Asymptotic results on the product of random probability matrices

Abstract

I study the product of independent identically distributed $D\times D$ random probability matrices. Some exact asymptotic results are obtained. I find that both the left and the right products approach exponentially to a probability matrix(asymptotic matrix) in which any two rows are the same. A parameter $λ$ is introduced for the exponential coefficient which can be used to describe the convergent rate of the products. $λ$ depends on the distribution of individual random matrices. I find $λ= 3/2$ for D=2 when each element of individual random probability matrices is uniformly distributed in [0,1]. In this case, each element of the asymptotic matrix follows a parabolic distribution function. The distribution function of the asymptotic matrix elements can be numerically shown to be non-universal. Numerical tests are carried out for a set of random probability matrices with a particular distribution function. I find that $λ$ increases monotonically from $\simeq 1.5$ to $\simeq 3$ as D increases from 3 to 99, and the distribution of random elements in the asymptotic products can be described by a Gaussian function with its mean to be 1/D.