Let $X_1,...,X_n$ be iid. Poisson distributed random variables with density $$f_\theta(x)=e^{-\theta}\frac{\theta^x}{x!},\ x\in\mathbb{N}$$
I computed the Likelihoodfunction for $\theta$:$$f(\underline{x},\theta)=e^{-n\theta}\frac{\theta^{\sum_{i=1}^n x_i}}{\prod_{i=1}^n x_i!}$$
Now i want to determine how $f(\underline{x},\theta)$ behaves asymptotically, but I cannot seem to make any progress. My attempts mainly revolved around this
Theorem: Let $\theta_0$ be the true value of the parameter $\theta$. Under certain regularity conditions (we will take them for granted) the log likelihood estimate $\hat\theta_n$ satisfies $$I_n(\theta)^{1/2}(\hat\theta_n - \theta_0)\rightarrow Z$$ where $Z$ has a $\mathcal{N}(0,1)$ distribution and convergence is in distribution.
I suspect there are simpler methods to do this.