Robustness of iterated function systems of Lipschitz maps

Loïc Hervé (1, 2) , James Ledoux (1, 2)

1 INSA Rennes - Institut National des Sciences Appliquées - Rennes
2 IRMAR - Institut de Recherche Mathématique de Rennes

Let {X n } n∈N be a X-valued iterated function system (IFS) of Lipschitz maps dened as: X 0 ∈ X and for n ≥ 1, X n := F (X n−1 , ϑ n), where {ϑ n } n≥1 are i.i.d.r.v. with common probability distribution ν and where F (•, •) is Lipschitz continuous in the rst variable. Under parametric perturbation of both F and ν, we are interested in the robustness of the V-geometrical ergodicity property of {X n } n∈N , of its invariant probability measure and nally of the probability distribution of X n. Specically, we propose a pattern of assumptions for studying such robustness properties for an IFS. This pattern is implemented for the autoregressive processes with ARCH errors and for the roundo IFS. Moreover, we provide a general set of assumptions, which cover the classical Feller-type hypotheses, for an IFS to be a V-geometrical ergodic process, together with an accurate bound for the rate of convergence.

Mots clés: Markov chain; Geometric ergodicity; Perturbation; Non-linear stochastic recursion; Autoregressive process

Domaines: Probabilités [math.PR]

Loïc Hervé, James Ledoux. Robustness of iterated function systems of Lipschitz maps. 2021. ⟨hal-03423198v1⟩