On the empirical spectral distribution of matrix perpetuities

Abstract

We study matrix perpetuities, that is, solutions to affine fixed-point equations of the form \[ \mathbf{X} \stackrel{d}{=} \mathbf{A}\,\mathbf{X} \,\mathbf{A}^\top+\mathbf{B},\qquad (\mathbf{A},\mathbf{B})\mbox{ and }\mathbf{X} \mbox{ are independent}, \] with particular emphasis on the empirical spectral distribution of the solution. We first establish existence and uniqueness results by relating the problem to classical vector perpetuities. Under orthogonal invariance, we then prove a compression identity for symmetric multiplicative convolution and show that principal submatrices of a matrix perpetuity are themselves lower-dimensional matrix perpetuities. For positive semidefinite, orthogonally invariant models, we prove a finite-dimensional spectral Kesten theorem: we obtain precise power-law tail asymptotics for the expected empirical spectral distribution, show that its tail is governed by the largest eigenvalue, and relate it explicitly to the tail of any diagonal entry. We also prove that, in the subcritical regime, the expected empirical spectral distribution of matrix perpetuities converges weakly, as the dimension tends to infinity, to the distribution of the corresponding free perpetuity. Our results are illustrated by matrix Beta prime perpetuities, for which explicit limiting spectral distributions are available.