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The Central Limit Theorem implies that near the center of mass we can approximate the binomial distribution with the normal distribution:

$$ P(B(n,p) \geq i) \approx P(Z \geq \frac{i - n p}{\sqrt{n p (1-p)}}) $$

where $Z$ is the standard normal.

I am interested in cases where $n \rightarrow \infty$ while $p$ remains constant. However, I am integrating a function over all integers $i$, so I cannot assume that $i$ itself is bounded. So the standard Central Limit Theorem, which only asserts that the above approximation holds pointwise in the limit, is not adequate for me.

Are there any references which give explicit (or asymptotic) error estimates for this type of approximation?

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Got something from the answer below? – Did Aug 13 '13 at 8:22
up vote 1 down vote accepted

The Berry–Esseen theorem gives an estimate for the normal approximation of the binomial distribution:

$$\sup_{x\in\mathbb R} \left|P\left(\frac{B(n,p)-np}{\sqrt{np(1-p)}} \le x\right)-\Phi(x)\right| \le \frac{C\rho}{\sqrt{n}}$$

with $C < 0.4748$ and $\rho=\frac{p^2+q^2}{\sqrt{pq}}$.

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For every fixed $x$ and $p$, the CLT asserts that $$ \lim\limits_{n\to\infty}\mathbb P\left(B(n,p)\geqslant np+x\sqrt{np(1-p)}\right)=\mathbb P(Z\geqslant x), $$ and not what you wrote.

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