Moderate Deviations for Linear Processes Generated by Martingale-Like Random Variables

Abstract

In this paper we study the moderate deviation principle for linear statistics of the type Sn=∑i∈Zcn,iξi, where cn,i are real numbers, and the variables ξi are in turn stationary martingale differences or dependent sequences satisfying projective criteria. As an application, we obtain the moderate deviation principle and its functional form for sums of a class of linear processes with dependent innovations that might exhibit long memory. A new notion of equivalence of the coefficients allows us to study the difficult case where the variance of Sn behaves slower than n. The main tools are: a new type of martingale approximations and moment and maximal inequalities that are important in themselves.