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The operator P has the eigenvalue 1 . Set λ 1 = 1 , let h ∗ = P 1 1 Ω and dµ = h ∗ dm
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.
Acta Applicandae Mathematicae – Springer Journals
Published: May 29, 2018
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