Convergence of Sums of Appell Polynomials with Infinite Variance

Abstract

We investigate the convergence of distributions of partial sums of Appell polynomials \(\mathcal{P}m(X_t )\) of a long-memory moving average process X t with i.i.d. innovations ξs in the case where the variance \(\mathcal{P}_m^2 (X_t ) = \infty \), and the distribution of #x03BE; 0 m belongs to the domain of attraction of an α-stable law with 1<α< 2. We prove that the limit distribution of partial sums of Appell polynomials is either an α-stable Levy process, or an mth order Hermite process, or the sum of two mutually independent processes depending on the values of α, m, and d, where 0<d<1/2 is the long-memory parameter of Xt.