[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Prove that if [math]\mat{P}[/math] is the transition matrix of an ergodic chain, then [math](1/2)(\mat {I} + \mat {P})[/math] is the transition matrix of a regular chain. Hint: Use Exercise.