LIMIT THEOREM FOR A CLASS OF SEQUENCE OF WEAKLY DEPENDENT RANDOM VARIABLES

In this paper, we consider a central limit theorem for the sequence of stationary m-dependent random variables, the variance of which is possibly infinite. Theorem. Let {Xn, n=l, 2,...} be a sequence of stationary m-dependent random variables with means zero. The following conditions are satisfied. (i) \[{M^2}\int_{{\text{|}}{X_1}| > M} {dP} /\int_{{X_1}| < M} {X_1^2} dP \to 0{\kern 1pt} {\kern 1pt} {\kern 1pt} (M \to \infty )\] (ii) \[\int_{\{ {X_1}| < M,|{X_i}| < M} {X_1^{}} {X_i}dP/\int_{|{X_1}| < M} {X_1^2} dP \to 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (M \to \infty )\] then there are constants Bsubsub>0, such that \[\frac{1}{{{B_n}}}\sum\limits_{i = 1}^n {{X_1}} \] converges in distribution N(0,1).     

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 par- ticular case,are introduced and studied.Large deviation principles and moderate devi- ation 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.     

On the Law of Large Numbers for Weakly Dependent Random Variables

In this paper, we present the Hsu–Robbins and Spitzer law of large numbers for m-dependent and -mixing random variables. In the main theorems, we do not assume that the random variables are identically distributed.     

On a Lower Bound of the Uniform Distance in the Central Limit Theorem for ϕ-Mixing Random Variables

We derive a lower bound of the uniform distance in the central limit theorem for real -mixing random variables under the finiteness of the eighth moments of summands. The main result of the present paper generalizes the corresponding author"s result obtained in 1997 for m-dependent random variables to the case of -mixing random variables.     

关于~*-mixing随机变量序列的强大数定律

利用任意随机变量序列的强大数定律讨论 * -mixing随机变量序列的强大数定律 ,得到了该序列的一个强极限定理 ,推广了经典的 * mixing随机变量序列的强大数定律 .同时讨论了 m相依序列和独立随机变量序列的强大数定律 .     

Central limit theorem for finitely-dependent random variables

For sequences of finitely-dependent random variables, under rather general hypotheses we establish estimates of integrals of the form where F_n(x) is the distribution function of the normalized sum of random variables; (x) is the standard normal distribution function. In the proof we use relations obtained by the method of C. Stein. The results are applicable, in particular, to m-dependent random variables and fields.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 93–97, 1988.     

Law of the iterated logarithm for sequences of dependent random variables

One obtains sufficient conditions for the applicability of the law of the iterated logarithm to sequences of m-dependent random variables possessing finite variances.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 186–194, 1980.The author is grateful to V. A. Egorov and L. V. Osipov for discussions on the topic of the paper and for useful remarks.     

Compound Poisson Approximation of the Number of Exceedances in Gaussian Sequences

Consider a finite sequence of Gaussian random variables. Count the number of exceedances of some level a, i.e. the number of values exceeding the level. Let this level and the length of the sequence increase simultaneously so that the expected number of exceedances remains fixed. It is well-known that if the long-range dependence is not too strong, the number of exceeding points converges in distribution to a Poisson distribution. However, for sequences with some individual large correlations, the Poisson convergence is slow due to clumping. Using Steins method we show that, at least for m-dependent sequences, the rate of convergence is improved by using compound Poisson as approximating distribution. An explicit bound for the convergence rate is derived for the compound Poisson approximation, and also for a subclass of the compound Poisson distribution, where only clumps of size two are considered. Results from numerical calculations and simulations are also presented.     

任意B值随机变量序列关于条件期望的一类强极限定理

利用随机变量的截尾方法和条件三级数定理,研究任意B值随机变量序列的极限性质,得到了一类关于条件期望的强极限定理和鞅差序列收敛定理,推广了与此相应的一些结果和若干经典的强大数定律.     

Lr Convergence for Arrays of Rowwise Negatively Sup eradditive Dep endent Random Variables

Let {Xnk, k≥1, n≥1} be an array of rowwise negatively superadditive depen-dent random variables and {an, n ≥ 1} be a sequence of positive real numbers such that an ↑ ∞. Under some suitable conditions, Lr convergence of a1n 1max≤j≤n ied. The results obtained in this paper generalize and improve some corresponding ones for negatively associated random variables and independent random variables. fl fl fl fl jP k=1 Xnk fl fl flfl is stud-ied. The results obtained in this paper generalize and improve some corresponding ones for negatively associated random variables and independent random variables.     

L_2-Cross-Validation最近邻中位数估计的相合性:m-相依情形

给出了ｍ－相依情形的Ｌ２－Ｃｒｏｓｓ－Ｖａｌｉｄａｔｉｏｎ最近邻中位数估计的弱相合性和用Ｌ_２－Ｃｒｏｓｓ－Ｖａｌｉｄａｔｉｏｎ方法选择的光滑参数的下界．     

NA随机变量列的有界重对数律

通过建立NA随机变量最大部分和的一些概率指数不等式，给出了具有不同分布的NA随机变量列有界重对数律的一些结果，因此推广了由R．Wittmann建立的独立随机变量的相关结果。     

相依随机变量阵列加权和的矩完全收敛性

本文研究了相依随机变量阵列加权和的矩完全收敛性.利用矩不等式和截尾法,建立了相依随机变量阵列加权和的矩完全收敛性的充分条件.将Volodin等(2004)及陈平炎等(2006)的关于独立随机变量阵列的结果推广到了负相协和负相依随机变量阵列的情形,推广并完善了Sung(2011),吴群英(2012)及郭明乐和祝东进(2012)的结果.     

关于任意随机可积序列的一个强收敛定理

本文研究了可积随机适应序列强收敛定理的问题.利用构造截尾停时以及鞅差序列的方法,获得了一类任意积随机适应序列的强收敛定理,推广了若干已知的结果.     

NA序列线性形式的强稳定性

本文研究了NA变量线性形式的强稳定性.利用随机变量的截尾术及强大数定律,得到了一般情况下不同分布NA变量具有强稳定性的充分条件,推广了NA列的Jamison型加权和具有强稳定性的充分条件.     

上尾渐近独立随机变量和的大偏差的渐近下界(英文)

本文研究了多元风险模型中服从长尾分布的带上尾渐近独立的随机变量和的大偏差渐近下界.利用大偏差的经典求法,得到了随机变量的非随机和和随机和的大偏差表达式,推广了独立同分布情形下的相关结论.     

A Law of the Iterated Logarithm for Randomly Stopped Sums of Heavy Tailed Random Vectors

 Let be an i.i.d. sequence of -valued random vectors belonging to the generalized domain of semistable attraction of some nonnormal law. Assume further that is a sequence of positive integer valued random variables such that for some for some discrete positive random variable D, where we do not assume that and are independent. Let . Then various laws of the iterated logarithm for the norm of as well as the radial projection onto a unit vector θ are presented.     

对数似然比与整值随机变量序列的一类强律

本文引进对数似然比作为整值随机变量序列相对于服从几何分布的独立随机变量序列的偏差的一种度量,并通过限制对数似然比给出了样本空间的一个子集.在此子集上得到了一类用不等式表示的强律,其中包含整值随机变量序列与相对熵密度及几何分布的熵函数有关的若干极限性质.     

A Law of the Iterated Logarithm for Randomly Stopped Sums of Heavy Tailed Random Vectors

 Let be an i.i.d. sequence of -valued random vectors belonging to the generalized domain of semistable attraction of some nonnormal law. Assume further that is a sequence of positive integer valued random variables such that for some for some discrete positive random variable D, where we do not assume that and are independent. Let . Then various laws of the iterated logarithm for the norm of as well as the radial projection onto a unit vector θ are presented. (Received 31 January 2000; in revised form 5 April 2000)     

整值随机变量序列与二重马氏链的比较及其极限性质

本文引进了对数似然比作为整值随机变量序列相对于二重马氏链的偏差的一种度量，并通过限制对数似然比给出了样本空间的某种子集。