相依序列加权和的几乎处处中心极限定理

该文讨论了非平稳负(正)相依序列加权和的几乎处处中心极限定理,改进并推广了相依序列几乎处处中心极限定理的相关结果.     

An Almost Sure Central Limit Theorem for Weighted Sums of Mixing Sequences

In this paper, we prove an almost sure central limit theorem for weighted sums of mixing sequences of random variables without stationary assumptions. We no longer restrict to logarithmic averages, but allow rather arbitrary weight sequences. This extends the earlier work on mixing random variables.     

On the convergence in law of almost all sums of independent random variables

The paper deals with sums of independent and identically distributed random variables defined on some probability space which are multiplied by random coefficients. These coefficients are the values of independent random variables defined on another probability space. We obtain conditions for the weak convergence of weighted sums, for almost all coefficients, to some infinitely divisible distribution. The limit distribution for these sums is found. Supported by the Russian Foundation for Fundamental Research (grant No. 93-011-16099). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

Some limit theorems for independent random processes at random points in time and random elements defined by them

The paper deals with random variables which are the values of independent identically distributed stochastic processes at random points in time. We obtain conditions for the weak convergence of their sums, at almost all points in time, to the same infinitely divisible distribution and describe the limit distribution for these sums. Also we obtain an analog of the Donsker theorem and limit theorems for empirical processes for such random variables. Supported by the Russian Foundation for Fundamental Research (grant No. 93-011-16099). Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

Central limit theorem for linearly dependent fields of continuous elements

Summary The central limit theorem for stationary linearly dependent sequences is extended for elements in the space of continuous functions on a compact metric space. The proof is based on a new estimate for exponential-type moments of sums of independent random variables.     

A Note on the Martingale Method of Proving the Central Limit Theorem for Stationary Sequences

Under appropriate assumptions, the martingale approximation method allows us to reduce the study of the asymptotic behavior of sums of random variables that form a stationary random sequence to a similar problem for sums of stationary martingale differences. In an early paper on the martingale method, the author have proposed certain sufficient conditions for the central limit theorem to hold. It is shown in the present note that these conditions, at least in one particular case, can be essentially relaxed. In the context of the central limit theorem for Markov chains, a similar observation was done in a recent Holzmann and author's work. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 124–132.     

A stationary,pairwise independent,absolutely regular sequence for which the central limit theorem fails

Summary A strictly stationary finite-state non-degenerate random sequence is constructed which satisfies pairwise independence and absolute regularity but fails to satisfy a central limit theorem. The mixing rate for absolute regularity is only slightly slower than that in a corresponding central limit theorem of Ibragimov.This work was partially supported by NSF grant DMS 86-00399     

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.     

On the convergence in the law of sums and maxima of independent random variables with random parameters

The paper deals with maxima and sums of independent random variables. These random variables are the values of independent identically distributed stochastic processes at a random point in time. We obtain conditions for their weak convergence, at almost all points in time to the same infinitely divisible distribution and describe the limit distribution for these sums. Some applications of these results to statistics are considered. This work was supported by the Russian Foundation for Fundamental Research (grant No. 93-011-16099). Research Institute of Mathematics and Mechanics, Kazan State University, 17 Universitetskaya St., Kazan, 420008, Russia. Translated from Lietuvos Matematikos Rinkinys, Vol. 5, No. 1, pp. 52–64, January–March, 1995. Translated by A. N. Chuprunov     

Limit distributions of sums of m-dependent Bernoulli random variables

Summary The set of limit distributions of row sums of a triangular array of Bernoulli random variables which is strictly stationary and m-dependent in each row is characterized. Necessary and sufficient conditions for the convergence of the row sums to a given limit distribution are found. The case of convergence to a Poisson distribution is given special attention.     

The central limit theorem for random perturbations of rotations

Summary.   We prove a functional central limit theorem for stationary random sequences given by the transformations

A RANDOM FUNCTIONAL CENTRAL LIMIT THEOREM FOR PROCESSES OF PRODUCT SUMS OF LINEAR PROCESSES GENERATED BY MARTINGALE DIFFERENCES＊＊＊

A random functional central limit theorem is obtained for processes of partial sums and product sums of linear processes generated by non-stationary martingale differences. It devel-ops and improves some corresponding results on processes of partial sums of linear processes generated by strictly stationary martingale differences, which can be found in [5].     

Convergence of two functionals of asymptotically normal sums of independent random variables

We consider two functionals of sums of independent random variables and demonstrate that the validity of the central limit theorem for the sums of independent random variables that enter the arguments of those functionals is a sufficient condition for one of the functionals and a necessary and sufficient condition for the other one to have a weak limit. Proceedings of the Seminar on Stability Problems for Stochastic Models. Moscow. Russia. 1996. Part II.     

Almost Sure Central Limit Theorem for Partial Sums of Markov Chain

The authors prove an almost sure central limit theorem for partial sums based on an irreducible and positive recurrent Markov chain using logarithmic means,which realizes the extension of the almost sure central limit theorem for partial sums from an i.i.d.sequence of random variables to a Markov chain.     

Central limit theorem and almost sure central limit theorem for the product of some partial sums

In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables.     

Toward the History of the Saint Petersburg School of Probability and Statistics. I. Limit Theorems for Sums of Independent Random Variables

This is the first in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg school of probability and statistics in the period from 1947 to 2017. It is devoted to limit theorems for sums of independent random variables—a traditional subject for St. Petersburg. It refers to the classical limit theorems: the law of large numbers, the central limit theorem, and the law of the iterated logarithm, as well as important relevant problems formulated in the second half of the twentieth century. The latter include the approximation of the distributions of sums of independent variables by infinitely divisible distributions, estimation of the accuracy of strong Gaussian approximation of such sums, and the limit theorems on the weak almost sure convergence of empirical measures generated by sequences of sums of independent random variables and vectors.     

Central limit theorem for positively associated stationary random fields

Positively associated stationary random fields on d-dimensional integral lattice arise in various models of mathematical statistics, percolation theory, statistical physics, and reliability theory. In this paper, we shall be concerned with a field with covariance functions satisfying a more general condition than summability. A criterion for the validity of the central limit theorem (CLT) for partial sums of a field from this class is established. The sums are taken over an increasing nest of parallelepipeds or cubes. The well-known conjecture of Newman stated that for an associated stationary random field the above condition on the covariance function should force the CLT to hold. As was shown by N. Herrndorf and A. P. Shashkin, this conjecture fails already for d = 1. In the present paper, the uniform integrability of the squared partial sums is shown as being of key importance for the CLT to hold. Thus, an extension of Lewis’s theorem proved for a sequence of random variables is obtained. Also, it is indicated how to modify Newman’s conjecture for any d. A representation of variances of partial sums of a field by means of slowly varying functions of several arguments is used in an essential way.     

A criterion of convergence of nonrandomly centered random sums of independent identically distributed random variables

Necessary and sufficient conditions are presented for the weak convergence of random sums of independent identically distributed random variables in the double array scheme. As corollaries, two criteria of the normal convergence of random sums are given. Supported by the Russian Foundation for Fundamental Research (grant No. 96-011-01919). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

整值随机变量序列发生次数与条件概率和的等价性

在没有独立性、平稳性和相依性假设的条件下,利用分析方法讨论了整值随机变量序列取某个数值的次数与随机变量序列取该数值的条件概率和的等价性问题,建立并证明了若干强极限定理.     

An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables

Summary In this paper we establish an almost sure invariance principle with an error termo((t log logt)^1/2) (ast) for partial sums of stationary ergodic martingale difference sequences taking values in a real separable Banach space. As partial sums of weakly dependent random variables can often be well approximated by martingales, this result also leads to almost sure invariance principles for a wide class of stationary ergodic sequences such as ø-mixing and -mixing sequences and functionals of such sequences. Compared with previous related work for vector valued random variables (starting with an article by Kuelbs and Philipp [27]), the present approach leads to a unification of the theory (at least for stationary sequences), moment conditions required by earlier authors are relaxed (only second order weak moments are needed), and our proofs are easier in that we do not employ estimates of the rate of convergence in the central limit theorem but merely the central limit theorem itself.