by **
Natalia Gorn**

**Preprint series:**
99-33 , Reports on Analysis

**Abstract:** Abstract:

The law of the iterated logarithm is one of the most important

limit theorems in the probability theory. In the dissertation

is the functional form of this law considered. We present a review

of the basic known results (Chapter 1) and also the following our new

results.

In Chapter 2 we consider the space of all continiuos functions endowed

with some lower semi-continuous norm and prove a simple necessary

and sufficient condition which should be imposed on this norm so that

the functional law of the iterated logarithm for the increments of

the Wiener process (generalized functional Revesz law) is valid

in the corresponding Banach space.

In Chapter 3 the cluster sets for partially observed processes are

studied. We describe the cluster set for partial sum process generated

by independent non-identically distributed random values satisfiyng some

condition of Lindeberg type (the Egorov condition). This condition is

also proved to be quite optimal.

In Chapter 4 we solve the problem of the exact convergence rate in

Chung's functional law for the so-called 'slowest points'. Our description

is closely related to the study of an interesting functional emerging from

the large deviation problem for the Wiener process in a strip.

**Keywords:** *functional law of the iterated logarithm, Wiener process, Revesz law, Chung's law*

**Notes:** Dissertation 15.04.99 (W. Linde)

**Upload:** 1999-06-09

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