Asymptotics of product of nonnegative 2-by-2 matrices with applications to random walks with asymptotically zero drifts

Abstract

Let $A_k{A}_{k-1}\cdots A_1$ be the product of some nonnegative 2-by-2 matrices. In general, its elements are hard to evaluate. Under some conditions, we show that $\mathrm{\forall }i,j\in \{1,2\},$ $(A_k{A}_{k-1}\cdots A_1{)}_{i,j}\sim c\varrho (A_k\left)\varrho \right(A_{k-1})\cdots \varrho (A_1)$ as $k\to \mathrm{\infty },$ where $\varrho \left(A_n\right)$ is the spectral radius of the matrix $A_n$ and $c\in (0,\mathrm{\infty })$ is some constant. Consequently, the elements of $A_k{A}_{k-1}\cdots A_1$ can be estimated. As applications, consider the maxima of certain excursions of (2,1) and (1,2) random walks with asymptotically zero drifts. We get some delicate limit theories which are quite different from those of simple random walks. Limit theories of both the tail and critical tail sequences of continued fractions play important roles in our studies.

Keywords:

• Product of nonnegative matrices
• random walk
• spectral radius
• tail of continued fraction

2020 Mathematics Subject Classifications:

• Primary 15B48
• Secondary 60G50
• 60J10

Acknowledgments

The authors thank the referees for their helpful comments and suggestions that improved the original manuscript. Also the authors would like to extend special thanks to Prof. W. M. Hong for introducing to us the Lamperti problem and Prof. Y. Chow for some useful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This project is supported by National Natural Science Foundation of China (Grant No. 11501008; 11601494).