Limsup results and LIL for finite dimensional Gaussian random fields

We establish strong limsup theorems related to the law of the iterated logarithm (LIL) for finite dimensional Gaussian random fields by using the second Borel-Cantelli lemma. Supported by KRF-2003-C00098.     

关于广义几何分布的强偏差定理

利用对数似然比作为一类整值随机变量序列相对于独立随机变量序列的偏差度量,在限定对数似然比的给定样本空间的子集上,建立并证明一类整值随机变量序列的强偏差定理,作为推论得到了此类分布的独立随机变量序列的若干强大数定律.     

Laws of iterated logarithm for transient random walks in random environments

We consider laws of iterated logarithm for one-dimensional transient random walks in random environments. A quenched law of iterated logarithm is presented for transient random walks in general ergodic random environments, including independent identically distributed environments and uniformly ergodic environments.     

A law of the iterated logarithm for Markov chains on ℕ0 associated with orthogonal polynomials

We derive laws of the iterated logarithm for Markov chains on the nonnegative integers whose transition probabilities are associated with a sequence of orthogonal polynomials. These laws can be applied to a large class of birth and death random walks and random walks on polynomial hypergroups. In particular, the results of our paper lead immediately to a law of the iterated logarithm for the growth of the distance of isotropic random walks on infinite distance-transitive graphs as well as on certain finitely generated semigroups from their starting points.     

On the local time of random walk on the 2-dimensional comb

We study the path behaviour of general random walks, and that of their local times, on the 2-dimensional comb lattice C² that is obtained from Z² by removing all horizontal edges off the x-axis. We prove strong approximation results for such random walks and also for their local times. Concentrating mainly on the latter, we establish strong and weak limit theorems, including Strassen-type laws of the iterated logarithm, Hirsch-type laws, and weak convergence results in terms of functional convergence in distribution.     

The Law of the Iterated Logarithm for Weakly Dependent Hilbert Space Valued Random Variables

We consider the bounded and compact laws of the iterated logarithm for weakly dependent Hilbert space valued random variables. Under optimal moment conditions, we prove the bounded and compact laws of the iterated logarithm for sequences of identically distributed Hilbert space valued random variables satisfying the uniform strong mixing condition.     

Limit theorems for random transformations and processes in random environments

I derive general relativized central limit theorems and laws of iterated logarithm for random transformations both via certain mixing assumptions and via the martingale differences approach. The results are applied to Markov chains in random environments, random subshifts of finite type, and random expanding in average transformations where I show that the conditions of the general theorems are satisfied and so the corresponding (fiberwise) central limit theorems and laws of iterated logarithm hold true in these cases. I consider also a continuous time version of such limit theorems for random suspensions which are continuous time random dynamical systems.

     

Integro-Local Limit Theorems for Compound Renewal Processes under Cramér’S Condition. I

We obtain integro-local limit theorems in the phase space for compound renewal processes under Cramér’s moment condition. These theorems apply in a domain analogous to Cramér’s zone of deviations for random walks. It includes the zone of normal and moderately large deviations. Under the same conditions we establish some integro-local theorems for finite-dimensional distributions of compound renewal processes.     

Strassen-type Laws of Iterated Logarithm for a Fractional Brownian Sheet

Abstract

We study Strassen-type laws of iterated logarithm for a fractional Brownian sheet including that for small time, which imply most of the former laws of the iterated logarithm and Strassen's laws for one-parameter and two-parameter Wiener processes.     

Moderate deviations and functional limits for random processes with stationary and independent increments

We first give a functional moderate deviation principle for random processes with stationary and independent increments under the Ledoux's condition. Then we apply the result to the functional limits for increments of the processes and obtain some Csorgo-Revesz type functional laws of the iterated logarithm.     

线性模型的最小二乘估计的中偏差及其重对数律

本文研究了线性模型的最小二乘估计的中偏差.通过估计Laplace渐近积分,得到了随机误差为取值于Rd的相互独立同分布随机变量情形下的中偏差与重对数律的结果.     

Law of the iterated logarithm for the periodogram

We consider the almost sure asymptotic behavior of the periodogram of stationary and ergodic sequences. Under mild conditions we establish that the limsup of the periodogram properly normalized identifies almost surely the spectral density function associated with the stationary process. Results for a specified frequency are also given. Our results also lead to the law of the iterated logarithm for the real and imaginary parts of the discrete Fourier transform. The proofs rely on martingale approximations combined with results from harmonic analysis and techniques from ergodic theory. Several applications to linear processes and their functionals, iterated random functions, mixing structures and Markov chains are also presented.     

Towards a universal self-normalized moderate deviation

This paper is an attempt to establish a universal moderate deviation for self-normalized sums of independent and identically distributed random variables without any moment condition. The exponent term in the moderate deviation is specified when the distribution is in the centered Feller class. An application to the law of the iterated logarithm is given.

     

Limsup results and a uniform LIL for partial sums of an LNQD sequence

In this paper, we establish some limsup results and a generalized uniform law of the iterated logarithm (LIL) for the increments of partial sums of a strictly stationary and linearly negative quadrant dependent (LNQD) sequence of random variables whose covariance coefficients decay polynomially.     

On the History of St. Petersburg School of Probability and Mathematical Statistics: II. Random Processes and Dependent Variables

This is the second paper in a series of reviews devoted to the scientific achievements of the Leningrad and St. Petersburg school of probability and mathematical statistics from 1947 to 2017. This paper is devoted to the works on limit theorems for dependent variables (in particular, Markov chains, sequences with mixing properties, and sequences admitting a martingale approximation) and to various aspects of the theory of random processes. We pay particular attention to Gaussian processes, including isoperimetric inequalities, estimates of the probabilities of small deviations in various norms, and the functional law of the iterated logarithm. We present a brief review and bibliography of the works on approximation of random fields with a parameter of growing dimension and probabilistic models of systems of sticky inelastic particles (including laws of large numbers and estimates for the probabilities of large deviations).     

Laws of the iterated logarithm for nonparametric density estimators

Summary We establish a law of the iterated logarithm for a triangular array of independent random variables, and apply it to obtain laws for a large class of nonparametric density estimators. We consider the case of Rosenblatt-Parzen kernel estimators, trigonometric series estimators and orthogonal polynomial estimators in detail, and point out that our technique has wider application.     

布朗运动在（r，p）－容度意义下的下极限性质

本文证明了布朗运动在（ｒ,ｐ）－容度意义下的一些基本下极限性质．     

Moderate deviations for Markovian occupation times

In this paper, we establish the moderate deviations for occupation times of Markov processes under the conditions given in Darling–Kac (1957. Trans. Amer. Math. Soc. 84, 444–458). When applied to the law of the iterated logarithm, our results generalize those obtained in Marcus–Rosen (1994a. Ann. Probab. 22, 626–658; 1994b. Ann. Inst. Henri Poincaré Probab. Statist. 30, 467–499) for Levy processes and random walks, and those obtained recently by the author (1999. Ann. Probab. 27, 1324–1346) for Harris recurrent Markov chains.     

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.     

Asymptotic behaviors for functionals of random dynamical systems

In this article, we consider asymptotic behaviors for functionals of dynamical systems with small random perturbations. First, we present a deviation inequality for Gaussian approximation of dynamical systems with small random perturbations under Hölder norms and establish the moderate deviation principle and the central limit theorem for the dynamical systems by the deviation inequality. Then, applying these results to forward-backward stochastic differential equations and diffusions in small time intervals, combining the delta method in large deviations, we give a moderate deviation principle for solutions of forward-backward stochastic differential equations with small random perturbations, and obtain the central limit theorem, the moderate deviation principle and the iterated logarithm law for functionals of diffusions in small time intervals.