Suppose $X_0,X_1,\cdots$ are iid $Poisson(\theta)$ r.v.
Define $Y_k = X_k I_{\{ X_{k-1} = 0 \}}$ for $k=1,2,3,\cdots$
Find the limit of $Var(\sqrt{n}\overline{Y_n})$ and asymptotic distribution of $\sqrt{n}\left(\bar{Y}_{n}-\mathrm{E} \bar{Y}_{n}\right)$.
It's not difficult to find $E(Y_k) = E(X_k)P(X_{k-1}=0) = \theta exp(-\theta)$. But when it comes to variance, I know that $Y_k$ is independent to $Y_{k+i}$ for $i>1$. However, I dont know how to calculate the covariance of $Y_k$ and $Y_{k+1}$. Besides, since $Y_k$ are not i.i.d. random variables, how can I use CLT to get the asymptotic distribution?