Proving Y is a Markov chain when Y_k = X_2k

[stochfreak]

Hello, This is my first question here. So, let's see how it goes. I need a proof to a simple problem. The proof seems so obvious that I am finding it hard to prove it. The question is as follows:

Let X = {X_n} (n=0 to infinity) denote a Markov Chain (MC) on a finite state space 'S' and with transition matrix 'P'. Consider the MC, Y = {Y_n} (n=0 to infinity), with the setting, Y_k = X_2k. Prove or disprove that Y is a MC.

On the face of it, it is apparent that Y must be a Markov Chain as it will have to satisfy the Markov property (because X satisfies it and Y is derived from it). But, how can we formally prove this?

 

[CompuChip]

Welcome to PF stochfreak.

You can just check the definition of the Markov property (what is it?) using that X satisfies it.