Let $\{X_n\}_{n=0,1,2\ldots}$ be an irreducible, positive recurrent DTMC with period $d \ge 2$ and one-step transition matrix $P$. Let $\{Y_n\}_{n =0,1,2,\ldots}$ be a DTMC with one-step transition matrix $P^d$

(a) Show that $\{Y_n\}$ has $d$ positive recurrent classes

(b) Show that $\{Y_n\}$ is aperiodic

(c) Prove the periodic extension of Basic Limit Theorem: $\lim_{n \to \infty} P^{(nd)} = d/m_{jj}$ where $m_{jj}$ is the mean recurrent time for state $j$

I really appreciate your help!

  • $\begingroup$ @AaronMontgomery I don't even know how to start this, for part (a) I think induction may be helpful but I still get confused how to connect these concepts $\endgroup$ – user370220 Oct 9 '17 at 13:15

Some hints for how to proceed:

(a) I don't think induction will be all that necessary or helpful here. Fix a state $x$; the $d$ classes will correspond to the periodicity of the original chain. That is, one class will be $\{y \colon p^{nd}(x,y) > 0$ for some $n\}$, another will be $\{y \colon p^{nd +1}(x,y) > 0$ for some $n\}$, then $\{y \colon p^{nd+2}(x,y) > 0$ for some $n\}$, etc. Your task is now to prove that this does indeed give an equivalence relation and that these classes are actually distinct.

(b) This should come straight from the definition of periodicity (or aperiodicity) of a Markov Chain.

(c) Use the Basic Theorem for Markov Chains on the chain $\{Y_n\}$; then, use the fact that one "step" in the $\{Y_n\}$ chain is equivalent to $d$ steps on the $\{X_n\}$ chain.

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