On Probability and Moment Inequalities for Supermartingales and Martingales

Abstract

Abstract. The probability inequality for max k≤n S k , where S k = k j=1 X j , is proved under the assumption that the sequence S k , k = 1, . . . , n is a supermartingale. This inequality is stated in terms of probabilities P(X j > y) and conditional variances of random variables X j , j = 1, . . . , n. As a simple consequence the well-known moment inequality due to Burkholder is deduced. Numerical bounds are given for constants in Burkholder's inequality.

Hoeffding’s inequality for supermartingales

“…We notice that inequality (23) improves an inequality of Fuk ([15], (3)). It also extends and improves Nagaev's inequality ( [22], (1.55)) which was obtained in the independent case.…”

On Probability and Moment Inequalities for Supermartingales and Martingales

“…Since inequalities (2) and (4) considerably differ in form, it is not a simple task to compare them. It is demonstrated in [25] that (4) does not follow from (2). Similar arguments show that it is impossible to deduce the Burkholder inequality (8), which is a generalization of the Rosenthal inequality [26], by means of that by Haeusler.…”

“…The proof of Theorem 1 contained in the primary version of this paper is published in [25]. In the suggested version the latter is modified, namely, two assertions from the previous text are released, which makes the presentation more transparent.…”

Probability and moment inequalities under dependence

“…Nagaev (2001) considered uniformly mixing processes, a very strong type of dependence condition. In Nagaev (2007) he considered martingales. Bertail and Clémençon (2010) dealt with functionals of positive recurrent geometrically ergodic Markov chains; see also Rio (2000).…”