Recurrence of multidimensional affine recursions in the critical case

TL;DR

This paper establishes recurrence conditions for multidimensional affine recursions in the critical case, covering various matrix types, based on moment criteria related to reverse norm control variables.

Contribution

It provides a general recurrence criterion for affine recursions in multiple dimensions under natural hypotheses, extending previous results to broader matrix classes.

Findings

Recurrence proven for similarities, invertible, rank 1, and non-negative coefficient matrices.

Criterion based on moment assumptions of reverse norm control variables.

Results unify and extend understanding of recurrence in multidimensional affine processes.

Abstract

We prove, under different natural hypotheses, that the random multidimensional affine recursion $X_n=A_nX_{n-1}+B_n\in\mathbb{R}^d, n \geq 1,$ is recurrent in the critical case. In particular we cover the cases where the matrices $A_n$ are similarities, invertible, rank 1 or with non negative coefficients. These results are a consequence of a criterion of recurrence for a large class of affine recursions on $\mathbb R^d$, based on some moment assumptions of the so-called ``reverse norm control random variable".