Maximal Inequalities and the Law of the Iterated Logarithm

Abstract

A supermartingale maximal inequality is derived. A maximal inequality is derived for arbitrary random variables $\{S_n, n \geqq 1\}$ (let $S_0 = 0$) satisfying $E\exp\lbrack u(S_{m + n} - S_m) \rbrack \leqq \exp(Knu^2)$ for all real $u$, all integers $m \geqq 0$ and $n \geqq 1$, and some constant $K$. These two maximal inequalities are used to derive upper half laws of the iterated logarithm for supermartingales, multiplicative random variables, and random variables not satisfying particular dependence assumptions.