Heavy tail properties of stationary solutions of multidimensional stochastic recursions

Abstract

We consider the following recurrence relation with random i.i.d. coefficients (an, bn):x n+1 = a n+1 xn + b n+1 where an ∈ GL(d, R), bn ∈ R d . Under natural conditions on (an, bn) this equation has a unique stationary solution, and its support is non-compact. We show that, in general, its law has a heavy tail behavior and we study the corresponding directions. This provides a natural construction of laws with heavy tails in great generality. Our main result extends to the general case the results previously obtained by H. Kesten in [16] under positivity or density assumptions, and the results recently developed in [17] in a special framework.1. Notation and problem General notationWe consider the d-dimensional vector space V = R d , endowed with the scalar product Let G = GL(V ) be the general linear group, and H = Af f (V ) the affine group. An element h ∈ H can be written in the formwhere a ∈ G, b ∈ V . In reduced form we write h = (b, a) and we observe that the projection map (b, a) → a is a homomorphism from H to G. We consider also the projection (b, a) → b of H on V and we observe that H can be written as a semi-direct product H = G ⋉ V where V denotes also the translation group of V .For a locally compact second countable (l.c.s.c.) space X, we denote by M 1 (X) the convex set of probability measures on X and we endow M 1 (X) with the weak

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

“…Then our main result implies the following Briefly, we say that ρ satisfies Pareto's asymptotics of index α (see [47], page 74). The convergence in Theorem C can be considered as a Cramér type estimate for the random variable R and was stated in [26].This statement gives the homogeneity at infinity of ρ, hence the measure C σ α ⊗ℓ α defined by the theorem can be interpreted as the "tail measure" of ρ. In the context of extreme value theory for the process X n , the convergence stated in the theorem implies that ρ has "multivariate regular variation" and this property plays an essential role in the theory (see [19], [47]).…”

“…For d = 1, positivity of C = C + + C − was proved in [20] using Levy's symmetrisation argument, positivity of C + and C − was tackled in [26] by a complex analytic method introduced in [11]. For d = 1 we have ν α + = δ 1 , ν α − = δ −1 and the precise form of Theorem C gives that the condition C + = 0 is equivalent to suppρ ⊂] − ∞, c] with c ∈ R. In the Appendix we give an approach to part of Theorem C using tools familiar in Analytic Number Theory like Wiener-Ikehara's theorem and a lemma of E. Landau but also results for Radon transforms of positive measures which are only valid for α ∉ N (see [5], [51]).…”

Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case

“…In the contracting case it was studied by Kesten [14] and later on by Le Page [15] and Guivarc'h [13], who applying quite involved techniques obtained results similar to (3). However the critical case, when the top Lapunov exponent is zero, seems to be still not well understood and up to now has been studied only in very restrictive settings.…”

Regular behavior at infinity of stationary measures of stochastic recursion on NA groups

“…The case when N = R n 0 and there is a group G ⊂ GL(n 0 ) acting on it was studied by many authors [11,17,18,23,24]. Then S = R n0 G and the action of G is assumed to be proximal and irreducible.…”