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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: $$lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{y_n - \lambda n}{\sqrt{ \lambda n}} \rightarrow N(0,1) $$

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    $\begingroup$ Does this answer your question? Showing MGF of Poisson converges to MGF of N(0,1) $\endgroup$ – BruceET May 18 at 16:14
  • $\begingroup$ 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). $\endgroup$ – whuber May 18 at 17:51
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Hint:

Try finding the limit of the characteristic function or moment generating function.

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

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