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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 $k$... As a first step towards modelling real time-series, we study a class of real-variable, bounded processes { X n , n ∈ N } $\{X_{n}, n\in \mathbb{N}\}$ defined by a deterministic k $k$ -term recurrence relation X n + k = φ ( X n , … , X n + k − 1 ) $X_{n+k} = \varphi (X _{n}, \ldots , X_{n+k-1})$ . These processes are noise-free. We immerse such a dynamical system into R k $\mathbb{R}^{k}$ in a slightly distorted way, which allows us to apply the multidimensional 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 $\varphi $ and by products of its first-order partial derivatives. They ensure that the induced transformation T $T$ is dilating. Under these conditions, T $T$ admits a greatest absolutely continuous invariant measure (ACIM). This implies the existence of an invariant density for X n $X_{n}$ , satisfying integral compatibility conditions. Moreover, if T $T$ is mixing, one obtains the exponential decay of correlations. 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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References (18)

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
DOI
10.1007/s10440-018-0192-z
Publisher site
See Article on Publisher Site

Abstract

As a first step towards modelling real time-series, we study a class of real-variable, bounded processes { X n , n ∈ N } $\{X_{n}, n\in \mathbb{N}\}$ defined by a deterministic k $k$ -term recurrence relation X n + k = φ ( X n , … , X n + k − 1 ) $X_{n+k} = \varphi (X _{n}, \ldots , X_{n+k-1})$ . These processes are noise-free. We immerse such a dynamical system into R k $\mathbb{R}^{k}$ in a slightly distorted way, which allows us to apply the multidimensional 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 $\varphi $ and by products of its first-order partial derivatives. They ensure that the induced transformation T $T$ is dilating. Under these conditions, T $T$ admits a greatest absolutely continuous invariant measure (ACIM). This implies the existence of an invariant density for X n $X_{n}$ , satisfying integral compatibility conditions. Moreover, if T $T$ is mixing, one obtains the exponential decay of correlations.

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: May 29, 2018

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