On the distribution of the limit of products of i. i. d. \(2\times 2\) random stochastic matrices (Q1295862)

Consider a set of \(n\) \(2\times 2\) stochastic matrices \(A\) with first column \((x_i,y_i)\) \((x_i\neq y_i)\), \(i=0,\dots, n-1\). Draw members of the set with probability \(p_i\), and form a matrix product in the order drawn. It is well known that the product converges weakly to a stochastic matrix with identical rows, and so has first column \((x,x)\). Sufficient conditions are given under which the distribution of \(x\) is continuous singular on \([0,1]\); for example \(a_0+ a_1+\cdots+ a_{n-1}< 1\) \((a_i:=|x_i- y_i|)\). This extends a known result (with the additional assumption that \(x_i< y_i)\) in the case \(n=2\) [see e.g. \textit{A. Nakassis}, J. Math. Anal. Appl. 70, 337-347 (1979; Zbl 0426.60012)].