Asymptotics of product of nonnegative 2-by-2 matrices with applications to random walks with asymptotically zero drifts. Issue 2 (22nd January 2023) - Wang, Hua-Ming; Sun, Hongyan
Abstract : Let A k A k − 1 ⋯ 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 ∀ i, j ∈ 1, 2 , ( A k A k − 1 ⋯ A 1 ) i, j ∼ c ϱ ( A k ) ϱ ( A k − 1 ) ⋯ ϱ ( A 1 ) as k → ∞, where ϱ ( A n ) is the spectral radius of the matrix A n and c ∈ ( 0, ∞ ) is some constant. Consequently, the elements of A k A k − 1 ⋯ 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.