Variables

In the paper, the complete convergence for arrays of rowwise Q-mixing random variables is studied. Some sufficient conditions for complete convergence for an array of row wise Q-mixing random variables without assumptions of identical distribution and stochastic domination are presented.

LARGE DEVIATIONS AND MODERATE DEVIATIONS FOR m-NEGATIVELY ASSOCIATED RANDOM VARIABLES

M-negatively associated random variables, which generalizes the classical one of negatively associated random variables and includes m-dependent sequences as its particular case, are introduced and studied. Large deviation principles and moderate deviation upper bounds for stationary m-negatively associated random variables are proved. Kolmogorov-type and Marcinkiewicz-type strong laws of large numbers as well as the three series theorem for m-negatively associated random variables are also given.

SOME LIMIT THEOREMS FOR WEIGHTED SUMS OF ARRAYS OF NOD RANDOM VARIABLES

In this paper the authors study the complete, weak and almost sure convergence for weighted sums of NOD random variables and obtain some new limit theorems for weighted sums of NOD random variables, which extend the corresponding theorems of Stout [1], Thrum [2] and Hu et al. [3].

On the Strong Law of Large Numbers for Non-Independent B-Valued Random Variables

This paper investigates some conditions which imply the strong laws of large numbers for Banach space valued random variable sequences. Some generalizations of the Marcinkiewicz-Zygmund theorem and the Hoffmann-J?rgensen and Pisier theorem are obtained. Key words strong law of large numbers - Banach space valued random variable sequence - p-smoothable Banach space CLC number O 211.4 - O 211.6 Foundation item: Supported by the National Natural Science Foundation of China (10071058)Biography: Gan Shi-xin (1939-), male, Professor, research direction: martingale theory, probability limiting theory and Banach space geometry theory.

Complete Convergence for (α,β)-Mixing Random Variables Sequences

The convergence properties are studied through the analysis of （α,β）-mixing random variables sequences in different situations. By using the corresponding inequality, a convergence theorem was presented, and some results for （α,β）-mixing random variables sequences with different distributions were obtained. The results extend the corresponding theorems of independent random variable sequences.

On Strong Law of Large Numbers for Random Sequence

This note is devoted to introduce a new concept of conditionally dominated random variables.Under suitable restrict conditions,a general strong law of large numbers for arbitrary continuous random variables is obtained.

Strong Limit Theorems for Arbitrary Fuzzy Stochastic Sequences

Based on fuzzy random variables, the concept of fuzzy stochastic sequences is defined. Strong limit theorems for fuzzy stochastic sequences are established. Some known results in non-fuzzy stochastic sequences are extended. In order to prove results of this paper, the notion of fuzzy martingale difference sequences is also introduced.

基于长度可变密钥的光学图像加密方法

为提升光学图像加密效果,设计了一种基于长度可变密钥的光学图像加密方法。基于混沌方程构成混沌序列均匀化普适算法,运用该算法生成均匀化伪随机序列,以此作为长度可变密钥,经空间光调制器调制后,转换此类密钥为伪随机相位掩模(PRPM),运用两个转换后PRPM双重随机相位编码初始光学图像,并结合变形分数傅里叶变换(AFrFT),相位转换编码后光学图像,获得加密光学图像,完成光学图像加密。实验结果显示,该方法可针对不同光学图像生成敏感性与唯一性密钥,实现不同光学图像的加密处理,加密效果显著,加密光学图像的像素频数分布均匀,且解密后光学图像与初始光学图像几乎一致,整体加解密效果理想,可有效保障光学图像的安全性,而且本方法可以在最短3 s的时间内完成光学图像的加密和解密,加解密效率更高,综合效果更佳。

Strong Stability of Linear Forms in φ-Mixing Random Variables

In this paper some new results of strong stability of linear forms in φ-mixing random variables are given. It is mainly proved that for a sequence of φ-mixing random variables {xn,n≥1} and two sequences of positive numbers {an,n≥1} and {bn,n≥1} there exist d dn∈R,n = 1,2,..., such that bn^-1∑i=1^naixi-dn→0 a.s.under some suitable conditions. The results extend and improve the corresponding theorems for independent identically distributed random variables.

Some Remarks for Sequences of Pairwise NQD Random Variables

We first obtain the Petrov theorem for pairwise NQD（negative quadrant dependent） random variables which may have different distributions.Some well-known results are improved and extended.Next,we give an example to clarify one of the important properties of sequences of pairwise NQD random variables,so that we can point out some mistakes that have appeared in recent published papers.

Laws of the Iterated Logarithm for-mixing Random Variables with Normal Distribution

Consider a ρ-mixing sequence of identically distributed random variables with the underlying dis- tribution in the domain of attraction of the normal distribution. This paper proves that law of the iterated logarithm holds for ρ-mixing sequences of random variables. Our results generalize and improve Theorems 1.2-1.3 of Qi and Cheng （1996） from the i.i.d, case to ρ-mixing sequences.

On a Conjecture of an Invariance Principle for Sequences of Associated Random Variables

A finite collection of random variables, X1} —, Xm, is said to be associated if any two coordinatewise nondecreasing functions /i and /a on Rm such that Jt = fj(Xi} , Xm~) has finite variance for j = l, 2, Oov(/i,fa)>0; an infinite collection is said to be associated if every finite subcollection is. associated. Thus the concept of "association" is introduced as dependence in probability (Esary, Proschan and Walkup (1967)). It is relative to a lot of practical models, such as the percolation models, the Ising models of statistical mechanics. Therefore, some people are

Precise Asymptotics in the Baum-Katz and Davis Laws of Large Numbers of ρ-mixing Sequences

Let {X,Xn;n ≥ 1} be a strictly stationary sequence of ρ-mixing random variables with mean zeros and finite variances. Set Sn =∑k=1^n Xk, Mn=maxk≤n｜Sk｜,n≥1.Suppose limn→∞ESn^2/n=：σ^2〉0 and ∑n^∞=1 ρ^2/d（2^n）〈∞,where d=2 if 1≤r〈2 and d〉r if r≥2.We prove that if E｜X｜^r 〈∞,for 1≤p〈2 and r〉p,then limε→0ε^2（r-p）/2-p ∑∞n=1 n^r/p-2 P{Mn≥εn^1/p}=2p/r-p ∑∞k=1（-1）^k/（2k＋1）^2（r-p）/（2-p）E｜Z｜^2（r-p）/2-p,where Z has a normal distribution with mean 0 and variance σ^2.