Asymptotics of product of nonnegative 2-by-2 matrices and maximum of a (2,1) random walk with asymptotically zero drift

Hongyan SUN, Hua-Ming WANG

Published 2020-04-28Version 1

Consider matrix product $A_kA_{k-1}\cdots A_1$ with $A_n,n\ge1$ some nonnegative 2-by-2 matrices. Under certain condition, we show that $\forall i,j= 1,2,$ $(A_kA_{k-1}\cdots A_1)_{i,j}\sim c\varrho(A_k)\varrho(A_{k-1})\cdots \varrho(A_1)$ as $k\rightarrow\infty$ with $\varrho(A_n)$ the spectral radius of $A_n$ and $0<c<\infty$ some constant. As an application, consider (2,1) random walk with asymptotically zero drift on the positive half line. Let $M$ be the maximum of an excursion starting from $2$ and ending at the point(either $0$ or $1$), where the walk hits the set $\{0,1\}$ for the first time. We study the distribution of $M$ and characterize its asymptotics, which are quite different from those of simple random walks.