Long-range dependence and Appell rank

Abstract

We study limit distributions of sums $S_N^{(G)} = \sum_{t=1}^N G(X_t)$ of nonlinear functions $G(x)$ in stationary variables of the form $X_t = Y_t + Z_t$, where ${Y_t}$ is a linear (moving average) sequence with long-range dependence, and ${Z_ t}$ is a (nonlinear) weakly dependent sequence. In particular, we consider the case when ${Y_ t}$ is Gaussian and either (1)${Z_t}$ is a weakly dependent multilinear form in Gaussian innovations, or (2) ${Z_t}$ is a finitely dependent functional in Gaussian innovations or (3)${Z_t}$ is weakly dependent and independent of $Y_t$ . 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 $k\geq 0$ such that $a_k = \partial^k E\{G(X_0+c)\}/\partial c^k|_{c=0 \not= 0$.