Maximal inequalities and an application under a weak dependence

We establish maximal moment inequalities of partial sums under-weak ψ dependence, which has been proposed by Doukhan and Louhichi P. Doukhan and S. Louhichi, A new weak dependence condition and application to moment inequality, Stochastic Process. Appl. 84 (1999), 313–342., to unify weak dependence such as mixing, association, Gaussian sequences and Bernoulli shifts. As an application of maximal moment inequalities, a functional central limit theorem is developed for linear processes with ψ weakly dependent innovations.