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

Let AkAk−1⋯A1 be the product of some nonnegative 2-by-2 matrices. In general, its elements are hard to evaluate. Under conditions, we show that ∀i,j∈{1,2}, (AkAk−1⋯A1)i,j∼cϱ(Ak)ϱ(Ak−1)⋯ϱ(A1) as k→∞, where ϱ(An) is spectral radius matrix An and c∈(0,∞) constant. Consequently, can estimated. As applications, consider maxima certain excursions (2,1) (1,2) random walks with asymptotically zero drifts. We get delicate limit theories which quite different from those simple walks. Limit both tail critical sequences continued fractions play important roles in our studies.