Tensor approach to mixed high-order moments of absorbing Markov chains

The moments of the number of the visits in an absorbing Markov chain are considered. The first moments and the non-mixed second moments of the number of the visits are calculated in classical textbooks such as the book of J. Kemeny and J. Snell “Finite Markov Chains”. The first moments and the non-mixed second moments can be easily expressed in a matrix form using the fundamental matrix of the absorbing Markov chain. Since the representation of the mixed moments of higher orders in a matrix form is not straightforward, if ever possible, they were not calculated. The gap is filled now. A tensor approach to the mixed high-order moments is proposed and compact closed-form expressions for the moments are discover.