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

Gerold Alsmeyer ()

Journal of Theoretical Probability, 2003, vol. 16, issue 1, 217-247

Abstract: 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 $$\mathbb{E}\log ^ + d(F_1 (x_0 ),x_0 ) 0, where L 1 denotes the Lipschitz constant of F 1. The same and also polynomial rates have been recently obtained in Alsmeyer and Fuh(1) by different methods. In this article, necessary and sufficient conditions are given for the positive Harris recurrence of (M n ) n≥0 on some absorbing subset $$\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.

Keywords: Iterated function system; Lipschitz map; Lipschitz constant; Liapunov exponent; Harris recurrence; total variation; f-ergodicity; geometric ergodicity; strictly contractive; drift condition; level γ ladder epoch (search for similar items in EconPapers)
Date: 2003
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1023/A:1022290807360 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:16:y:2003:i:1:d:10.1023_a:1022290807360

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1023/A:1022290807360

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().