[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]

Consider the Markov chain with general [math]2 \times 2[/math]

transition matrix

[[math]] \mat {P} = \pmatrix{ 1 - a & a \cr b & 1 - b}\ . [[/math]]

• Under what conditions is [math]\mat{P}[/math] absorbing?
• Under what conditions is [math]\mat{P}[/math] ergodic but not regular?
• Under what conditions is [math]\mat{P}[/math] regular?