Exact convergence rates for the bounded law of the iterated logarithm in Hilbert space

Summary In a previous paper we obtained upper and lower class type results refining the bounded LIL for sums of iid Hilbert space valued mean zero random variables, whose covariance operators satisfy certain regularity assumptions. We now establish precise convergence rates for the bounded LIL in the non-regular case. It turns out that the almost sure behavior in this case is entirely different from the behavior in the previous situation.Supported in part by NSF Grant DMS 90-05804     

A note on the law of the iterated logarithm in Hilbert space

Summary Kuelbs (1975) established a Kolmogorov-Erdös-Petrowski type integral test for lower and upper classes in the law of the iterated logarithm for sums of i.i.d. Hilbert space valued Gaussian mean zero random variables. We show that this integral test remains valid for sums of i.i.d. pregaussian mean zero random variables satisfying an additional (very mild) assumption.     

Law of the iterated logarithm for U-statistics

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 1, pp. 97–100, January, 1989.     

Continuity properties of Hilbert space valued martingales

Necessary and sufficient conditions for Hölder continuity of Hilbert space valued martingales are given in terms of the associated quadratic variation. As an application one obtains a sufficient condition for a mild solution of a stochastic evolution equation to have a continuous version if the semigroup governing this equation is analytic. Further we derive Levy's modulus of continuity for the Hilbert space valued stochastic integral with the Wiener process as integrator and obtain a generalization of the loglog law for that integral.     

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.     

On Gaussian approximation of Hilbert space valued discrete time martingales

Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 33, No. 4, pp. 476–491, October–December, 1993.     

On the law of the iterated logarithm for trigonometric series with bounded gaps

Let (n _k ) _k??1 be an increasing sequence of positive integers. Bobkov and G?tze proved that if the distribution of $$\label{distr}\frac{\cos 2\pi n_1 x + \cdots +\cos 2 \pi n_N x}{\sqrt{N}}\qquad\quad\quad\quad (1)$$ converges to a Gaussian distribution, then the value of the variance is bounded from above by 1/2 ? lim sup k/(2n _k ). In particular it is impossible that for a sequence (n _k ) _k??1 with bounded gaps (i.e. n _k+1 ? n _k ?? c for some constant c) the distribution of (1) converges to a Gaussian distribution with variance ?? ²?=?1/2 or larger. In this paper we show that the situation is considerably different in the case of the law of the iterated logarithm. We prove the existence of an increasing sequence of positive integers satisfying $$n_{k+1} - n_k \leq 2$$ such that $$\limsup_{N \to \infty}\frac{\sum_{k=1}^N \cos 2 \pi n_k x}{\sqrt{2N \log \log N}} = +\infty \quad {a.e.}$$      

Reproducing kernel Hilbert spaces and the law of the iterated logarithm for Gaussian processes

In this paper, we find the limit set of a sequence (2 log n)^?1/2 X _n (t), n≧3) of Gaussian processes in C [0,1], where the processes X _n (t) are defined on the same probability space and have the same distribution. Our result generalizes the theorems of Oodaira and Strassen, and we also apply it to obtain limit theorems for stationary Gaussian processes, moving averages of the type \(\int\limits_0^t {f\left( {t - s} \right)dW\left( s \right)} \) , where W(s) is the standard Wiener process, and other Gaussian processes. Using certain properties of the unit ball of the reproducing kernel Hubert space of X _n (t), we derive the usual law of the iterated logarithm for Gaussian processes. The case of multidimensional time is also considered.     

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...     

Laws of the iterated logarithm for locally square integrable martingales

Three types of laws of the iterated logarithm （LIL） for locally square integrable martingales with continuous parameter are considered by a discretization approach. By this approach, a lower bound of LIL and a number of FLIL are obtained, and Chung LIL is extended.     

Exponential estimate for the law of the iterated logarithm in banach space

Obninsk Institute of Atomic Energy. Translated from Matematicheskie Zametki, Vol. 56, No. 5, pp. 98–107, November, 1994.     

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 .

     

Functional law of the iterated logarithm for fields and its applications

For a Wiener field with an arbitrary finite number of parameters, we construct the law of the iterated logarithm in the functional form. We consider the problem for random fields of a certain type to reside within curvilinear boundaries without assuming that the Cairoli—Walsh condition is satisfied. Donetsk University, Donetsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 883–894, July, 1997.     

A compact law of the iterated logarithm for UH-statistics

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