Convergence of Sums of Appell Polynomials with Infinite Variance

We investigate the convergence of distributions of partial sums of Appell polynomials \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{P}m(X_t )$$ \end{document} of a long-memory moving average process Xt with i.i.d. innovations ξs in the case where the variance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{P}_m^2 (X_t ) = \infty $$ \end{document}, and the distribution of #x03BE;0m 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 Lévy 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.