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Let $X_a$ be a random variable Poisson distributed with intensity $a$. That is $$\mathbb{P}(X_a=k)= e^{-a} a^k / (k!)$$ for any $k\in \mathbb{N}$. Let $$Y_a=(X-a)/\sqrt{a}$$ the normalization of $X_a$ such that it has mean $0$ and variance $1$. Let $F_{Y_a}$ its cumulative distribution function. That is $$F_{Y_a}(t)=\mathbb{P}(Y_a\leq t).$$ Let $N$ be the standard normal distribution and $F_N$ its cumulative distribution function. That is $$F_N(t)=\mathbb{P}(N\leq t) = \int_{-\infty}^t \frac{1}{\sqrt{2\pi}} e^{-\frac12 t^2} \mathrm{d}t.$$ Wikipedia says

For sufficiently large values of $a$, (say $a$>1000), the normal distribution with mean $a$ and variance $a$ (standard deviation $\sqrt{a}$) is an excellent approximation to the Poisson distribution.

My question is: How strong is the convergence of $Y_a$ against $N$ when $a\to\infty$?

Do we only have convergence in distribution : $F_{Y_a}(t) \to N(t)$ when $a\to\infty$ for any $t\in\mathbb{R}$.

Or can we say more? In particular I would like to have an upper bound (depending on $a$) for $$ \int_{-\infty}^{\infty} |F_{Y_a}(t)-F_N(t)| \mathrm{d}t.$$ I would be even more happy if someone comes up with an upper bound for $$ \int_{-\infty}^{\infty} |F_{Y_a}(t)-F_N(t)|^p \mathrm{d}t \quad \text{with }p>1.$$ What is a good reference for this kind of results?

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    $\begingroup$ Possibly related? The Berry-Esseen Theoem en.wikipedia.org/wiki/Berry–Esseen_theorem $\endgroup$
    – GEdgar
    Jun 2 '15 at 17:55
  • $\begingroup$ Thanks GEdgar. I did not know about it. This is the kind of results I am looking for. $\endgroup$ Jun 2 '15 at 18:08
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    $\begingroup$ For $p=1$, see L1 bounds in normal approximation by Larry Goldstein, and the references therein. $\endgroup$
    – Did
    Jun 3 '15 at 6:39

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