A zero-one law for invariant measures and a local limit theorem for coefficients of random walks on the general linear group

TL;DR

This paper proves a zero-one law for stationary measures on algebraic sets and applies it to establish a local limit theorem for the coefficients of random walks on the general linear group, advancing understanding of their asymptotic behavior.

Contribution

It introduces a zero-one law for invariant measures on algebraic sets and derives a local limit theorem for coefficients of random walks on GL(n), extending previous results.

Findings

Zero-one law for stationary measures on algebraic sets

Local limit theorem for coefficients of random walks on GL(n)

Generalization of Furstenberg and Guivarc'h-Le Page results

Abstract

We prove a zero-one law for the stationary measure for algebraic sets generalizing the results of Furstenberg [13] and Guivarc'h and Le Page [20]. As an application, we establish a local limit theorem for the coefficients of random walks on the general linear group.