Let $\theta \in (0,1)$ and $(X_n)_{n\ge0}$ be a markov chain with state space $\mathbb Z$, initial distribution $\delta_0$ and transition matrix $P$ with $$p_{i,i+1}=\theta, \ \ \ p_{i,i-1}=1-\theta$$
All the other entries are zero.
Define $Y_n:=X_{2n}$ for $n\ge0$. Then $(Y_n)_{n\ge0}$ is a markov chain with state space $J=2 \mathbb Z$.
I want to determine the transition matrix $\tilde P$ of $Y:$
$\tilde P_{ij} =\mathbb P[Y_{n+1}=j|Y_n=i]=\mathbb P[X_{2n+2}=j|X_{2n}=i]=\begin{cases}2\theta(1-\theta),&&j=i\\\theta^2,&&j=i+2\\(1-\theta)^2,&&j=i-2\\0,&&\mathrm{otherwise}\end{cases}$
Is this correct? I don't understand why $J=2\mathbb Z$ instead of just $\mathbb Z$. The $i$ and $j$ in the formula above are elements of $\mathbb Z$, aren't they?