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Erschienen in:
Definition of Long Memory Kapitel lesen Erstes Kapitel lesen

2013 | OriginalPaper | Buchkapitel

3. Mathematical Concepts

verfasst von : Jan Beran, Yuanhua Feng, Sucharita Ghosh, Rafal Kulik

Erschienen in: Long-Memory Processes

Verlag: Springer Berlin Heidelberg

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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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Vorheriges Kapitel Origins and Generation of Long Memory
Nächstes Kapitel Limit Theorems
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Metadaten
Titel
Mathematical Concepts
verfasst von
Jan Beran
Yuanhua Feng
Sucharita Ghosh
Rafal Kulik
Copyright-Jahr
2013
Verlag
Springer Berlin Heidelberg
Buch
Long-Memory Processes

Print ISBN: 978-3-642-35511-0

Electronic ISBN: 978-3-642-35512-7

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-3-642-35512-7_3

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