Law of large numbers for the spectral radius of random matrix products

TL;DR

This paper establishes strong and weak laws of large numbers for the spectral radius of i.i.d. random matrix products in complex general linear groups, without requiring irreducibility.

Contribution

It proves the law of large numbers for the spectral radius of i.i.d. random matrix products under minimal moment conditions, without irreducibility assumptions.

Findings

Spectral radius satisfies a strong law of large numbers with finite second moment.

Spectral radius satisfies a weak law of large numbers with finite first moment.

No irreducibility assumption needed for the results.

Abstract

We prove that the spectral radius of an i.i.d.\ random walk on $\GL_d(\C)$ satisfies a strong law of large numbers under finite second moment assumption and a weak law of large numbers under finite first moment. No irreducibility assumption is supposed.