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An L 2 convergence theorem for random affine mappings

Part of:Limit theoremsProbability 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

Information

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.

References

[1] Barnsley, M. F., Berger, M. A. and Soner, H. M. (1988) Mixing Markov chains and their images. Prob. Eng. Inf. Sci. 2, 387–414.CrossRefGoogle Scholar
[2] Berger, M. A. and Amit, Y. (1988) Products of random affine maps. Preprint.Google Scholar
[3] Berger, M. A. and Soner, H. M. (1988) Random walks generated by affine mappings. J. Theoret. Prob. 3, 239–254.CrossRefGoogle Scholar
[4] Bickel, P. J. and Freedman, D. A. (1981) Some asymptotic theory for the bootstrap. Ann. Statist. 9, 1196–1217.CrossRefGoogle Scholar
[5] Brandt, A. (1986) The stochastic equation Yn + 1 = AnYn + Bn with stationary coefficients. Adv. Appl. Prob. 18, 211–220.Google Scholar
[6] Elton, J. H. (1987) A multiplicative ergodic theorem for Lipschitz maps. Preprint.CrossRefGoogle Scholar
[7] Fürstenberg, H. and Kesten, H. (1960) Products of random matrices. Ann. Math. Statist. 31, 457–469.CrossRefGoogle Scholar
[8] Karlsen, H. A. (1990) Existence of moments in a stationary stochastic difference equation. Adv. Appl. Prob. 22, 129–146.CrossRefGoogle Scholar
[9] Kellerer, H. G. (1993) Ergodic behaviour of affine recursions, Parts I to III. Preprint.Google Scholar
[10] Kesten, H. (1973) Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248.CrossRefGoogle Scholar
[11] Kingman, J. F. C. (1973) Subadditive ergodic theory. Ann. Prob. 1, 883–909.CrossRefGoogle Scholar
[12] Letac, G. (1986) A contraction principle for certain Markov chains and its applications. Cont. Math. 50, 263–273.CrossRefGoogle Scholar
[13] Major, P. (1978) On the invariance principle for sums of independent, identically distributed random variables. J. Multivariate Anal. 8, 457–517.CrossRefGoogle Scholar
[14] Rösler, U. (1992) A fixed point theorem for distributions. Stoch. Proc. Appl. 37, 195–214.CrossRefGoogle Scholar
[15] Vervaat, W. (1979) On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob, 11, 750–783.CrossRefGoogle Scholar