Empirical processes of long-memory sequences

Abstract

Asymptotic expansions of long-memory sequences indexed by piecewise differentiable functionals are investigated, and upper bounds of outer expectations of these functionals are given. These results differ strikingly from the classical theories of empirical processes of independent random variables. Our results go beyond earlier ones by allowing wider classes of function as well as by presenting sharper bounds, and thus provide a more versatile approach for related statistical inferences. A complete characterization of empirical processes for the class of indicator functions is presented, and an application to $M$-estimation is discussed.