Long-range dependence and appell rank

Abstract

We study limit distributions of sums S(G)N = ∑Nt=1 G(Xt) of nonlinear functions G(x) in stationary variables of the form Xt = Yt + Zt, where {Yt} is a linear (moving average) sequence with long-range dependence, and {Zt} is a (nonlinear) weakly dependent sequence. In particular, we consider the case when {Yt} is Gaussian and either (1) {Zt} is a weakly dependent multilinear form in Gaussian innovations, or (2) [Zt] is a finitely dependent functional in Gaussian innovations or (3) {Zt} is weakly dependent and independent of {Yt}. We show in all three cases that the limit distribution of S(G)N is determined by the Appell rank of G(x), or the lowest k ≥ 0 such that ak = ∂kE{G(X0 + c)}/∂ck|c=0 ≠ 0.