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

Let be the product of some nonnegative 2-by-2 matrices. In general, its elements are hard to evaluate. Under some conditions, we show that as where is the spectral radius of the matrix and is some constant. Consequently, the elements of can be estimated. As applications, consider the maxima of certa...

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Bibliographic Details
Published in:Linear & multilinear algebra 2023-01, Vol.71 (2), p.150-177
Main Authors: Wang, Hua-Ming, Sun, Hongyan
Format: Article
Language:English
Subjects:
Asymptotic properties
Primary 15B48
Product of nonnegative matrices
Random walk
Secondary 60G50
spectral radius
tail of continued fraction
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Summary:Let be the product of some nonnegative 2-by-2 matrices. In general, its elements are hard to evaluate. Under some conditions, we show that as where is the spectral radius of the matrix and is some constant. Consequently, the elements of 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.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2021.2022083