Ergodicity of $\mathbf{p}$-Majorizing Quadratic Stochastic Operators

#### M. Saburov

2018, v.24, Issue 1, 131-150

ABSTRACT

A scrambling square stochastic matrix plays an important role in the theory of the classical Markov chain. One of the classical results states that a row-stochastic matrix is strongly ergodic if and only if its some power is a scrambling matrix. In this paper, we deal with the similar problem for a cubic stochastic matrix. We introduce a notion of $\mathbf{p}$-majorizing quadratic stochastic operators and study the strong ergodicity of $\mathbf{p}$-majorizing quadratic stochastic operators associated with scrambling, Sarymsakov, and Wolfowitz cubic stochastic matrices.

Keywords: quadratic stochastic operator; $p$-majorizing operator; scrambling, Sarymsakov, and Wolfowitz cubic stochastic matrix; strong ergodicity