Functional limit theorem for the empirical process of a class of Bernoulli shifts with long memory

Abstract. We prove a functional central limit theorem for the empirical pro-cess of a stationary process Xt = Yt + Vt, where Yt is a long memory mov-ing average in i.i.d. r.v.’s ζs, s ≤ t, and Vt = V (ζt, ζt−1,...) is a weakly de-pendent nonlinear Bernoulli shift. Conditions of weak dependence of Vt are written in terms of L2−norms of shift-cut differences V (ζt,..., ζt−n, 0,..., ) − V (ζt,..., ζt−n+1, 0,...). Examples of Bernoulli shifts are discussed. The limit empirical process is a degenerated process of the form f(x)Z, where f is the marginal p.d.f. of X0 and Z is a standard normal r.v. The proof is based on a uniform reduction principle for the empirical process.