Editing Central limit theorem (section)

===Relation to the law of large numbers===
The [[law of large numbers]] as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behavior of {{math|S<sub>{{mvar|n}}</sub>}} as {{mvar|n}} approaches infinity?" In mathematical analysis, [[asymptotic series]] are one of the most popular tools employed to approach such questions.

Suppose we have an asymptotic expansion of <math display="inline">f(n)</math>:

<math display="block">f(n)= a_1 \varphi_{1}(n)+a_2 \varphi_{2}(n)+O\big(\varphi_{3}(n)\big) \qquad  (n \to \infty).</math>

Dividing both parts by {{math|''φ''<sub>1</sub>(''n'')}} and taking the limit will produce {{math|''a''<sub>1</sub>}}, the coefficient of the highest-order term in the expansion, which represents the rate at which {{math|''f''(''n'')}} changes in its leading term.

<math display="block">\lim_{n\to\infty} \frac{f(n)}{\varphi_{1}(n)} = a_1.</math>

Informally, one can say: "{{math|''f''(''n'')}} grows approximately as {{math|''a''<sub>1</sub>''φ''<sub>1</sub>(''n'')}}". Taking the difference between {{math|''f''(''n'')}} and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about {{math|''f''(''n'')}}:

<math display="block">\lim_{n\to\infty} \frac{f(n)-a_1 \varphi_{1}(n)}{\varphi_{2}(n)} = a_2 .</math>

Here one can say that the difference between the function and its approximation grows approximately as {{math|''a''<sub>2</sub>''φ''<sub>2</sub>(''n'')}}.  The idea is that dividing the function by appropriate normalizing functions, and looking at the limiting behavior of the result, can tell us much about the limiting behavior of the original function itself.

Informally, something along these lines happens when the sum, {{mvar|S<sub>n</sub>}}, of independent identically distributed random variables, {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}}, is studied in classical probability theory.{{Citation needed|date=April 2012}}  If each {{mvar|X<sub>i</sub>}} has finite mean {{mvar|μ}}, then by the law of large numbers, {{math|{{sfrac|''S<sub>n</sub>''|''n''}} → ''μ''}}.<ref>{{cite book|last=Rosenthal |first=Jeffrey Seth |date=2000 |title=A First Look at Rigorous Probability Theory |publisher=World Scientific |isbn=981-02-4322-7 |at=Theorem 5.3.4, p. 47}}</ref>  If in addition each {{mvar|X<sub>i</sub>}} has finite variance {{math|''σ''<sup>2</sup>}}, then by the central limit theorem,

<math display="block"> \frac{S_n-n\mu}{\sqrt{n}} \to \xi ,</math>

where {{mvar|ξ}} is distributed as {{math|''N''(0,''σ''<sup>2</sup>)}}.  This provides values of the first two constants in the informal expansion

<math display="block">S_n \approx \mu n+\xi \sqrt{n}. </math>

In the case where the {{mvar|X<sub>i</sub>}} do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors:

<math display="block">\frac{S_n-a_n}{b_n} \rightarrow \Xi,</math>

or informally

<math display="block">S_n \approx a_n+\Xi b_n. </math>

Distributions {{math|Ξ}} which can arise in this way are called ''[[stable distribution|stable]]''.<ref>{{cite book|last=Johnson |first=Oliver Thomas |date=2004 |title=Information Theory and the Central Limit Theorem |publisher=Imperial College Press |isbn= 1-86094-473-6 |page= 88}}</ref>  Clearly, the normal distribution is stable, but there are also other stable distributions, such as the [[Cauchy distribution]], for which the mean or variance are not defined.  The scaling factor {{mvar|b<sub>n</sub>}} may be proportional to {{mvar|n<sup>c</sup>}}, for any {{math|''c'' ≥ {{sfrac|1|2}}}}; it may also be multiplied by a [[slowly varying function]] of {{mvar|n}}.<ref name=Uchaikin>{{cite book |first1=Vladimir V. |last1=Uchaikin |first2=V.M. |last2=Zolotarev |year=1999 |title=Chance and Stability: Stable distributions and their applications |publisher=VSP |isbn=90-6764-301-7 |pages=61–62}}</ref><ref>{{cite book|last1=Borodin |first1=A. N. |last2=Ibragimov |first2=I. A. |last3=Sudakov |first3=V. N. |date=1995 |title=Limit Theorems for Functionals of Random Walks |publisher=AMS Bookstore |isbn= 0-8218-0438-3 |at=Theorem 1.1, p. 8}}</ref>

The [[law of the iterated logarithm]] specifies what is happening "in between" the [[law of large numbers]] and the central limit theorem. Specifically it says that the normalizing function {{math|{{sqrt|''n'' log log ''n''}}}}, intermediate in size between {{mvar|n}} of the law of large numbers and {{math|{{sqrt|''n''}}}} of the central limit theorem, provides a non-trivial limiting behavior.