Editing Central limit theorem (section)

===Convergence to the limit===
The central limit theorem gives only an [[asymptotic distribution]]. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.{{citation needed|reason=Not immediately obvious, I didn't find a source via google|date=July 2016}}

The convergence in the central limit theorem is [[uniform convergence|uniform]] because the limiting cumulative distribution function is continuous. If the third central [[Moment (mathematics)|moment]] <math display="inline">\operatorname{E}\left[(X_1 - \mu)^3\right]</math> exists and is finite, then the speed of convergence is at least on the order of <math display="inline">1 / \sqrt{n}</math> (see [[Berry–Esseen theorem]]). [[Stein's method]]<ref name="stein1972">{{Cite journal| last = Stein |first=C. |author-link=Charles Stein (statistician)| title = A bound for the error in the normal approximation to the distribution of a sum of dependent random variables| journal = Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability| pages= 583–602| year = 1972|volume=6 |issue=2 | mr=402873 | zbl = 0278.60026| url=http://projecteuclid.org/euclid.bsmsp/1200514239 }}</ref> can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.<ref>{{Cite book|  title = Normal approximation by Stein's method|  publisher = Springer| year = 2011|last1=Chen |first1=L. H. Y. |last2=Goldstein |first2=L. |last3=Shao |first3=Q. M. |isbn = 978-3-642-15006-7}}</ref>

The convergence to the normal distribution is monotonic, in the sense that the [[information entropy|entropy]] of <math display="inline">Z_n</math> increases [[monotonic function|monotonically]] to that of the normal distribution.<ref name=ABBN/>

The central limit theorem applies in particular to sums of independent and identically distributed [[discrete random variable]]s.  A sum of [[discrete random variable]]s is still a [[discrete random variable]], so that we are confronted with a sequence of [[discrete random variable]]s whose cumulative probability distribution function converges towards a cumulative probability distribution function corresponding to a continuous variable (namely that of the [[normal distribution]]).  This means that if we build a [[histogram]] of the realizations of the sum of {{mvar|n}} independent identical discrete variables, the piecewise-linear curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as {{mvar|n}} approaches infinity; this relation is known as [[de Moivre–Laplace theorem]]. The [[binomial distribution]] article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values.