I'm trying to do this exercise from Probability by John Walsh:
Let ${X_{n}, n = 0,1, 2, ... }$ be a Markov chain whose transition probabilities MAY NOT be stationary. Define $X'_{n}$ to be the n-tuple $X'=(X_{0},. .. , X_{n})$. Show that $(X'_{n})$ is a Markov chain with STATIONARY transition probabilities. (What is its state space?)
If I apply the stationarity condition I should obtain something like:
$P(X'_{n+1}=j|X'_{n}=i)=P(X'_{1}=j|X'_{0}=i).$
But, for example, the values that would take $X'_{n}$ are n-tuples meanwhile the values for $X'_{0}$ are just 1-tuples.
I want to find a "common" state space for the chain and show the claimed stationary condition.
Thanks in advance.