Local limit theorem

LOCAL LIMIT THEOREM

In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properlynormalized sum tends toward a normal distribution (informally a bell curve) even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920,^[1] thereby serving as a bridge between classical and modern probability theory.

If {\textstyle X_{1},X_{2},\dots ,X_{n}}  are {\textstyle n}  random samples drawn from a population with overall mean {\textstyle \mu }  and finite variance {\textstyle \sigma ^{2}} , and if {\textstyle {\bar {X}}_{n}}  is the sample mean, then the limiting form of the distribution, {\textstyle Z=\lim _{n\to \infty }{\sqrt {n}}{\left({\frac {{\bar {X}}_{n}-\mu }{\sigma }}\right)}} , is a standard normal distribution.^[2]

For example, suppose that a sample is obtained containing many observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic mean of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the probability distribution of the average will closely approximate a normal distribution. A simple example of this is that if one flips a coin many times, the probability of getting a given number of heads will approach a normal distribution, with the mean equal to half the total number of flips. At the limit of an infinite number of flips, it will equal a normal distribution.

The central limit theorem has several variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, if they comply with certain conditions.

Let {\textstyle \{X_{1},\ldots ,X_{n}\}}  be a random sample of size {\textstyle n}  — that is, a sequence of independent and identically distributed (i.i.d.) random variables drawn from a distribution of expected value given by {\textstyle \mu }  and finite variance given by {\textstyle \sigma ^{2}} . Suppose we are interested in the sample average

{\displaystyle {\bar {X}}_{n}\equiv {\frac {X_{1}+\cdots +X_{n}}{n}}}

of these random variables. By the law of large numbers, the sample averages converge almost surely (and therefore also converge in probability) to the expected value {\textstyle \mu }  as {\textstyle n\to \infty } . The classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number {\textstyle \mu }  during this convergence. More precisely, it states that as {\textstyle n}  gets larger, the distribution of the difference between the sample average {\textstyle {\bar {X}}_{n}}  and its limit {\textstyle \mu } , when multiplied by the factor {\textstyle {\sqrt {n}}}  (that is {\textstyle {\sqrt {n}}({\bar {X}}_{n}-\mu )} ) approximates the normal distribution with mean 0 and variance {\textstyle \sigma ^{2}} . For large enough n, the distribution of {\textstyle {\bar {X}}_{n}}  is close to the normal distribution with mean {\textstyle \mu }  and variance {\textstyle \sigma ^{2}/n} . The usefulness of the theorem is that the distribution of {\textstyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}  approaches normality regardless of the shape of the distribution of the individual {\textstyle X_{i}} . Formally, the theorem can be stated as follows: