Asymptotic results on the product of random probability matrices

I study the product of independent identically distributed D x D random probability matrices. Some exact asymptotic results are obtained. I find that both the left and the right products approach exponentially 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 the individual random matrices. I find λ = 3/2 for D = 2 when each element of the 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 shown numerically 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 ≃1.5 to ≃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 a mean of 1/D.