Finding an error estimation for the De Moivre–Laplace theorem Context for my question: For one part of my thesis I try to find an upper bound for the error in the normal approximation of the binomial distribution following the standard proof of the De Moivre–Laplace theorem with Stirling's formula. To make it concrete: Let $B_n$ be binomially distributed and let $N$ have the standardized normal distribution. I want to find an upper bound for $$\epsilon_n = \sup_{a<b} \left|\mathcal P\left(a\le \frac{B_n-np}{\sqrt{np(1-p)}} \le b\right)-\mathcal P(a \le N \le b)\right|$$
I want to compare this error with the best known error estimation of the Berry-Essee theorem for the binomial distribution. So far I have only found a proof which shows that $\epsilon_n \in O\left(\frac 1{\sqrt n}\right)$. See this proof by Márton Balázs and Bálint Tóth (which also just considered $\left|\mathcal P\left(a\le \frac{B_n-np}{\sqrt{np(1-p)}} \le b\right)-\mathcal P(a \le N \le b)\right|$ without the supremum). Other proofs do not investigate the error at all (see for example this proof on Wikipedia).
My Question: Do you know any proof in a textbook / paper / article where the theorem by De Moivre and Laplace is proved with Stirling's formula and the total error is estimated? The value of any occurring constants in the error estimate shall also be calculated. Can you point me to this proof?
Update: I reasked the question on MO, see https://mathoverflow.net/questions/219253/finding-an-error-estimation-for-the-de-moivre-laplace-theorem-with-stirlings-fo
 A: In fact, it follows form Balazs and Toth's paper that 
$$
\sup_{a_n\le c<d\le b_n}\left|\mathcal P\left(c\le \frac{B_n-np}{\sqrt{np(1-p)}} \le d\right)-\mathcal P(c \le N \le d)\right| \le C \frac{|a_n|^3+|b_n|^3+1}{\sqrt{n}}.
$$
On the other hand, by Chernoff's (or Berstein's, or Hoeffding's) inequality, for $x\ge b_n>p$ we have 
$$
\mathcal P\left(\frac{B_n-np}{\sqrt{np(1-p)}} \ge x\right) \le Ke^{-Kx^2}\le Ke^{-K b_n^2},
$$
and a similar inequality may be written for $x\le a_n<-p$. Moreover, $\mathcal P\left(|N| \ge x\right)\le Me^{-Mx^2}$. Therefore, taking $a_n =  - c\sqrt{\log n}$, $b_n =  c\sqrt{\log n}$ with $c>0$ large enough, we get 
$$
\sup_{a<b} \left|\mathcal P\left(a\le \frac{B_n-np}{\sqrt{np(1-p)}} \le b\right)-\mathcal P(a \le N \le b)\right|  = \mathcal O\left(\frac{\log^{3/2} n}{\sqrt{n}}\right),
$$
which is not sharp, of course, but at least uniform.
A: (Note:
I had the wrong author.
This is now corrected.)
IIRC,
Uspensky's book "Introduction to Mathematical Probability"
(published maybe 1937)
has a proof of the
central limit theorem
with explicit bounds on
the error term.
The result is quite complicated.
A: The best two-sided inequalities for the binomial distribution function may be found in
Theory Probab. Appl., 57(3), 539–544. (6 pages)
A Complete Proof of Universal Inequalities for the Distribution Function of the Binomial Law
Published online: 04 September 2013
Keywords
binomial distribution function, two-sided estimates, corrected normal approximations
Publisher: Society for Industrial and Applied Mathematics
A. M. Zubkov and A. A. Serov
https://doi.org/10.1137/S0040585X97986138
We present a new form and a short complete proof of explicit two-sided estimates for the distribution function $F_{n,p}(k)$ of the binomial law with parameters $n,p$ from [D. Alfers and H. Dinges, Z. Wahrsch. Verw. Geb., 65 (1984), pp. 399--420]. These inequalities are universal (valid for all values of parameters and argument) and exact (namely, the upper bound for $F_{n,p}(k)$ is the lower bound for $F_{n,p}(k+1)$). Such estimates allow to bound any quantile of the binomial law by two subsequent integers that it contains.
© 2013, Society for Industrial and Applied Mathematics
