Long-range dependence and Appell rank

We study limit distributions of sums S-N((G)) = Sigma(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-N((G)) is determined by the Appell rank of G(x), or the lowest k greater than or equal to 0 such that a(k) = a(k) E{G(X-0 + c)}/ac(k)\(c=0) not equal 0.