A CHARACTERIZATION OF REVERSIBLE MARKOV-CHAINS BY A ROTATIONAL REPRESENTATION

Let P = (p(ij)), i,j = 1,2,...,n be the matrix of a recurrent Markov chain with stationary vector upsilon > 0 and let R = (r(ij)), i, j = 1,2,...,n be a matrix, where r(ij) = upsilon-(i)p(ij). If R is a symmetric matrix, we improve Alpern's rotational representation of P. By this representation we characterize the reversible Markov chains.