# Basic Limit Theorem proof for a markov chain with period $\ge 2$

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 $j$

(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.
(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.