Cubic stochastic matrices and their probability interpretation

This paper considers associative multiplications of cubic matrices generalizing the ordinary multiplication of matrices. Cubic analogues of stochastic matrices are introduced and their probabilistic interpretations are given. Cubic stationary stochastic matrices are described and the proposition on convergence of a cubic stochastic matrix to a stationary one is proved. We introduce the notion of the Markov interaction process which generalizes the notion of a Markov process and show that the notion of ergodicity of such a process is naturally related with the associative multiplication of cubic matrices.