Some asymptotic results related to the law of iterated logarithm for Brownian motion

For 0<<1, let . The questions addressed in this paper are motivated by a result due to Strassen: almost surely, lim sup _t U ((t))=1–exp{–4(^–1)–1}. We show that Strassen's result is closely related to a large deviations principle for the family of random variablesU (t), t>0. Also, when =1,U (t)0 almost surely and we obtain some bounds on the rate of convergence. Finally, we prove an analogous limit theorem for discounted averages of the form as 0, whereD is a suitable discount function. These results also hold for symmetric random walks.     

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 in multiphase queueing systems

Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 35, No. 3, pp. 360–366, July–September, 1995.     

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.     

An iterated logarithm law related to decimal and continued fraction expansions

For an irrational number x and n ≥ 1, we denote by k _n (x) the exact number of partial quotients in the continued fraction expansion of x given by the first n decimals of x. G. Lochs proved that for almost all x, with respect to the Lebesgue measure In this paper, we prove that an iterated logarithm law for {k _n (x): n ≥ 1}, more precisely, for almost all x, for some constant σ > 0. Author’s address: Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P.R. China     

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...     

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.