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

• Published in: Ergodic theory and dynamical systems Vol. 28; no. 2; pp. 423 - 446
• Main Authors: GUIVARC’H, Y., LE PAGE, EMILE
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.04.2008; Cambridge University Press (CUP)
• Subjects: Mathematics; Probability; CENTRAL-LIMIT THEOREM; MATRICES; PIECEWISE MONOTONIC TRANSFORMATIONS
• ISSN: 0143-3857; 1469-4417
• DOI: 10.1017/S0143385707001010

We consider a random walk on the affine group of the real line, we denote by P the corresponding Markov operator on $\mathbb {R}$, 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.