Noncentral Limit Theorems and Appell Polynomials

Let Xᵢ be a stationary moving average with long-range dependence. Suppose EXᵢ = 0 and EX²ⁿᵢ < ∞ for some n ≥ 2. When the Xᵢ are Gaussian, then the Hermite polynomials play a fundamental role in the study of noncentral limit theorems for functions of Xᵢ. When the Xᵢ are not Gaussian, the relevant polynomials are Appell polynomials. They satisfy a multinomial-type expansion that can be used to establish noncentral limit theorems.