Introduction to Probability · Problem 11.22

Random Walks — Grinstead & Snell

Mathematics — Probability · markov-chains-regular

Problem

Show that if

\mathbf{P}

is the transition matrix of a regular Markov chain, and

\mathbf{W}

is the matrix each of whose rows is the fixed probability vector corresponding to

\mathbf{P}

, then

\mathbf{P}\mathbf{W} = \mathbf{W}

, and

\mathbf{W}^k = \mathbf{W}

for all positive integers

k