I'm currently working on an practice question from my notes. But I'm not quite understanding the idea of how to prove that something is a Markov chain.
Let {$X_i$}, $i = 1,2,...$, be a Markov chain on $E =$ {$0,1,2,3$}, with transition matrix $$P= \begin{pmatrix} 1/4 & 0 & 1/2 & 1/4 \\ 0 & 1/5 & 0 & 4/5 \\ 0 & 1 & 0 & 0 \\ 1/3 & 1/3 & 0 & 1/3 \\ \end{pmatrix}$$ A new process is defined by $Z_n = 0$ if $X_n = 0$ or $1$ and $Z_n = X_n$ if $X_n = 2$ or $3$.
Firstly, how do I find $P(Z_{n+1}=2|Z_n=0, Z_{n-1}=2)$ and $P(Z_{n+1}=2|Z_n=0, Z_{n-1}=3)$? Could anyone show me how this is done?
Secondly, the question also requires me to answer if {$Z_n$} is a Markov chain.
The approach I would like to take to the second part of the question would be to show that the value of $Z_{n+1}$ only depends on $Z_n$ and not $Z_{n-1}$, hence in this case $$P(Z_{n+1}=2|Z_n=0, Z_{n-1}=2) = P(Z_{n+1}=2|Z_n=0, Z_{n-1}=3)$$
If this is true then it would imply that {$Z_n$} is Markov chain as it $Z_{n+1}$ is only dependent on $Z_n$ and not $Z_{n-1}$.
Am I on the right track? If not, how do I go about showing whether or not {$Z_n$} is a Markov chain?