Bounded laws of the iterated logarithm for quadratic forms in Gaussian random variables

In this paper we prove bounded laws of the iterated logarithm for Gaussian quadratic forms. The underlying sequence of Gaussian variables is assumed to satisfy quite general conditions on its covariance structure. Basic tools are maximal inequalities of exponential type for sums of dependent random variables which may be of own interest. Several examples illustrate the sharpness of the results. In a particular section the bounded law of the iterated logarithm is shown for quadratic variation of Brownian motion.     

B值独立随机元重对数律收敛速度的一般形式

本文讨论了Ｂ值独立同分布（ｉｉｄ）随机元重对数律收敛速度的一般形式,使得Ｄａｖｉｓ＾「１」及Ｇｕｔ＾「２,３」中的一些结果成为特款,同时减弱了Ｄａｖｉｓ结果中的矩条件,并且得到了Ｂ值ｉｉｄ随机元满足有界重对数律的一个充分性条件。作为应用,我们给出了随机足标和的相应结果。     

On the law of the iterated logarithm for sequences of dependent random variables

Sufficient conditions for the applicability of the law of the iterated logarithm to sequences of dependent random variables are obtained. As a corollary, a theorem on the law of the iterated logarithm for a sequence of m-orthogonal random variables is proved.     

在通常矩条件下B—值随机元配重迭对数律不成立的一个反例

本文讨论B－值随机元配重迭对数律,给出了在通常矩条件下B－值随机元配重迭对数律不成立的一个反例,文中还指出,当随机元范数矩的条件弱于二阶矩时,配重迭对数律不成立。     

线性过程的强逼近和重对数律

本文讨论由独立同分布随机变量列产生的线性过程的泛函型重对数律和强逼近, 同时又给出由NA随机变量列产生的线性过程的重对数律.     

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.     

The law of the iterated logarithm for positively dependent random variables

The Levy's type maximal inequality is a key to establish the law of the iterated logarithm for associated random variables. Unfortunately, this type inequality cannot be obtained for a generalization of association, i.e., linear positive quadrant dependence, because of their special dependence structure. The purpose of this paper is to provide a different approach to obtain a law of the iterated logarithm for a sequence of linear positive quadrant dependent random variables.     

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.     

Law of iterated logarithm for NA sequences with non-identical distributions

Based on a law of the iterated logarithm for independent random variables sequences, an iterated logarithm theorem for NA sequences with non-identical distributions is obtained. The proof is based on a Kolmogrov-type exponential inequality.     

A LIL for independent non-identically distributed random variables in Banach space and its applications

In this paper,we prove a general law of the iterated logarithm (LIL) for independent non-identically distributed B-valued random variables.As an interesting application,we obtain the law of the iterated logarithm for the empirical covariance of Hilbertian autoregressive processes.     

On the Upper Limit of a Random Sequence and the Law of the Iterated Logarithm

We obtain some results concerning the upper limit of a random sequence and the law of the iterated logarithm for sums of independent random variables.     

The law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges

We prove both the validity and the sharpness of the law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges.     

相伴的高斯随机变量序列的一个强不变原理

本文利用鞅的Skorohod表示, 在序列是高斯的且序列的协方差系数以幂指数速度递减的条件下,证明了相伴高斯随机变量序列的一个强不变原理\bd 作为推论得到了相伴高斯随机变量序列的重对数律和钟重对数律     

One moment estimate for the supremum of normalized sums in the law of the iterated logarithm

For a sequence of independent random elements in a Banach space, we obtain an upper bound for moments of the supremum of normalized sums in the law of the iterated logarithm by using an estimate for moments in the law of large numbers. An example of their application to the law of the iterated logarithm in Banach lattices is given. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 653–665, May, 2006.     

On the law of the iterated logarithm for increments of sums of independent random variables

We obtain the law of the iterated logarithm for increments of sums of independent random variables. Our results generalize the Kolmogorov theorem and the Hartman—Wintner theorem on the law of the iterated logarithm. Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 174–186.     

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.     

Generalized Law of the Iterated Logarithm and Its Convergence Rate

Abstract

This article considers the partial sums from a sequence of independent and identically distributed random variables. It is well-known that the Hartman-Wintner law of the iterated logarithm holds if and only if the second moment exists. This article studies the generalized law of the iterated logarithm for the partial sums when they are normalized by a sequence of constants that are regularly varying with index 1/2. As a result, two equivalent conditions for the law are obtained.     

The small-time Chung-Wichura law for L��vy processes with non-vanishing Brownian component

We give a ??small time?? functional version of Chung??s ??other?? law of the iterated logarithm for Lévy processes with non-vanishing Brownian component. This is an analogue of the ??other?? law of the iterated logarithm at ??large times?? for Lévy processes and random walks with finite variance, as extended to a functional version by Wichura. As one of many possible applications, we mention a functional law for a two-sided passage time process.     

不完全信息随机截尾模型的MLE 的Chung 重对数律

在一定条件下,证明不完全信息随机截尾模型的MLE 满足 Chung重对数律. 作为其推论得到:不完全信息随机截尾试验下,指数分布和Weibull 分布的MLE 满足Chung 重对数律.     

On the Law of Iterated Logarithm for the Maximum Scheme in Ideal Banach Spaces

Ukrainian Mathematical Journal - We obtain asymptotic estimates in the law of iterated logarithm for the extreme values of a sequence of independent random variables in Banach spaces.