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Mathematical Concepts

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Long-Memory Processes

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Abstract

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.

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Authors and Affiliations

  1. Dept. of Mathematics and Statistics, University of Konstanz, Konstanz, Germany

    Jan Beran

  2. Faculty of Business Administration and Economics, University of Paderborn, Paderborn, Germany

    Yuanhua Feng

  3. Swiss Federal Research Institute WSL, Birmensdorf, Switzerland

    Sucharita Ghosh

  4. Dept. of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada

    Rafal Kulik

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Beran, J., Feng, Y., Ghosh, S., Kulik, R. (2013). Mathematical Concepts. In: Long-Memory Processes. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35512-7_3

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