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

We prove a functional central limit theorem for the empirical process of a stationary process X_t=Y_t+V_t, where Y_t is a long memory moving average in i.i.d. r.v.s _s, s t, and V_t=V (_t, _t-1,...) is a weakly dependent nonlinear Bernoulli shift. Conditions of weak dependence of V_t are written in terms of L²-norms of shift-cut differences V (_t, _t-n, 0,...,) – V(_t,...,_t-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.