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

TL;DR

This paper studies the spectral gap and extreme value behavior of multivariate affine stochastic recursions, showing convergence to classical extreme value laws and analyzing clustering of large values.

Contribution

It introduces a spectral gap approach to analyze extreme values in multivariate affine recursions, providing explicit convergence results and clustering properties.

Findings

Large values follow classical extreme value laws with a non-trivial extremal index

Convergence to Fréchet, exponential, and stable laws established

Spectral gap property is key to analyzing the recursion's extremal behavior

Abstract

We consider a general multivariate affine stochastic recursion and the associated Markov chain on $\mathbb R^{d}$. We assume a natural geometric condition which implies existence of an unbounded stationary solution and we show that the large values of the associated stationary process follow extreme value properties of classical type, with a non trivial extremal index. We develop some explicit consequences such as convergence to Fr{\'e}chet's law or to an exponential law, as well as convergence to a stable law. The proof is based on a spectral gap property for the action of associated positive operators on spaces of regular functions with slow growth, and on the clustering properties of large values in the recursion.