Consider a Markov chain with n states and a symmetric tridiagonal transition probability matrix P. Let C be a class of vectors { p(0) } such that half the elements of p(0) are zero and another half are 1. The problem is to find the vector p(0) in C such that the value ||p(0)P^N-p(∞)|| is the minimum for any large nonnegative integer N, where p(∞) = (1/2, 1/2, . . . , 1/2) and p(∞) is called a stationary concentration vector The solution is stated as follows. The optimal vectors in a sense defined in this paper are of the form (p_0,p_1 . . . ,p_<n-1>) which satisfy the following conditions: p_k =p_<3-k>, p_k =p_<4g+k> for 0 ≦ k ≦ 3 and 0 ≦ g ≦ 2^<m-2>-1 where n = 2^m (m ≧ 2). The optimal vectors can be represented as repetition of 0110 or 1001 . This problem was related to models for mixing process of particulate solids. The optimal solutions have been conjectured through the computer simulation conducted by Akao et al. The objective of this paper is to support the mathematical background for their experimental results and to show an interesting property of Markov chains.
View full abstract