Stable laws and spectral gap properties for affine random walks

Abstract

We consider a general multidimensional affine recursion with corresponding Markov operator P and a unique P -stationary measure. We show spectral gap properties on Hölder spaces for the corresponding Fourier operators and we deduce convergence to stable laws for the Birkhoff sums along the recursion. The parameters of the stable laws are expressed in terms of basic quantities depending essentially on the matricial multiplicative part of P . Spectral gap properties of P and homogeneity at infinity of the P -stationary measure play an important role in the proofs.

On spectral gap properties and extreme value theory for multivariate affine stochastic recursions

“…For another approach, not using spectral gap properties, see chapter 4 of the recent book [5]. Here we give new proofs of the results given in ( [10], [15]), following and completing the point process approach of [7] in the case of affine stochastic recursions.…”

“…For a Lipschitz function f on V with non negative real part we define the Fourier-Laplace operator P f by P f ϕ(x) = P (ϕexp(−f )). In [10], spectral gap properties for Fourier operators were studied for…”

“…The proof depends on the two lemmas given below, and of calculations analogous to those of [10] for Fourier operators.…”

“…The convergence to stable laws of the normalized sums n Σ i=1 X i under (c-e) was shown in ( [10], [15]) where explicit formulae for the corresponding characteristic functions were given and non degeneracy of the limiting laws was proved. It was observed that these formulae involved the asymptotic tail Λ of ρ, as well as the occupation measure…”

“…In these studies we follow closely the approaches previously developed in ( [2], [7]) in the context of extreme value theory for general stationary processes, in particular we make full use of the concepts of tail and cluster processes introduced in [2]. This allow us to recover, in a natural setting, the characteristic functions of the above α-stable laws, as described in [15] if d = 1 and in [10] if d > 1, completing thereby the results of ([1], [2], [7]). Furthermore, in view of our results we observe that, with respect to the Q-invariant measure Λ 0 , the potential theory of the associated linear random walk enters essentially in the description of the extreme value asymptotics for the affine random walk X n , through excursions, occupation measures and capacities.…”