A Maximal Inequality and Dependent Strong Laws

Abstract

This paper contains a general dependent extension of Doob's inequality for martingales, $E(\max_{i\leqq n} S_i^2) \leqq 4ES_n^2$. This inequality is then used to extend the martingale convergence theorem for $L_2$ bounded variables, and to prove strong laws under dependent assumptions. Strong and $\varphi$-mixing variables are shown to satisfy the conditions of these theorems and hence strong laws are proved as well for these.