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

Abstract

International audienceWe consider a random walk on the affine group of the real line, we denote by P the corresponding Markov operator on 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 alpha is an element of]0, 2[ or to a normal law. The corresponding analysis is based on the spectral properties of two families of associated transfer operators P-t, T-t. The operator P-t is a Fourier operator and is considered here as a perturbation of the Markov operator P = P-0 of the random walk. The operator T-t is related to P-t by a symmetry of Heisenberg type and is also considered as a perturbation of the Markov operator T-0 = 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

Stable limits for sums of dependent infinite variance random variables

“…Remark 7 Analogs of of Proposition 3 have recently been proved in Guivarc'h and Le Page [30] in the one-dimensional case and in Buraczewski et al [9], Theorem 1.6, also in the multivariate case. The results are formulated for a non-stationary version of the process (X n ) starting at some fixed value X 0 = x.…”

On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains

“…Moreover, it holds |x| t ν (dx) < ∞ for any t ∈ [0, α). We shall verify that hypotheses M1, M2, M3, M4 and M5 are satisfied with the function f (x) = x. Hypothesis M1 is obvious and hypotheses M2 and M3 follow from Theorem 1 and Proposition 4 in [17]. If δ > 0 is such that 2 + 2δ ≤ α, by simple calculations we obtain…”

“…'s with values in (0, ∞) × R of the same distribution µ and x 0 = x. Following Guivarc'h and Le Page [17], we assume the conditions:…”

Stable laws and spectral gap properties for affine random walks

“…Basic aspects of these special processes continue to hold in the general case of X x n , and give a heuristic guide for the study of the affine random walk X x n . On the other hand, independently of any density condition for µ, the conjunction of these two different processes give rise to new properties, in particular spectral gap properties for P (Cf [5,21]) and homogeneity at infinity for the P -stationary measure(Cf [6,17,22]). …”

“…Here we consider relation (1.1) in the case where [suppμ] is "large", a case which is generic and opposite to the case of [5]. We will need the detailed information on the stationary law η of P given in [22] and summarised in Theorem 2.4 below; also as in [5,21], a basic role will be played by the spectral properties of the Fourier operators P v (v ∈ R) defined by P v ϕ = P (X v ϕ), where X v (x) = e i v,x . Furthermore, the homogeneity at infinity of η plays an essential role and implies that the dominant eigenvalue of P v has an asymptotic expansion at 0 in terms of fractional powers of |v|.…”