Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples

TL;DR

This paper develops new inequalities of Rosenthal type for the moments of the maximum of partial sums in stationary processes, extending existing results to include martingales and their generalizations, with applications demonstrated.

Contribution

It introduces novel Rosenthal-type inequalities for stationary sequences, generalizing maximal inequalities for martingales using projection norms and dyadic induction.

Findings

New maximal inequality generalizing Doob's inequality

Extensions of moment inequalities to stationary processes

Applications to various stochastic process examples

Abstract

The aim of this paper is to propose new Rosenthal-type inequalities for moments of order higher than 2 of the maximum of partial sums of stationary sequences including martingales and their generalizations. As in the recent results by Peligrad et al. [Proc. Amer. Math. Soc. 135 (2007) 541-550] and Rio [J. Theoret. Probab. 22 (2009) 146-163], the estimates of the moments are expressed in terms of the norms of projections of partial sums. The proofs of the results are essentially based on a new maximal inequality generalizing the Doob maximal inequality for martingales and dyadic induction. Various applications are also provided.