Long-range dependence and Appell rank. (English) Zbl 1130.60306

Summary: 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 \(a_k = \partial^k E\{G(X_0+c)\}/\partial c^k|_{c=0} \not= 0\).


60F05 Central limit and other weak theorems
60G15 Gaussian processes
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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