The law of the iterated logarithm for tail sums

Abstract

The main purpose of this thesis is to derive the law of the iterated logarithm for tail sums in various contexts in analysis. The various contexts are sums of Rademacher functions, general dyadic martingales, independent random variables and lacunary trigonometric series. We name the law of the iterated logarithm for tail sums as tail law of the iterated logarithm. We first establish the tail law of the iterated logarithm for sums of Rademacher functions and obtain both upper and lower bound in it. Sum of Rademacher functions is a nicely behaved dyadic martingale. With the ideas from the Rademacher case, we then establish the tail law of the iterated logarithm for general dyadic martingales. We obtain both upper and lower bound in the case of martingales. A lower bound is obtained for the law of the iterated logarithm for tail sums of bounded symmetric independent random variables. Lacunary trigonometric series exhibit many of the properties of partial sums of independent random variables. So we finally obtain a lower bound for the tail law of the iterated logarithm for lacunary trigonometric series introduced by Salem and Zygmund.