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Discrete probability measures on 2 × 2 stochastic matrices and a functional equation on [0, 1]

Mukherjea A., Ratti J. S.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 2

359--372

In this paper, we consider the following natural problem: suppose μ1 and μ2 are two probability measures with finite supports S(μ1), S(μ2) respectively, such that |S(μ1)| = |S(μ2)| and S(μ1) U S(μ2) ⊂ 2 × 2 stochastic matrices, and μn1 (the n-th convolution power of μ1 under matrix multiplication), as well as μn 2 , converges weakly to the same probability measure λ, where S(λ) ⊂ 2 × 2 stochastic matrices with rank one. Then when does it follow that μ1 = μ2? What if S(μ1) = S(μ2)? In other words, can two different random walks, in this context, have the same invariant probability measure? Here, we consider related problems.

Uncoupling measures and eigenvalues of stochastic matrices

Pokarowski P.

Journal of Applied Analysis

1998

Vol. 4, nr 2

259-267

This paper gives bounds for the uncoupling measures of a stochastic matrix P in terms of its eigenvalues. The proofs are combinatorial. We use the Matrix-Tree Theorem which represents principal minors of I - P as sums of weights of directed forests.