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.
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