Mathematical Concepts

In this chapter we present some mathematical concepts that are useful when deriving limit theorems for long-memory processes. We start with a general description of univariate orthogonal polynomials in Sect. 3.1, with particular emphasis on Hermite polynomials in Sect. 3.1.2. Under suitable conditions, a function G can be expanded into a series $$G(x)=\sum_{j=0}^{\infty}g_j H_j(x) $$ with respect to an orthogonal basis consisting of Hermite polynomials H j (⋅) ( $j\in\mathbb{N}$ ). Such expansions are used to study sequences G(X t ) where X t ( $t\in\mathbb{Z}$ ) is a Gaussian process with long memory (see Sect. 4.2.3 ). Hermite polynomials can also be extended to the multivariate case. This is discussed in Sect. 3.2.