Maximal Inequalities for Dependent Random Variables

Abstract

Maximal inequalities play a crucial role in many probabilistic limit theorem; for instance, the law of large numbers, the law of the iterated logarithm, the martingale limit theorem and the central limit theorem. Let X1, X2, … be random variables with partial sums S k = X1 + ⋯ + X k . Then a maximal inequality gives conditions ensuring that the maximal partial sum M n = max1 ≤ i ≤ n S i is of the same order as the last sum S n . In the literature there exist large number of maximal inequalities if X1, X2, … are independent but much fewer for dependent random variables. In this paper, I shall focus on random variables X1, X2, … having some weak dependence properties; such as positive and negative In-correlation, mixing conditions and weak martingale conditions.