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

Abstract

Let $M$ and $M_n,n\ge 1$ be nonnegative 2-by-2 matrices such that ${\lim }_{n\to \infty }{M}_n=M.$ 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 $M_k,k\ge 1,$ as $n\to \infty .$

14 pages; we add some new results in this edition

References in corpus (1)

• Two-type linear fractional branching processes in varying environments with asymptotically constant mean matrices

Cited by in corpus (1)

• Regeneration of branching processes with immigration in varying environments