A study on moment inequalities under a weak dependence

Abstract

We establish Roussas–Ioannides-type inequalities [Roussas, G. G., & Ioannides, D. A. (1987). Moment inequalities for mixing sequences of random variables. Stochastic Analysis and Applications, 5, 61–120] under general ψ-weak dependence proposed by Doukhan and Louhichi [Doukhan, P., & Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities. Stochastic Processes and their Applications, 84, 313–342], which unifies weak dependence such as mixing, association, Gaussian sequences and Bernoulli shifts. Simple applications of the inequalities extend many important moment inequalities available in the literature for mixing sequences to those for ψ-weakly dependent sequences. As an illustration, the established inequalities are applied to extend the result for moment bound of partial sum under strong mixing by Cox and Kim [Cox, D. D., & Kim, T. Y. (1995). Moment bounds for mixing random variables useful in nonparametric function estimation. Stochastic Process and their Applications, 56, 151–158] to the class of ψ-weakly dependent processes.