Convergence to Stable Laws for Products of Random Matrices

TL;DR

This paper proves a generalized Central Limit Theorem for the log-norm of products of i.i.d. random matrices under certain algebraic and moment conditions, extending to higher rank cases with Cartan projections.

Contribution

It establishes a new weak law of large numbers for the difference between log-norms and sums, based on pivotal times, under minimal moment assumptions.

Findings

Log-norms satisfy a GCLT under algebraic and moment conditions.

A weak law of large numbers for the difference in log-norms is proven.

Results extend to higher rank via Cartan projections.

Abstract

Under reasonable algebraic assumptions and under an infinite second order moment assumption, we show that the logarithm of the norm (log-norm) of a product of random i.i.d. matrices with entries in $\mathbb{R}$ or in any other local field satisfies a generalized Central Limit Theorem (GCLT) in the sense of Paul L\'evi. The proof is based on a weak law of large number for the difference $\Delta_n$ between the log-norm of the product of the first $n$ matrices and the sum of their log-norms. This weak law of large numbers morally says that $\Delta_n$ behaves like a sum of i.i.d. random variables that have a finite moment of order $2q$ as long as the log-norm of each matrices has a finite moment of order $q$ for a given $q > 0$. This gain of moment is the central result of the present paper and is based on the construction of pivotal times. Moreover, these results admit a nice higher rank extension when one looks at the full Cartan projection instead of the log-norm.