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

### MSC:

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

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