A Maximal Inequality and Dependent Strong Laws

This paper contains a general dependent extension of Doob's inequality for martingales, E (₈ ₍ Sᵢ²) 4ESₙ². This inequality is then used to extend the martingale convergence theorem for L₂ bounded variables, and to prove strong laws under dependent assumptions. Strong and -mixing variables are shown to satisfy the conditions of these theorems and hence strong laws are proved as well for these.