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An L2 convergence theorem for random affine mappings

Part of: Limit theorems Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  14 July 2016

Robert M. Burton  and
Uwe Rösler
Robert M. Burton*
Affiliation:
Oregon State University
Uwe Rösler*
Affiliation:
Universität Kiel
*
∗Postal address: Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA.
∗∗Postal address: Mathematisches Seminar, Universität Kiel, Ludewig-Meyn-Str. 4, 24118 Kiel, Germany.
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Abstract

We consider the composition of random i.i.d. affine maps of a Hilbert space to itself. We show convergence of the nth composition of these maps in the Wasserstein metric via a contraction argument. The contraction condition involves the operator norm of the expectation of a bilinear form. This is contrasted with the usual contraction condition of a negative Lyapunov exponent. Our condition is stronger and easier to check. In addition, our condition allows us to conclude convergence of second moments as well as convergence in distribution.

Keywords

ITERATED FUNCTION SYSTEMRANDOM MATRICESLYAPUNOV EXPONENTCONTRACTIONHILBERT SPACE

MSC classification

Primary: 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case)
Secondary: 60F25: $L^p$-limit theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 183 - 192
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Supported in part by AFOSR Grant 91-0215 and NSF Grant DMS-9103738.

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