Asymptotics of entries of products of nonnegative 2-by-2 matrices

TL;DR

This paper studies the asymptotic behavior of entries in products of nonnegative 2-by-2 matrices, showing they relate to continued fraction tails under mild conditions, which aids in estimating these entries in applications.

Contribution

It establishes a link between matrix product entries and continued fraction tails, providing asymptotic estimates under mild assumptions.

Findings

Entries of matrix products are asymptotically proportional to continued fraction tail products.

The results apply to sequences of matrices converging to a limit.

Provides a method for estimating matrix product entries in practical scenarios.

Abstract

Let $M$ and $M_n,n\ge1$ be nonnegative 2-by-2 matrices such that $\lim_{n\rightarrow\infty}M_n=M.$ It is usually hard to estimate the entries of $M_{k+1}\cdots M_{k+n}$ which are useful in many applications. In this paper, under a mild condition, we show that up to a multiplication of some positive constants, entries of $M_{k+1}\cdots M_{k+n}$ are asymptotically the same as $\xi_{k+1}\cdots \xi_{k+n},$ the product of the tails of a continued fraction which is related to the matrices $M_k,k\ge1,$ as $n\rightarrow\infty.$