Mathematical Concepts

Jan Beran, Yuanhua Feng, Sucharita Ghosh and Rafal Kulik
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Jan Beran: University of Konstanz, Dept. of Mathematics and Statistics
Yuanhua Feng: University of Paderborn, Faculty of Business Administration and Economics
Sucharita Ghosh: Swiss Federal Research Institute WSL
Rafal Kulik: University of Ottawa, Dept. of Mathematics and Statistics

Chapter Chapter 3 in Long-Memory Processes, 2013, pp 107-208 from Springer

Abstract: 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.

Keywords: Derive Limit Theorems; Linear Fractional Stable Motion (LFSM); Wick Product; Gaussian Random Measure; Diagram Formula (search for similar items in EconPapers)
Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-35512-7_3

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DOI: 10.1007/978-3-642-35512-7_3

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