Noncentral Limit Theorems and Appell Polynomials

Abstract

Let $X_i$ be a stationary moving average with long-range dependence. Suppose $EX_i = 0$ and $EX^{2n}_i < \infty$ for some $n \geq 2$. When the $X_i$ are Gaussian, then the Hermite polynomials play a fundamental role in the study of noncentral limit theorems for functions of $X_i$. When the $X_i$ are not Gaussian, the relevant polynomials are Appell polynomials. They satisfy a multinomial-type expansion that can be used to establish noncentral limit theorems.