Exponential Decay of Correlations for a Real-Valued Dynamical System Generated by a k $k$ Dimensional System

Exponential Decay of Correlations for a Real-Valued Dynamical System Generated by a k $k$... Acta Appl Math https://doi.org/10.1007/s10440-018-0192-z Exponential Decay of Correlations for a Real-Valued Dynamical System Generated by a k Dimensional System 1 1 1 Lisette Jager · Jules Maes · Alain Ninet Received: 13 November 2017 / Accepted: 19 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract As a first step towards modelling real time-series, we study a class of real- variable, bounded processes {X ,n ∈ N} defined by a deterministic k-term recurrence re- lation X = ϕ(X ,...,X ). These processes are noise-free. We immerse such a dy- n+k n n+k−1 namical system into R in a slightly distorted way, which allows us to apply the multidimen- sional techniques introduced by Saussol (Isr. J. Math. 116:223–248, 2000) for deterministic transformations. The hypotheses we need are, most of them, purely analytic and consist in estimates satisfied by the function ϕ and by products of its first-order partial derivatives. They ensure that the induced transformation T is dilating. Under these conditions, T admits a greatest absolutely continuous invariant measure (ACIM). This implies the existence of an invariant density for X , satisfying integral compatibility conditions. Moreover, if T is mixing, one obtains the exponential decay of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Exponential Decay of Correlations for a Real-Valued Dynamical System Generated by a k $k$ Dimensional System

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
Subject
Mathematics; Computational Mathematics and Numerical Analysis; Applications of Mathematics; Partial Differential Equations; Probability Theory and Stochastic Processes; Calculus of Variations and Optimal Control; Optimization
ISSN
0167-8019
eISSN
1572-9036
D.O.I.
10.1007/s10440-018-0192-z
Publisher site
See Article on Publisher Site

Abstract

Acta Appl Math https://doi.org/10.1007/s10440-018-0192-z Exponential Decay of Correlations for a Real-Valued Dynamical System Generated by a k Dimensional System 1 1 1 Lisette Jager · Jules Maes · Alain Ninet Received: 13 November 2017 / Accepted: 19 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract As a first step towards modelling real time-series, we study a class of real- variable, bounded processes {X ,n ∈ N} defined by a deterministic k-term recurrence re- lation X = ϕ(X ,...,X ). These processes are noise-free. We immerse such a dy- n+k n n+k−1 namical system into R in a slightly distorted way, which allows us to apply the multidimen- sional techniques introduced by Saussol (Isr. J. Math. 116:223–248, 2000) for deterministic transformations. The hypotheses we need are, most of them, purely analytic and consist in estimates satisfied by the function ϕ and by products of its first-order partial derivatives. They ensure that the induced transformation T is dilating. Under these conditions, T admits a greatest absolutely continuous invariant measure (ACIM). This implies the existence of an invariant density for X , satisfying integral compatibility conditions. Moreover, if T is mixing, one obtains the exponential decay of

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: May 29, 2018

References

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