Abstract
We establish some maximal inequalities for demimartingales which generalize the result of Wang (2004). The maximal inequality for demimartingales is used as a key inequality to establish other results including Doob's type maximal inequality, strong law of large numbers, strong growth rate, and integrability of supremum for demimartingales, which generalize and improve partial results of Christofides (2000) and Prakasa Rao (2007).
Highlights
1.1 for all coordinatewise nondecreasing functions f such that the expectation is defined
It is seen that the partial sum of a sequence of mean zero strongly positive dependent random variables is a demimartingale by the inequality 3 in Zheng 7, that is, for all n ≥ 1, E f S1, . . . , Sn Sn 1 − Sn E f X1, X1 X2, . . . , X1 X2 · · · Xn Xn 1 ≥ 0 1.5 for all coordinatewise nondecreasing functions f such that the expectation is defined
In Theorem 2.5, if we assume that g x is a nonnegative and nondecreasing convex function on R with g 0 0, the condition “E g Sk−1 p ≤ E g Sk p for each k ≥ 1” is satisfied
Summary
It is seen that the partial sum of a sequence of mean zero strongly positive dependent random variables is a demimartingale by the inequality 3 in Zheng 7 , that is, for all n ≥ 1, E f S1, . Lemma 1.9 see Christofides 3, Corollary 2.4, Theorem 2.1 . 1.14 ii Let {Sn, n ≥ 1} be a demisubmartingale and {ck, k ≥ 1} a nonincreasing sequence of positive numbers.
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