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The classical de Moivre-Laplace theorem states that we can approximate the normal distribution by discrete binomial distribution:

$${n \choose k} p^k q^{n-k} \simeq \frac{1}{\sqrt{2 \pi npq}}e^{-(k-np)^2 / (2npq)}.$$

My question is: are there more precise, quantitative versions of this theorem in the literature? Are there good estimates how to measure the error? I am unfortunately not familiar with the subject but need a result of this type.

Of course there is always the option of going through existing proofs and checking the details, and turning them from "soft" to "hard", but I suspect this has to be already done. And maybe this is not optimal, maybe there are good accessible ways.

Can someone point me a good reference in this direction?

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Note that $p+q=1.$ –  Will Jagy Sep 29 '11 at 20:07
    
And positive. Maybe I was writing the question a bit quickly assuming the result is too well-known... –  András Bátkai Sep 29 '11 at 20:17
    
The way I was taught it, the really good approximation is by taking the integral of an appropriate normal PDF from $ k - (1/2)$ to $k + (1/2).$ I guess you are saying something a bit different. –  Will Jagy Sep 29 '11 at 20:17
    
Yes, you are doing something different. In beginning statistics classes, the idea is that the normal CDF is available in printed tables and in software (essentially the error function erf). So you take the fixed distribution with the same mean and variance as your binomial distribution. The integer parameter $k$ does not appear in the description of which normal distribution. –  Will Jagy Sep 29 '11 at 20:22
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Littlewood wrote a famous paper about very accurate estimates for the tails of the distribution, which might not be what you want (but it might). I haven't read it and can't access it, but you could start from this paper: On Littlewood's Estimate for the Binomial Distribution, B.D. McKay Adv. Appl. Prob. 21, 475-478 (1989) and related papers. –  Zen Harper Sep 30 '11 at 3:54

4 Answers 4

up vote 3 down vote accepted

Firstly, I think by "qualitative" you mean "quantitative". Secondly, while there is a huge literature on the quantitative versions of the central limit theorem, the canonical results can be found in Feller's Vol 2. For the center of the distribution there is the Berry-Esseen theorem, for the tails there is the large deviations theory, the introduction to which is also covered by Feller.

EDIT If you really care about the specific approximation of the binomial by the normal (or vice versa) you are just talking about the higher terms in the Stirling approximation to the factorial (and hence to the binomial coefficients). You can read all about it in, eg, Graham/Knuth/Patashnik's Concrete Math.

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Thank you. Unfortunately, applying these results you only get a $1/\sqrt{n}$ term, if I interpret it correctly. I was hoping that the special structure of the problem could lead to sharper results. (I edited the typo, thank you...) –  András Bátkai Sep 29 '11 at 20:40
    
But I will have a closer look at Feller. –  András Bátkai Sep 29 '11 at 20:41
    
See the edit for enhanced answer... –  Igor Rivin Sep 29 '11 at 20:53
    
Thanks for the edited version! Yes, This is what I am after somehow... –  András Bátkai Sep 29 '11 at 20:56

You just want a local limit theorem for a sum of i.i.d. Bernoulli random variables. A standard reference (not just for Bernoulli r.v.!) is "Sums of Independent Random Variables" by Petrov, in particular Chapter VII, §3.

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You might be interested in this paper (a very precise estimate, apparently overlooked by most people!)

J. E. Littlewood, On the probability in the tail of a binomial distribution, Adv. Appl. Prob. 1 (1969) 43–72.

revisited and corrected by McKay

Brendan D. McKay, On Littlewood's Estimate for the Binomial Distribution, Advances in Applied Probability, Vol. 21, No. 2 (Jun., 1989), pp. 475-478

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http://www.johndcook.com/normal_approx_to_binomial.html

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Thanks, that is a very nice summary of the results you can also get from the wikipedia page of the normal distribution. I was hoping there is something deeper. –  András Bátkai Sep 29 '11 at 20:03
    
also stats stackexchange –  psd Sep 29 '11 at 20:21

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