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2018 | OriginalPaper | Chapter

4. Stationarity

Author : Paul Doukhan

Published in: Stochastic Models for Time Series

Publisher: Springer International Publishing

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Abstract

Some bases for the theory of time series are given below. The chapter deals with the widely used assumption of stationarity which yields a simpler theory for time series. This concept is widely considered in Rosenblatt (Stationary processes and random fields, Birkhäuser, Boston, 1985) and in Brockwell and Davis (Time series: theory and methods, 2nd edn., Springer series in statistics, Springer, New York, 1991). The latter reference is more involved with linear time series.

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previous chapter Estimation Concepts
next chapter Gaussian Chaos
Footnotes
1
Nile data may be found on: https:datamarket.com/dataset22w8mean-annual-nile-flow-1871-1970.
 
2
Use a triangular scheme, by successive extraction of convergent subsequences. Choose a denumerable and dense sequence \((\lambda _k)_k\) in \([-\pi ,\pi ]\).
Here \(\phi _{k+1}(n)\) will denote a subsequence of \(\phi _k(n)\) such that \(G_{\phi _{k+1}(n)}(\lambda _{k+1})\) converges as \(n\rightarrow \infty \).
Setting \(G_{\phi (n)}=G_{\phi _{n}}(n)\) allows to end the proof.
 
3
Recall that monotonic functions admit limits on the left and on the right at each point, the non-empty open intervals \((f(x-), f(x+))\) are disjoint in \(\mathbb {R}\). Choose a rational number in each of them to conclude.
 
4
This process is the centred Gaussian process indexed on \(\mathbb {R}^+\) with the covariance \(\mathbb {E}W(s)W(t)=s\wedge t\).
 
5
Landau notations:
\(v_n=o(u_n)\) if \(\lim _n(v_n/u_n)=0\) in case \(u_n\ne 0\) for all n,
\(v_n=\mathcal{O}(u_n)\) if there exists \(C>0\) such that \(|v_n|\le C|u_n|\) for all n.
 
Metadata
Title
Stationarity
Author
Paul Doukhan
Copyright Year
2018
Book
Stochastic Models for Time Series

Print ISBN: 978-3-319-76937-0

Electronic ISBN: 978-3-319-76938-7

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-76938-7_4

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