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.     

On Strassen's law of the iterated logarithm

Summary Kolmogorov's law of the iterated logarithm has been sharpened by Strassen who proved a more refined theorem by using tools from functional analysis. The present paper gives a classical proof of Strassen's theorem, using a method along the lines of Kolmogorov's original approach. At the same time the result proved here is more general since a) the random variables involved need not have the same distributions, b) the condition of independence is weakened and c) instead of Kolmogorov's growth condition on the random variables, only a mild restriction on their moments of order l3 is needed.     

On the other law of the iterated logarithm

Summary A general integral test is established which refines the Jain-Pruitt Chung LIL for iid random variables. As a corollary we obtain that Chung's integral test for Brownian motion is valid for partial sums of iid random variables satisfyingEX ²1{|X|t}=O((LLt) ^–1) ast.Supported in part by NSF grant DMS 90-05804     

On the Makarov law of the iterated logarithm

We obtain considerable improvement of Makarov's estimate of the boundary behavior of a general conformal mapping from the unit disk to a simply connected domain in the complex plane. We apply the result to improve Makarov's comparison of harmonic measure with Hausdorff measure on simply connected domains.

     

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.     

On the functional law of the iterated logarithm for trimmed sums

New conditions for the functional law of the iterated logarithm for trimmed sums of symmetric independent identically distributed random variables are obtained. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 119–125. Translated by N. B. Lebedinskaya.     

On the law of the iterated logarithm for permuted lacunary sequences

It is known that for any smooth periodic function f the sequence (f(2 ^k x)) _k≥1 behaves like a sequence of i.i.d. random variables; for example, it satisfies the central limit theorem and the law of the iterated logarithm. Recently Fukuyama showed that permuting (f(2 ^k x)) _k≥1 can ruin the validity of the law of the iterated logarithm, a very surprising result. In this paper we present an optimal condition on (n _k ) _k≥1, formulated in terms of the number of solutions of certain Diophantine equations, which ensures the validity of the law of the iterated logarithm for any permutation of the sequence (f(n _k x)) _k≥1. A similar result is proved for the discrepancy of the sequence ({n _k x}) _k≥1, where {·} denotes the fractional part.     

On a law of the iterated logarithm for sums mod 1 with application to Benford's law

Summary Let Z _n be the sum mod 1 of n i.i.d.r.v. and let 1_[0,x](·) be the indicator function of the interval [0, x]. Then the sequence 1_[0,x](Z _n ) does not converge for any x. But if arithmetic means are applied then under suitable suppositions convergence with probability one is obtained for all x as well-known. In the present paper the rate of this convergence is shown to be of order n ^-1/2 (loglogn)^1/2 by using estimates of the remainder term in the CLT for m-dependent r.v.     

A general law of iterated logarithm

Probability Theory and Related Fields - We show that the law of iterated logarithm holds for a sequence of independent random variables (X n ) provided (i) $$\sum\limits_{n = 1}^\infty {(s_n^2...