Abstract
As a first step towards modelling real time-series, we study a class of real-variable, bounded processes \(\{X_{n}, n\in \mathbb{N}\}\) defined by a deterministic \(k\)-term recurrence relation \(X_{n+k} = \varphi (X _{n}, \ldots , X_{n+k-1})\). These processes are noise-free. We immerse such a dynamical system into \(\mathbb{R}^{k}\) 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 \(\varphi \) 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 \(X_{n}\), satisfying integral compatibility conditions. Moreover, if \(T\) is mixing, one obtains the exponential decay of correlations.
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Notes
If \(\varphi _{j}\) is \(C^{2}\) on \(B_{{\varepsilon }_{1}}(\overline{O _{j}})\), it is necessarily \(C^{1,\alpha }\) on \(B_{{\varepsilon }_{1}}(\overline{O _{j}})\) with \(\alpha = 1\).
In favorable cases, the geometrical hypothesis can be replaced by the following one, stronger but much simpler: for all points \((u_{1}, u_{2}, \ldots , u_{k})\) and \((v_{1}, u_{2}, \ldots , u_{k})\) in \(B_{{\varepsilon }_{1}}(\overline{O_{j}})\), the segment \([(u_{1}, u _{2}, \ldots , u_{k}),(v_{1}, u_{2}, \ldots , u_{k})]\) is contained in \(B_{{\varepsilon }_{1}}(\overline{O_{j}})\).
Which is equivalent to: if 1 is the only modulus-1 eigenvalue of \(P\) and if, additionally, it is simple.
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Jager, L., Maes, J. & Ninet, A. Exponential Decay of Correlations for a Real-Valued Dynamical System Generated by a \(k\) Dimensional System. Acta Appl Math 160, 21–34 (2019). https://doi.org/10.1007/s10440-018-0192-z
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DOI: https://doi.org/10.1007/s10440-018-0192-z