Recurrence of multidimensional affine recursions in the critical case

2024/08/07 by R. Aoun, Richard Aoun, Aoun, Richard +6
Mathematics · #math.PR

paper · pdf · doi:10.48550/arxiv.2408.03853

Abstract

We prove, under different natural hypotheses, that the random multidimensional affine recursion X_n=A_nX_n-1+B_n∈ℝ^d, n ≥ 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".

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