On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results

Abstract

A result by Elton⁽⁶⁾ states that an iterated function system

$$M_n = F_n (M_{n - 1} ),{\text{ }}n \geqslant 1,$$

of i.i.d. random Lipschitz maps F ₁,F ₂,... on a locally compact, complete separable metric space $(\mathbb{X},d)$ converges weakly to its unique stationary distribution π if the pertinent Liapunov exponent is a.s. negative and $\mathbb{E}\log ^ + d(F_1 (x_0 ),x_0 ) < \infty$ for some $x_0 \in \mathbb{X}$. Diaconis and Freedman⁽⁵⁾ showed the convergence rate be geometric in the Prokhorov metric if $\mathbb{E}L_1^p < \infty {\text{ and }}\mathbb{E}d(F_1 (x_0 ),x_0 )^p < \infty$ for some p>0, where L ₁ denotes the Lipschitz constant of F ₁. The same and also polynomial rates have been recently obtained in Alsmeyer and Fuh⁽¹⁾ by different methods. In this article, necessary and sufficient conditions are given for the positive Harris recurrence of (M _n ) _n≥0 on some absorbing subset $\mathbb{H}{\text{ of }}\mathbb{X}$. If $\mathbb{H} = \mathbb{X}$ and the support of π has nonempty interior, we further show that the same respective moment conditions ensuring the weak convergence rate results mentioned above now lead to polynomial, respectively geometric rate results for the convergence to π in total variation ∥⋅∥ or f-norm ∥⋅∥ _f , f(x)=1+d(x,x ₀)^η for some η∈(0,p]. The results are applied to various examples that have been discussed in the literature, including the Beta walk, multivariate ARMA models and matrix recursions.