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2015 | OriginalPaper | Chapter

2. Geometry of Second-Order Random Processes

Authors : Anders Lindquist, Giorgio Picci

Published in: Linear Stochastic Systems

Publisher: Springer Berlin Heidelberg

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Abstract

In this book, modeling and estimation problems of random processes are treated in a unified geometric framework. For this, we need some basic facts about the Hilbert space theory of stochastic vector processes that have finite second order moments and are stationary in the wide sense. Such a process \(\{y(t)\}_{t\in \mathbb{Z}}\) is a collection of real or complex-valued random variables \(y_{k}(t),k = 1,2,\ldots,m,t \in \mathbb{Z}\), which generate a Hilbert space H with inner product
$$\displaystyle{\langle \xi,\eta \rangle =\mathop{ \mathrm{E}}\nolimits \{\xi \bar{\eta }\},}$$
where bar denotes conjugation. This Hilbert space is endowed with a shift, i.e., a unitary operator \(\mathcal{U}:\, \mathbf{H} \rightarrow \mathbf{H}\) with the property that
$$\displaystyle{y_{k}(t + 1) = \mathcal{U}y_{k}(t),\quad k = 1,2,\ldots,t \in \mathbb{Z}.}$$
In this chapter we introduce some basic geometric facts for such Hilbert spaces. Although we shall assume that the reader has some knowledge of elementary Hilbert space theory, for the benefit of the reader, some relevant facts are collected in Appendix B.2.

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next chapter Spectral Representation of Stationary Processes
Footnotes
1
Measure-theoretic probability theory (and concepts such as σ-algebra) will be used very seldom in this book, and most of the book can be read without deeper knowledge of it.
 
2
For the same reason the squared norm of the difference \(\|x_{k} -\hat{ x}_{k}\|^{2}\) is a variance.
 
3
See Appendix B.1 in the appendix for the definition of the Moore-Penrose pseudoinverse.
 
4
In the Russian literature a compact operator is called completely continuous.
 
5
Provided the Hilbert space is separable, which is a standard assumption in this book.
 
6
A statistic is sufficient with respect to a statistical model if no other statistic calculated from the same sample provides additional information about the model parameters [83, 94, 134, 249].
 
7
Here, to conform with the standard terminology in the literature, we should add the attribute “wide sense”, but since we shall never talk about “strict sense” properties in this book, we shall refrain from doing so.
 
8
We denote by \(\mathbb{R}_{{\ast}}^{m\times n}\) the space of full rank m × n real matrices.
 
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Metadata
Title
Geometry of Second-Order Random Processes
Authors
Anders Lindquist
Giorgio Picci
Copyright Year
2015
Publisher
Springer Berlin Heidelberg
Book
Linear Stochastic Systems

Print ISBN: 978-3-662-45749-8

Electronic ISBN: 978-3-662-45750-4

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-662-45750-4_2