[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 a random walk on a circle of circumference [math]n[/math].

The walker takes one unit step clockwise with probability [math]p[/math] and one unit counterclockwise with probability [math]q = 1 - p[/math]. Modify the program ErgodicChain to allow you to input [math]n[/math] and [math]p[/math] and compute the basic quantities for this chain.

• For which values of [math]n[/math] is this chain regular? ergodic?
• What is the limiting vector [math]\mat{w}[/math]?
• Find the mean first passage matrix for [math]n = 5[/math] and [math]p = .5[/math]. Verify that [math]m_{ij} = d(n - d)[/math], where [math]d[/math] is the clockwise distance from [math]i[/math] to [math]j[/math].