On Actions on Cubic Stochastic Matrices

#### I. Paniello

2017, v.23, №2, 325-348

ABSTRACT

We consider the set of ($n\times n\times n$) cubic stochastic matrices of type (1,2) together with different multiplication rules that not only retain their stochastic
properties but also endow this set with an associative semigroup structure. Then we introduce different actions of the semigroup of nonnegative column stochastic $n\times n$ matrices on the set of cubic stochastic matrices of type (1,2) and study how these actions translate to the
cubic matrix slices and marginal distributions. Actions introduced here provide an algebraic framework where considering different changes affecting the transition probabilities ruling certain biological populations.

Keywords: stochastic matrix, stochastic group, group action, marginal distribution, bivariate Markov chain