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There is a similar question to mine here, but it never received an answer.

Let $\left(X_n\right)_{n\geq 0}$ be an irreducible Markov chain on the state space $I=\{1,2,3,4,5\}$ with intial distribution $\lambda$ and one-step transition matrix $P$. Let $Y_n=X_{3n}$.

  1. Is $\left(Y_n\right)_{n\geq 1}$ a Markov chain? What are its initial distribution and one-step transition matrix?
  2. Is $\left(Y_n\right)_{n\geq 1}$ irreducible?
  3. Is $\left(Y_n\right)_{n\geq 1}$ reversible if $\left(X_n\right)_{n\geq 0}$ is?
  4. Let $T=\sup\{n:\sum_{i=1}^n Y_i\leq 20\}$. Is $\left(Y_{T+n}\right)_{n\geq 1}$ a Markov chain?
  5. Suppose $\left(X_n\right)_{n\geq 0}$ has period 4. What is the period of $\left(Y_n\right)_{n\geq 1}$?

My Attempt

  1. $\left(Y_n\right)_{n\geq 1}$ is a Markov chain since $$P(Y_{n+1}=i_{n+1}\mid Y_1=i_1,\dots, Y_n=i_n)=P(X_{3n+3}=i_{n+1}\mid X_3=i_1,\dots, X_{3n}=i_n)=P(X_{3n+3}=i_{n+1}\mid X_{3n}=i_n)=P(Y_{n+1}=i_{n+1}\mid Y_n=i_n).$$ It has intial distribution $\lambda P^3$ and one-step transition matrix $P^3$.
  2. Not necessarily. If for example $\left(X_n\right)_{n\geq 0}$ is given by the graph below, then $\left(Y_n\right)_{n\geq 0}$ won't be irreducible.
  3. I think the answer is no, but I couldn't come up with a counterexample.
  4. I think the answer is no since $T$ is not a stopping time, but again, I am having trouble coming up with a proof or a counterexample.
  5. I think $\left(Y_n\right)_{n\geq 1}$ will have period 4 as well becuase if $i_1i_2i_3i_4$ is a 4-cycle in the graph of $\left(X_n\right)_{n\geq 0}$, then $i_4i_3i_2i_1$ is a 4-cycle in the graph of $\left(Y_n\right)_{n\geq 1}$.

Any hints or comments on my solutions are appreciated!

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