CLT with explicit variance for products of random singular matrices related to Hill's equation

TL;DR

This paper establishes a CLT for products of certain random singular matrices linked to Hill's equation, providing explicit variance formulas and novel proof techniques involving $m$-dependent sequences.

Contribution

It introduces an explicit variance formula for the CLT of matrix products and connects the problem to $m$-dependent sequence theory for the first time.

Findings

Explicit variance formula derived for the CLT.

Application to specific examples with exact calculations.

New connection to $m$-dependent sequences established.

Abstract

We prove a central limit theorem (CLT) for the product of a class of random singular matrices related to a random Hill's equation studied by Adams$\unicode{x2013}$Bloch$\unicode{x2013}$Lagarias. The CLT features an explicit formula for the variance in terms of the distribution of the matrix entries and this allows for exact calculation in some examples. Our proof relies on a novel connection to the theory of $m$-dependent sequences which also leads to an interesting and precise nondegeneracy condition.