How to apply the Central Limit Theorem to a sample Poisson distribution

Question

I have a random variable, $X$, that follows a population distribution of the Poisson type, $Po(\lambda)$, with an unknown $\lambda$ parameter. In an experiment, $k=100$ events were measured in a given time interval, each event having a very small probability to happen. I want to estimate the population value of $\lambda$ using the $k$ we found to estimate it, $\hat\lambda=k$. How can I apply the Central Limit Theorem to find the sample distribution of this estimator?

Answer 1

$\hat{k}$ is the number of events in the considered time interval $[0, T]$ i.e., $$ \hat{k} = \sum_{i=1}^{T - \Delta t}I[t_i \le X_i \le t_i+\Delta t] = \sum_{i=1}^nI_i, $$ where $\Delta t = 1/T$, so by the CLT $$ T^{-1/2}\left(\sum^T I_i - \mathbb{E}\sum^T I_i\right) \xrightarrow{D} N(0, \operatorname{var}(I_i)), $$ where $ \operatorname{var}(I_i) = \lambda \Delta t = \lambda/T$. As such, the sample distribution is given by $$ \hat{k} \sim^{approx.} N(\lambda, \lambda). $$