Long-range dependence and Appell rank

We study limit distributions of sums S-N((G)) = Sigma(t=1)(N) G(X-t) of non linear 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 dependen ce, and {Z(t)} is a (nonlinear) weakly dependent sequence. In particular, w e consider the case when {Y-t} is Gaussian and either (1) {Z(t)} is a weakl y dependent multilinear form in Gaussian innovations, or (2) {Z(t)} is a fi nitely 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 li mit distribution of S-N((G)) is determined by the Appell rank of G(x), or t he 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.