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.
Citation
D. L. McLeish. "A Maximal Inequality and Dependent Strong Laws." Ann. Probab. 3 (5) 829 - 839, October, 1975. https://doi.org/10.1214/aop/1176996269
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