Laws of the iterated logarithm and an almost sure invariance principle for mixing <Emphasis Type="Italic">B</Emphasis>-valued random variables and autoregressive processes

Under optimal moment conditions, we prove the compact law of the iterated logarithm and the almost sure invariance principle for ψ-mixing random variables with values in type 2 Banach spaces. These results, together with the bounded law of the iterated logarithm proved earlier by author, allow us to prove the same kind of results for the Banach space valued autoregressive processes with ψ-mixing innovations. The results for autoregressive processes can be considered as asymptotic properties of the estimator of mean.     

Convergence en loi et lois du logarithme itéré pour les vecteurs gaussiens

In this paper we prove a Strassen version of the law of the iterated logarithm for some sequences of weakly asymptotically independant Banach space valued gaussian random variables which converge in distribution, and we prove that the central limit theorem implies the functional form of the law of the iterated logarithm for the partial sums of certain Banach space valued gaussian sequences.Furthermore we give conditions for the convergence in distribution of sequences of gaussian random variables and gaussian stochastic processes, and these conditions permit us to prove that our results generalize in the gaussian case all similar results known to the authors at present.     

On the law of the iterated logarithm

An analogue of the law of the iterated logarithm for Brownian motion in Banach spaces is proved where the expression √2 loglog s is replaced by a positive non-decreasing function satisfying certain conditions.     

Exponential inequalities and the law of the iterated logarithm in the unbounded forecasting game

We study the law of the iterated logarithm in the framework of game-theoretic probability of Shafer and Vovk. We investigate hedges under which a game-theoretic version of the upper bound of the law of the iterated logarithm holds without any condition on Reality’s moves in the unbounded forecasting game. We prove that in the unbounded forecasting game with an exponential hedge, Skeptic can force the upper bound of the law of the iterated logarithm without conditions on Reality’s moves. We give two examples such a hedge. For proving these results we derive exponential inequalities in the game-theoretic framework which may be of independent interest. Finally, we give related results for measure-theoretic probability which improve the results of Liu and Watbled (Stochastic Processes and their Applications 119:3101–3132, 2009).     

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.     

On Almost Sure Behavior of Sums of Independent Banach Space Valued Random Variables

Levy's strong law of large numbers is extended to the Banach space and Chover type laws of the iterated logarithm are proved for random variables which do not necessarilly belong to the domain of normal attraction of a stable law. Also characterizations of Banach spaces in which conditions on the summands imply the above strong laws are given.     

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.     

LIL behavior for weakly dependent random variables in Banach spaces

Given a sequence of identically distributed ψ-mixing random variables {X _n ; n ≧ 1} with values in a type 2 Banach space B, under certain conditions, the law of the iterated logarithm for this sequence is obtained without second moment.     

Finitary codes and the law of the iterated logarithm

Summary We investigate the validity of the law of the iterated logarithm for processes which are obtained from mixing processes via finitary codes with finite moments of order greater than one.     

Moments of the maximum of normed partial sums of random variables with multidimensional indices

Summary For a set of i.i.d. random variables indexed by the positive integer d-dimensional lattice points we give conditions for the existence of moments of the supremum of normed partial sums, thereby obtaining results related to the Kolmogorov-Marcinkiewicz strong law of large numbers and the law of the iterated logarithm.     

On the Law of the Iterated Logarithm without Assumptions of the Existence of Moments

Sufficient conditions are given for the applicability of the law of the iterated logarithm to sequences of independent random variables. These conditions do not include assumptions on the existence of any moments. Bibliography: 5 titles.     

On the functional law of the iterated logarithm for partially observed sums of random variables

We consider a partial-sum process generated by a sequence of nonidentically distributed independent random variables. Assuming that this process is available for observation along an arbitrary time sequence, we fill the gaps by linear interpolation and prove the functional law of the iterated logarithm (FLIL) for sample paths obtained in this way. Assuming that the V. A. Egorov condition holds, we show that FLIL is valid, while under other conditions sufficient for the usual law of the iterated logarithm FLIL may fail. Bibliography: 16 titles. Translated, fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 73–95.     

Bounded law of the iterated logarithm for multidimensional martingales normalized by matrices

We investigate a bounded law of the iterated logarithm for matrix-normalized weighted sums of martingale differences in R ^d . We consider the normalization of matrices inverse to the covariance matrices of these sums by square roots. This result is used for the proof of the bounded law of the iterated logarithm for martingales with arbitrary matrix normalization. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 7, pp. 1006–1008, July, 2006.     

The Strassen law of iterated logarithm for combinatorial assemblies

In [13], we investigated one-dimensional laws of iterated logarithm for additive functions defined on a class of combinatorial assemblies. In this paper, we obtain a functional law of iterated logarithm. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 211–219, April–June, 2007.     

An estimate for the convergence rate of the robbins-monro process

A bound for the convergence rate is given for the Robbins-Monro process in the form of the iterated logarithm as well as more precise conditions for the iterated logarithm law to be applicable to the procedures of stochastic approximation. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 207, pp. 109–114, 1993. Translated by the author     

A finitely additive version of kolmogorov’s law of the iterated logarithm

In their paperSome finitely additive probability (to appear in Ann. Probability), Roger A. Purves and William D. Sudderth introduced the measurable strategy idea. In this paper, we first generalize the measurable strategy idea to the more general sigma-fields of subsets ofX and prove an important theorem. Then, based on this theorem, we state and prove a finitely additive version of Kolmogorov’s law of the iterated logarithm and a finitely additive version of Hartman and Wintner’s law of the iterated logarithm in a finitely additive setting. This research was written with the partial support of the U.S. Army Grant DA-ARO-D-31-124-70-G-102.     

扩散过程在Holder范数下的局部 Strassen重对数律

在该文中,作者应用扩散过程在Holder范数下的大偏差得到了扩散过程在Holder范数下的局部Strassen重对数律. 并且还得到了重Ito积分的泛函重对数律.     

The Law of the Iterated Logarithm and Central Limit Theorem for L-Statistics

The Chung–Smirnov law of the iterated logarithm and the Finkelstein functional law of the iterated logarithm for empirical processes are used to establish new results on the central limit theorem, the law of the iterated logarithm, and the strong law of large numbers for L-statistics with certain bounded and smooth weight functions. These results are used to obtain necessary and sufficient conditions for almost sure convergence and for convergence in distribution of some well-known L-statistics and U-statistics, including Gini's mean difference statistic. A law of the logarithm for weighted sums of order statistics is also presented.     

B值强混合随机变量几何加权级数的广义重对数律

设{X,Xn;n≥0}是一取值于可分Banach空间中的同分布Φ~*-混合随机变量序列,并记其几何加权级数为ξ(β)=sum from n=0 to ∞β~nX_n,其中0<β<1.在X的二阶矩可能不存在的条件下,建立了ξ(β)的一个广义重对数律.     

On the law of the iterated logarithm for Gaussian processes

We present some optimal conditions for the compact law of the iterated logarithm of a sequence of jointly Gaussian processes in different situations. We also discuss the local law of the iterated logarithm for Gaussian processes indexed by arbitrary index sets, in particular for self-similar Gaussian processes. We apply these results to obtain the law of the iterated logarithm for compositions of Gaussian processes. Research partially supported by NSF Grant DMS-93-02583.