# Properties of the Markov chain $X_{3n}$

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!