Probability and moment inequalities for sums of weakly dependent random variables, with applications

Doukhan and Louhichi [P. Doukhan, S. Louhichi, A new weak dependence condition and application to moment inequalities, Stochastic Process. Appl. 84 (1999) 313–342] introduced a new concept of weak dependence which is more general than mixing. Such conditions are particularly well suited for deriving estimates for the cumulants of sums of random variables. We employ such cumulant estimates to derive inequalities of Bernstein and Rosenthal type which both improve on previous results. Furthermore, we consider several classes of processes and show that they fulfill appropriate weak dependence conditions. We also sketch applications of our inequalities in probability and statistics.