Functional Limit Theorem for the Empirical Process of a Class of Bernoulli Shifts with Long Memory

Abstract

Abstract. We prove a functional central limit theorem for the empirical pro-cess of a stationary process X t = Y t + V t , where Y t is a long memory mov-ing average in i.i.d. r.v.'s  s ;s  t, and V t = V ( t ; t 1 ;:::) is a weakly de-pendent nonlinear Bernoulli shift. Conditions of weak dependence of V t arewritten in terms of L 2 norms of shift-cut di erences V ( t ;:::; t n ;0;:::;) V ( t ;:::; t n +1 ;0;:::). Examples of Bernoulli shifts are discussed. The limitempirical process is a degenerated process of the form f (x )Z , where f is themarginal p.d.f. of X 0 and Z is a standard normal r.v. The proof is based on auniform reduction principle for the empirical process. 1 Introduction Time series analysis has important statistical applications in various elds. For example,nonlinear times series are used to model crashes in nancial markets.The main object of times series analysis is the study of short-range dependent random se-quences for which the usual Donsker and the Empirical Functional Limit Theorems (EFLT)hold with appropriate modi cations. Rosenblatt (1961), in his seminal work, and after-wards, Taqqu (1975), Dobrushin and Major (1979) and other authors found that alternative1