Suppose we have a sample of $n$ random variables $X_1, \ldots,X_n$ where each $X_i$ has a Poisson distribution with parameter $\lambda$. Define $\theta=\exp\{-\lambda\}$ and $$ \tilde{\theta}_n=\frac{1}{n}\sum_{i=1}^{n}\mathbb{1}_{\{0\}}(X_i) $$
Is there a limit distribution for $\tilde{\theta}_n$? Please any help will be appreciated.
As you have already mentioned, $\mathbb E[\mathbb 1_{\{0\}}(X)]=\exp\{-\lambda\}=\theta$. Thus, you only have to work out the variance $\mathbb V[\mathbb 1_{\{0\}}(X)]=\sigma^2$ and then you can use CLT to show that $\sqrt{n}(\tilde\theta_n-\theta)\overset{d}{\to} \mathcal N(0, \sigma^2)$.