[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]

Let [math]\mat{P}[/math] be a regular [math]r \times r[/math] transition matrix

and [math]\mat{y}[/math] any [math]r[/math]-component column vector. Show that the value of the limiting constant vector for [math]\mat {P}^n \mat {y}[/math] is [math]\mat{w}\mat{y}[/math].