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I was looking for some facts on the probability theory and I found this exercise on Billingsley's "Probability and Measure" book (exercise 27.3, page 379). It doesn't look like a hard exercise but I'm having a hard time trying to prove the general case. Here it goes:

Exercise: Let $Y_\lambda$ be a Poisson random variable with mean $\lambda$ on a probability space $(\Omega, \Sigma,P)$. Show that $$P\left( \dfrac{Y_\lambda - \lambda}{\sqrt{\lambda}} \leq t\right) \rightarrow \dfrac{1}{\sqrt{2\pi}}\int_{-\infty}^t e^{-\frac{x^2}{2}} dx$$ as $\lambda\rightarrow\infty$.

I have a partial solution in the sense that if we consider $\lambda\in\mathbb{N}$ and make $P_i=Y_1, \ i\in\mathbb{N}$, ie, $P_i$ a Poisson random variable with mean 1. Now observe that $$\dfrac{Y_\lambda - \lambda}{\sqrt{\lambda}} = \dfrac{\sum_{i=1}^{\lambda} P_i - \lambda \cdot 1}{\sqrt{\lambda\cdot 1}} $$

The result on this special case follows using the CLT for $\{P_i=Y_1\}$.

Do you guys have any idea of how to solve it for any $\lambda$? I'm pretty sure that the general solution is realated to this case but I can't see how.

Thank you guys in advance!!

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up vote 4 down vote accepted

You could use characteristic functions and the Lévy continuity theorem.

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