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Let $P$ be an irreducible transition matrix, with period $d$. Consider transition matrix $P^k$.

In terms of $d$ and $k$, how many communicating classes does $P^k$ have and what is the period of each state?

So far I've drawn out various Markov chains and have noticed that $P^k$ has gcd($d,k$) communicating classes with period either:

a) 1 if $d$ divides $k$

or

b) period $d$ if $d$ does not divide $k$

This may be wrong, but I'm not sure how to proceed with a rigorous proof

1 Answers 1

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That's not right. You'll want to consider $\gcd(d,k)$.

EDIT: Let $T_i$, $i = 0, \ldots, d-1$ be set of the states $x$ such that the number of steps to go from a particular state $x_0$ to $x$ is congruent to $i$ mod $d$. Thus if you are in $T_i$ now, next step takes you to $T_{i+1}$ if $i < d-1$, or $T_0$ if $i = d-1$.

Let $g = \gcd(d,k)$. In a multiple of $d$ steps you can go from, say, $T_0$ to $T_n$ if $n$ is a multiple of $g$, but not otherwise. So there should be $g$ classes. Each should have period $d/g$.

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    I edited the above questions and found the right trend But i still do not know how to approach the proof – 2017-02-14