Limit Distributions of Products of Independent and Identically Distributed Random 2 × 2 Stochastic Matrices: A Treatment with the Reciprocal of the Golden Ratio

Abstract

Consider a sequence (Xn)n≥1 of i.i.d. 2×2 stochastic matrices with each Xn distributed as μ. This μ is described as follows. Let (Cn,Dn)T denote the first column of Xn and for a given real r with 0<r<1, let r−1Cn and r−1Dn each be Bernoulli distributions with parameters p1 and p2, respectively, and 0<p1,p2<1. Clearly, the weak limit of the sequence μn, namely λ, is known to exist, whose support is contained in the set of all 2×2 rank one stochastic matrices. In a previous paper, we considered 0<r≤12 and obtained λ explicitly. We showed that λ is supported countably on many points, each with positive λ-mass. Of course, the case 0<r≤12 is tractable, but the case r>12 is very challenging. Considering the extreme nontriviality of this case, we stick to a very special such r, namely, r=5−12 (the reciprocal of the golden ratio), briefly mention the challenges in this nontrivial case, and completely identify λ for a very special situation.