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.     

Strong approximation for sums of asymptotically negatively dependent Gaussian sequences with applications

By applying the Skorohod martingale embedding method, a strong approximation theorem for partial sums of asymptotically negatively dependent (AND) Gaussian sequences, under polynomial decay rates, is established. As applications, the law of the iterated logarithm, the Chung-type law of the iterated logarithm and the almost sure central limit theorem for AND Gaussian sequences are derived.     

A strong approximation theorem for positively dependent Gaussian sequences and its applications

Under the polynomial decay rates and finite second moment, a strong approximation theorem for partial sums of linear positive quadrant dependent Gaussian sequences is derived. As applications, the law of the iterated logarithm, the Chung-type law of the iterated logarithm and the almost sure central limit theorem are obtained with minimal conditions.     

A metric discrepancy result for the Hardy–Littlewood–Pólya sequences

The exact law of the iterated logarithm for discrepancies of the Hardy– Littlewood–Pólya sequences is proved. The exact constant in the law of the iterated logarithm is explicitly computed in the case when the Hardy–Littlewood–Pólya sequence consists of odd numbers.     

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.     

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     

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.     

A Berry-Esseen Theorem and a Law of the Iterated Logarithm for Asymptotically Negatively Associated Sequences

Negatively associated sequences have been studied extensively in recent years, Asymptotically negative association is a generalization of negative association, In this paper a Berry Esseen theorem and a law of the iterated logarithm are obtained for asymptotically negatively associated sequences.     

A functional law of the iterated logarithm for associated sequences

Recently, a functional central limit theorem and a Berry-Essen Theorem have been demonstrated for classes or associated random variables. Using these results, and similar results for multiplicative sequences, we show a functional law of the iterated logarithm for associated sequences satisfying a rate requirement.     

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.     

关于非强平稳NA列的重对数律及其应用

本文讨论了非强平稳ＮＡ（ＮｅｙａｔｉｖｅｌｙＡｓｓｏｃｉａｔｅｄ）列的若干重对数律,并用以估计线性模型中ＮＡ误差的方差估计的一类强收敛速度．     

加权U—统计量的一个重对数律

证明了关于独立同分布随机变量序列的加权Ｕ－统计量的一个重对数律,类似于献「３」证明了一个加权Ｕ－统计量的解耦不等式。     

A note on some laws of the iterated logarithm

Some function space laws of the iterated logarithm for Brownian motion with values in finite and infinite dimensional vector spaces are shown to follow from Hincin's classical law of the iterated logarithm and some martingale techniques. A law of the iterated logarithm for Brownian motion in a differentible manifold is also stated.     

Transformations in functional iterated logarithm laws and regular variation

Summary It is shown that functional iterated logarithm (log log) laws for geometric subsequences imply the corresponding laws for full sequences, and that the converse is not true. The implication is proved by simple algebraic arguments of regular variation type.Supported in part by a NATO grant for international collaboration in research     

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.     

NA随机变量序列的最大部分和不等式及有界重对数律

本文给出了NA随机变量序列关于最大部分和的概率不等式及矩不等式,并获得了NA随机变量序列的Teicher型和Egorov型有界重对数律等.     

正相协列乘积和的重对数律与强大数律

证明了强平稳正相协列乘积和的重对数律与不同分布正相协列乘积和的强大数律,指出了部分和服从强大数律但乘积和未必服从强大数律这一事实,并讨论了定理2中一个条件的必要性．     

CONVERGENCE OF THE CLIPPED SAMPLE AUTOCORRELATION AND AUTOCOVARIANCE

1.IntroductionClassicalmethodstoestimatetheautocorrelationandautocovarianceofastationarylineartimeseriesworkwellforgoodobservations,buttheyaresensitivetotheexistenceofoutliers.[1]gayeamethodofestimatinganARprocessbycountingthenumberofcertainrunsintheassociatedclippedbinaryseries.Althoughinsodoingsomeefficiencyislost,theproceduregivesgoodestimatesagainstoutliers.LetX.,n=0,11,12,'1beareallinearGaussiansequencewithzeromeanandautocorrelationspj~EXoXj/EXI,j=0,1,'landXu,n~0,11,',theclipped…     

Generalization of the law of the iterated logarithm for associated random fields

A variant of the law of the iterated logarithm for associated fields for which the indexing set for partial sums can be arbitrarily unbounded is proved. Depending on the structure of this set, an explicit value of the upper limit in the law of the iterated logarithm is given.     

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.