# Proof Poisson converges to Normal [closed]

I am looking for a formal proof that, with the CLT transformation, a random variable $$Y \sim POI(\lambda)$$ converges to a normal distribution ($$Z\sim N(0,1)$$). I believe this can be formulated as: $$lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{y_n - \lambda n}{\sqrt{ \lambda n}} \rightarrow N(0,1)$$

• Does this answer your question? Showing MGF of Poisson converges to MGF of N(0,1) – BruceET May 18 at 16:14
• What does "$y_n$" represent?? Evidently it's intended to be a sequence of random variables of mean and variance $\lambda n,$ perhaps related to your "$Y,$" but even then the sum diverges if the $y_n$ are independent. Perhaps you will find the related question at stats.stackexchange.com/questions/383620 helpful in formulating your question (and maybe even in answering it yourself). – whuber May 18 at 17:51

Or (if allowed) just apply the central limit theorem, as $$Y_n$$ has the same distribution as the sum of $$n$$ independent copies of $$Y$$