Spectral gap properties for linear random walks and Pareto’s asymptotics for affine stochastic recursions

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]). …”

“…A remarkable property of η is its "homogeneity at infinity", a property which was first observed in [31] for the tails of η, extended to the general case in [34] and further developed in [1,6,13], under special conditions. See [17] for a survey of [34] as well for a precise description of the homogeneity property of η, proved in a special case in [6] and in a generic case in [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|.…”

“…Here we follow the general line of [21,5]. With respect to these papers, new arguments are needed for the analysis of relation (1.1), in the generic case considered below(See [22]). The asymptotics of products of random matrices (See [18,3,23]) will play an important role, and we need to give corresponding notations.…”

“…Condition C will be assumed in our results (compare with condition (H) of [5]), except if the contrary is specified. We observe that condition i-p for [supp ρ] is valid on an open dense set in weak topology of measures ρ on G. It follows that condition C is open in the weak topology of probability measures on H. Conditions C 1 and C 3 are used to prove homogeneity at infinity of η, a property which depends on the spectral gap properties of twisted convolution operators defined byμ on the projective space of V (Cf [22]). Condition C 2 plays the basic role in the homogeneity at infinity of η.…”