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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. [ABSTRACT FROM AUTHOR]

Consider a (2, 1) random walk in an i.i.d. random environment, whose environment involves certain parameter. We construct an M-estimator for the environment parameter which can be written as functionals of a multitype branching process with immigration in a random environment (BPIRE). Because the offspring distributions of the involved multitype BPIRE are of the linear fractional type, the limit invariant distribution of the multitype BPIRE can be computed explicitly. As a result, we get the weak consistency of the M-estimator. Our result is a generalization of Comets et al. [Stochastic Process. Appl.2014, 124, 268–288]. [ABSTRACT FROM AUTHOR]