Higher Order Markov Chains, Transition Probability Tensors and Stochastic Tensors

Abstract: 

Markov chain, named after Russian mathematician Andrey Markov, is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Markov chains have wide application in science and engineering. If the probability of each event depends on several previous states, then we have higher order Markov chains. It turns out that nonnegative tensors play an important role in higher order Markov chains. The basic nonnegative tensor in the higher order Markov chains is the transition probability tensor.

In this talk, we describe some basic concepts of higher order Markov chains, discuss conditions to ensure existence and uniqueness of the limiting probability distribution of the transition probability tensor of a higher Markov chain, and learn some related tensors, namely stochastic tensors, multi-stochastic tensors and plane stochastic tensors.