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On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks

Published online by Cambridge University Press:  01 April 2008

Y. GUIVARC’H  and
EMILE LE PAGE
Y. GUIVARC’H
Affiliation:
IRMAR, CNRS Rennes I, Université de Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France (email: [email protected])
EMILE LE PAGE
Affiliation:
LMAM, Université de Bretagne Sud, Campus de Tohannic, BP 573, 56000 Vannes-Cedex, France (email: [email protected])
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Abstract

We consider a random walk on the affine group of the real line, we denote by P the corresponding Markov operator on , and we study the Birkhoff sums associated with its trajectories. We show that, depending on the parameters of the random walk, the normalized Birkhoff sums converge in law to a stable law of exponent α∈ ]0,2[ or to a normal law. The corresponding analysis is based on the spectral properties of two families of associated transfer operators Pt,Tt. The operator Pt is a Fourier operator and is considered here as a perturbation of the Markov operator P=P0 of the random walk. The operator Tt is related to Pt by a symmetry of Heisenberg type and is also considered as a perturbation of the Markov operator T0=T. We prove that these operators have an isolated dominant eigenvalue which has an asymptotic expansion involving fractional powers of t. The parameters of this expansion have simple expressions in terms of tails and moments of the stationary measures of P and T.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008 , pp. 423 - 446
Copyright
Copyright © Cambridge University Press 2008

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