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

Consider a sequence ( X n ) n ≥ 1 of i.i.d. 2 × 2 stochastic matrices with each X n distributed as μ . This μ is described as follows. Let ( C n , D n ) T denote the first column of X n and for a given real r with 0 < r < 1 , let r − 1 C n and r − 1 D n each be Bernoulli distributions with parameters p 1 and p 2 , respectively, and 0 < p 1 , p 2 < 1 . Clearly, the weak limit of the sequence μ n , 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 ≤ 1 2 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 ≤ 1 2 is tractable, but the case r > 1 2 is very challenging. Considering the extreme nontriviality of this case, we stick to a very special such r , namely, r = 5 − 1 2 (the reciprocal of the golden ratio), briefly mention the challenges in this nontrivial case, and completely identify λ for a very special situation.