Metric discrepancy results for alternating geometric progressions

The law of the iterated logarithm for discrepancies of {θ ^k x} is proved for θ < ?1. When θ is not a power root of rational number, the limsup equals to 1/2. When θ is an odd degree power root of rational number, the limsup constants for ordinary discrepancy and star discrepancy are not identical.     

The law of the iterated logarithm for wavelet series

We prove a law of iterated logarlthm for wavelet series:lin sup/n→∞f^a n/√S^2n(f)loglogSn(f)≤C holds almost everywhere on {x∈R^n; S(f) =∞}.     

The law of iterated logarithm for logarithmic combinatorial assemblies

The strong convergence of dependent random variables is analyzed and the law of iterated logarithm for real additive functions defined on the class of combinatorial assemblies is obtained. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 532–547, October–December, 2006.     

The functional law of the iterated logarithm for truncated sums

The functional law of the iterated logarithm (FLIL) is obtained for truncated sums $S_n = \sum _{j = l}^n X_j I\{ X_j^{\text{2}} \leqslant b_n \} $ of independent symmetric random variables X_j, 1<-j≤n, b_n≤∞. Considering the random normalization $T_n^{1/{\text{2}}} = \left( {\sum\limits_{j = 1}^n {X_j^{\text{2}} } I\{ X_j^{\text{2}} \leqslant b_n \} } \right)^{1/{\text{2}}} ,$ we obtain an upper estimate in the FLIL, using only the condition that T_n→∞ almost surely. These results are useful in studying trimmed sums. Bibliography: 9 titles.     

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.     

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.     

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...     

A law of the iterated logarithm for arithmetic functions

Let be a sequence of centered iid random variables. Let be a strongly additive arithmetic function such that and put . If and satisfies a Lindeberg-type condition, we prove the following law of the iterated logarithm:

We also prove the validity of the corresponding weighted strong law of large numbers in .

     

The law of the iterated logarithm for normalized empirical distribution function

This paper deals with the law of the iterated logarithm and its analogues for sup , where the sup is taken on an interval of the form (a _n ,b _n ),(0a _n <b _n 1). Under certain conditions on a _n and b _n the corresponding lim sup results will be proved.     

A compact law of the iterated logarithm for UH-statistics

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 5, pp. 672–676, May, 1989.