Given that $(X_n)_{n\geq 0}$ is a Markov chain on the state space I={1,2,3,4,5}, initial distribution $\lambda$ and one-step transition probability matrix P.
Show that $(Y_n)_{n\geq 1}$ with $Y_n = X_{kn}$ is a Markov chain. What is the initial distribution?
For the first part I have the following derivation: \begin{equation} \begin{aligned} P(Y_{n+1}=i_{n+1}|Y_1=i_1, Y_2=i_2, ... ,Y_n=i_n) &= P(X_{k(n+1)}=i_{n+1}|X_k=i_1, Y_{2k}=i_2, ... ,Y_{kn}=i_n) \\ &=P(X_{k(n+1)}=i_{n+1}|X_{kn}=i_n)\\ &= P(Y_{n+1}=i_{n+1}|Y_n=i_n) \end{aligned} \end{equation}
and $P(X_{kn+k}=j|X_{kn}=i)= (P^k)_{ij}$
-How can I find the initial distribution of $Y_n = X_{kn}$?
-For the same Markov Chains, how can I check if irreducibility of one implies irreducibility of the other? and reversibility?