Strassen-type invariance principles for exchangeable sequences

We prove strong invariance principles for exchangeable sequences of real random variables. Under the same conditions as those used for the central limit theorem, we obtain Strassen's invariance principle.     

A Nonconventional Invariance Principle for Random Fields

In Kifer and Varadhan (Ann Probab, to appear), we obtained a nonconventional invariance principle (functional central limit theorem) for sufficiently fast mixing stochastic processes with discrete and continuous time. In this article, we derive a nonconventional invariance principle for sufficiently well-mixing random fields.     

Weighted Invariance Principle for Banach Space-Valued Random Variables

A weighted weak invariance principle for nonseparable Banach space-valued functions is described via asymptotic behavior of a weighted Wiener process. It is proved that, unlike the usual weak invariance principle, the weighted variant cannot be characterized via validity of a central limit theorem in a Banach space. A strong invariance principle is introduced in the present context and used to prove the weighted weak invariance principle that we seek herewith. The result then is applied to empirical processes.     

A Note on Mixing Properties of Certain Associated Processes

We prove that under appropriate conditions a sequence of associated, integer-valued random variables is also strongly mixing. The almost sure invariance principle and the central limit theorem for integer-valued associated processes are also discussed.     

有限级亚纯函数的不变原理

研究有限级亚纯函数的统计性质,证明了有限级亚纯函数的几乎必然不变原理,如中心极限定理、重对数定律和泛函中心极限定理等统计定律是几乎必然不变原理的推论.     

有限级亚纯函数的不变原理

研究有限级亚纯函数的统计性质,证明了有限级亚纯函数的几乎必然不变原理,如中心极限定理、重对数定律和泛函中心极限定理等统计定律是几乎必然不变原理的推论.     

Almost sure invariance principles via martingale approximation

In this paper, we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale with stationary differences. The results are exploited to further investigate the central limit theorem and its invariance principle started at a point, the almost sure central limit theorem, as well as the law of the iterated logarithm via almost sure approximation with a Brownian motion, improving the results available in the literature. The conditions are well suited for a variety of examples; they are easy to verify, for instance, for linear processes and functions of Bernoulli shifts.     

关于随机环境中的马尔可夫过程的简介——纪念李国平院士吴新谋教授诞辰100周年

该文系统地介绍随机环境中的马尔可夫过程. 共4章, 第一章介绍依时的随机环境中的马尔可夫链(MCTRE), 包括MCTRE的存在性及等价描述; 状态分类; 遍历理论及不变测度; p-θ 链的中心极限定理和不变原理. 第二章介绍依时的随机环境中的马尔可夫过程(MPTRE), 包括MPTRE的基本概念; 随机环境中的q -过程存在唯一性; 时齐的q -过程;MPTRE的构造及等价性定理.第三章介绍依时的随机环境中的分枝链(MBCRE), 包括有限维的和无穷维的MBCRE的模型和基本概念; 它们的灭绝概念;两极分化; 增殖率等.第四章介绍依时依空的随机环境中的马尔可夫链(MCSTRE), 包括MCSTRE的基本概念、构造; 依时依空的随机环境中的随机徘徊(RWSTRE)的中心极限定理、不变原理.     

U-statistics in danach spaces

U-statistics in Banach spaces are considered and thoroughly investigated. The martingale structure, estimates of moments, the law of large numbers, the central limit theorem, the invariance principle, estimates of the rate of convergence, and large deviations are established     

Strong approximations for partial sums of i.i.d.B-valued r.v.'s in the domain of attraction of a Gaussian law

Summary We obtain a strong approximation theorem for partial sums of i.i.d.d-dimensional r.v.'s with possibly infinite second moments. Using this result, we can extend Philipp's strong invariance principle for partial sums of i.i.d.B-valued r.v.'s satisfying the central limit theorem toB-valued r.v.'s which are only in the domain of attraction of a Gaussian law. This new strong invariance principle implies a compact as well as a functional law of the iterated logarithm which improve some recent results of Kuelbs (1985).     

An invariance principle for lattices of dependent random variables

Summary It is shown that for -mixing arrays of Banach space valued random vectors the central limit theorem implies the invariance principle. Applying this result to lattices of random variables a higher dimensional invariance principle under dependence assumptions is obtained.Dedicated to Professor Leopold Schmetterer     

Invariance principles for Diophantine approximation of formal Laurent series over a finite base field

In a recent paper, the first and third author proved a central limit theorem for the number of coprime solutions of the Diophantine approximation problem for formal Laurent series in the setting of the classical theorem of Khintchine. In this note, we consider a more general setting and show that even an invariance principle holds, thereby improving upon earlier work of the second author. Our result yields two consequences: (i) the functional central limit theorem and (ii) the functional law of the iterated logarithm. The latter is a refinement of Khintchine's theorem for formal Laurent series. Despite a lot of research efforts, the corresponding results for Diophantine approximation of real numbers have not been established yet.     

On the Weak Invariance Principle for Stationary Sequences under Projective Criteria

In this paper, we study the central limit theorem and its weak invariance principle for sums of a stationary sequence of random variables, via a martingale decomposition. Our conditions involve the conditional expectation of sums of random variables with respect to the distant past. The results contribute to the clarification of the central limit question for stationary sequences. Magda Peligrad is supported in part by a Charles Phelps Taft research support grant at the Univeristy of Cincinnati and the NSA grant H98230-05-1-0066.     

Limit theorems and Markov approximations for chaotic dynamical systems

Summary We prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients. We also develop techniques of Markov approximations for dynamical systems. We apply our results to expanding interval maps, Axiom A diffeomorphisms, chaotic billiards and hyperbolic attractors.     

Maximum of partial sums and an invariance principle for a class of weak 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 and the convergence of the moments in the central limit theorem.

     

Stochastic approximation with dependent noise

In this work we derive the usual limit laws (weak and strong convergence, central limit theorem, invariance principle) for stochastic approximation with stationary noise. The idea is to introduce an artificial sequence, related to the SA scheme, but which clearly obeys the desired limit law. This sequence is subtracted from the SA scheme and the remainder, which behaves more or less deterministically, is shown to vanish using simple limit arguments.     

Central limit theorem for associated random variables

In this paper, we investigate an functional central limit theorem for a nonstatioaryd-parameter array of associated random variables applying the criterion of the tightness condition in Bickel and Wichura[1971]. Our results imply an extension to the nonstatioary case of invariance principle of Burton and Kim(1988) and analogous results for thed-dimensional associated random measure. These results are also applied to show a new functional central limit theorem for Poisson cluster random variables.     

Uniform CLT for Markov chains and its invariance principle: A martingale approach

The convergence of stochastic processes indexed by parameters which are elements of a metric space is investigated in the context of an invariance principle of the uniform central limit theorem (UCLT) for stationary Markov chains. We assume the integrability condition on metric entropy with bracketing. An eventual uniform equicontinuity result is developed which essentially gives the invariance principle of the UCLT. We translate the problem into that of a martingale difference sequence as in Gordin and Lifsic.⁽⁷⁾ Then we use the chaining argument with stratification adapted from that of Ossiander.⁽¹¹⁾ The results of this paper generalize those of Levental⁽¹⁰⁾ and Ossiander.⁽¹¹⁾     

The central limit theorem for Markov chains with normal transition operators, started at a point

The central limit theorem and the invariance principle, proved by Kipnis and Varadhan for reversible stationary ergodic Markov chains with respect to the stationary law, are established with respect to the law of the chain started at a fixed point, almost surely, under a slight reinforcing of their spectral assumption. The result is valid also for stationary ergodic chains whose transition operator is normal. Received: 28 March 2000 / Revised version: 25 July 2000 /?Published online: 15 February 2001     

Fluctuations of shapes of large areas under paths of random walks

Summary We discuss statistical properties of random walks conditioned by fixing a large area under their paths. We prove the functional central limit theorem (invariance principle) for these conditional distributions. The limiting Gaussian measure coincides with the conditional probability distribution of certain timenonhomogeneous Gaussian random process obtained by an integral transformation of the white noise. From the point of view of statistical mechanics the studied problem is the problem of describing the fluctuations of the phase boundary in the one-dimensional SOS-model.