See also: TCL

The theorem of the central limit (or: centered limit; one finds in Franglais the frequent name of central limit theorem ) is a whole of results on the weak convergence of a continuation of random variable in Probabilité. Intuitively, according to these results, any sum of independent and identically distributed random variables tends towards a certain random variable. The most known result and most important is simply called “theorem of the central limit” who relates to a sum of random variables of which the number tends towards the infinite one.

In the simplest case, considered below for the demonstration of the theorem, these variables are independent and have the same average and the same variance. In general, the sum grows indefinitely at the same time as the number of terms. To try to obtain a finished result, it is necessary to center this sum by withdrawing its average to him and to reduce it by dividing it by its standard deviation. Under rather broad conditions, the law of probability converges then towards a normal Loi unit. The omnipresence of the normal law is explained by the fact why many phenomena considered as random are due to the superposition of many causes.

Convergence is ensured in this simple case by the existence of the moment of order 3. There exist several generalizations which do not require identical laws but call upon conditions which ensure that none the variables exerts an influence significantly more important than the others. Such are the condition of Lindeberg and the condition of Lyapounov . Other generalizations authorize even a “weak” dependence. Moreover, one generalization due to Gnedenko and Kolmogorov stipulate that the sum of a certain number of random variables with a tail of decreasing distribution according to 1| X |^α+1 with 0  <  α  <  2 (having thus an infinite variance) tends towards a Loi of Levy symmetrical and stable when the number of variables increases. This article will be limited to the theorem of the central limit concerning the laws with finished variance.

“The” theorem of the central limit

That is to say X ₁, X ₂… a continuation of random variables definite on the same space of probability, according to the same law D and independent. Let us suppose that the hope $\backslash \; driven$ and the standard deviation $\backslash \; sigma$ of D exist and are finished ($\backslash \; sigma\; \backslash \; neq\; 0$).

Let us consider the sum S _N   =  X ₁  +  …   +  X _N . Then the hope of S _N is N μ and its standard deviation is worth σ N ^½ . Moreover, to speak in an abstract way, the law of S _N tightens towards the normal law NR ( N μ, σ² N ) when N tends towards the infinite one.

In order to clarify this idea of convergence, we will pose

$Z\_n\; =\; \backslash \; frac\; \{S\_n\; -\; N\; \backslash \; driven\}\; \{\backslash \; sigma\; \backslash \; sqrt\; \{N\}\}.$

so that the hope and the standard deviation of $Z\_n$ are worth 0 and 1 respectively.

Then the law of Z _N converges towards the reduced centered normal law NR (0,1) when N tends towards the infinite one (it is about the convergence in law). That means that if Φ is the Fonction of distribution of NR (0,1), then for any reality Z :

$\backslash \; lim\_\; \{N\; \backslash \; to\; \backslash \; infty\}\; \backslash \; mbox\; \{P\}\; (Z\_n\; \backslash \; Z)\; =\; \backslash \; Phi\; (Z),$

or, in an equivalent way:

$\backslash \; lim\_\; \{N\; \backslash \; to\; \backslash \; infty\}\; \backslash \; mbox\; \{P\}\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; overline\; \{X\}\; \_n-\; \backslash \; driven\}\; \{\backslash \; sigma\; \backslash \; sqrt\; \{N\}\}\; \backslash \; Leq\; Z\; \backslash \; right)\; =\; \backslash \; Phi\; (Z)$

where

$\backslash \; overline\; \{X\}\; \_n=S\_n/n=\; (X\_1+\; \backslash \; cdots+X\_n)\; /n$

Demonstration of the theorem of the central limit

For a theorem of such an importance in statistics and of probability applied, there exists a particularly simple demonstration using the characteristic functions. This demonstration resembles that of one of the laws of the great numbers. For a random variable Y of hope 0 and variance 1, the function characteristic of Y admits the limited development:

$\backslash \; varphi\_Y\; (T)\; =\; 1\; -\; \{t^2\; \backslash \; over\; 2\}\; +\; O\; (t^2),\; \backslash \; quad\; T\; \backslash \; rightarrow\; 0.$

If Y _I is worth $\backslash \; frac\; \{X\_i\; -\; \backslash \; driven\}\; \{\backslash \; sigma\}$, it is easy to see that the average centered reduced of the observations X₁, X₂,…, X _N is simply:

$Z\_n\; =\; \backslash \; frac\; \{\backslash \; overline\; \{X\}\; \_n-\; \backslash \; driven\}\; \{\backslash \; sigma\; \backslash \; sqrt\; \{N\}\}\; =\; \backslash \; sum\_\; \{i=1\}\; ^n\; \{Y\_i\; \backslash \; over\; \backslash \; sqrt\; \{N\}\}.$

According to the elementary properties of the characteristic functions, the function characteristic of Z _N is

$\backslash \; left\; \backslash \; over\; \backslash \; sqrt\; \{N\}\}\; \backslash \; right)\; \backslash \; right^n\; =\; \backslash \; left\; 1\; -\; \{t^2\; \backslash \; over\; 2n\}\; +\; O\; \backslash \; left\; (\{t^2\; \backslash \; over\; N\}\; \backslash \; right)\; \backslash \; right]\; ^n\; \backslash ,\; \backslash \; rightarrow\; \backslash ,\; e^\; \{-\; t^2/2\}$ when $n\; \backslash \; to\; +\; \backslash \; infty.$

But this limit is the function characteristic of the reduced centered normal law NR (0,1), from where one deduces the theorem from the central limit thanks to the theorem of continuity of Levy, who affirms that the convergence of the characteristic functions implies convergence in law.

Convergence towards the limit

If moment of order 3rd - μ) ³ exists and is finished, then convergence is uniform and the speed of convergence is at least of order 1 N ^½ (see the theorem of Berry-Esseen).

Images of a law smoothed by summation which show the distribution of the original law and three successive summations (obtained by Convolution):

In the practical applications, this theorem makes it possible in particular to replace a sum of random variables of number rather large but finished by a normal approximation, generally easier to handle. It is thus interesting to see how the sum approaches the limit. The terms used are explained in Random variable.

A sum of variable continuous is a continuous variable which one can compare the density of probability with that of the normal limit. With a sum of variable discrete , it is sometimes convenient to define a pseudo-density of probability but the most effective tool is the function of probability represented by a diagram in sticks. One can graphically note a certain coherence between the two diagrams, difficult to interpret. In this case, it is more effective to compare the functions of distribution .

In addition, the normal approximation is particularly effective in the vicinity of the central values. Some even say that as regards convergence towards the normal law, the infinite one often starts to six.

The precision is degraded as one moves away from these central values. It is particularly true for a sum of positive variables by nature: the normal law always reveals negative values with low but nonzero probabilities. Even if it is less shocking, that remains true in all circumstances: whereas any physical size is necessarily limited, the normal law which covers an infinite interval is only one useful approximation.

Lastly, for a given number of terms of the sum, the normal approximation is of as much better than the distribution is more symmetrical.

Application to the mathematical statistics

This theorem of probabilities has an interpretation in Statistique mathematics. The latter associates a law of probability with a population. Each element extracted the population is thus regarded as a random variable and, by joining together a number N of these presumedly independent variables, one obtains a sample. The sum of these random variables divided by N gives a new named variable the empirical average. This one, once reduced, tends towards a reduced normal variable when N tends towards the infinite one.

Other formulations of the theorem

Densities of probability

The Densité of probability of the sum of several independent variables is obtained by convolution their densities (if those exist). Thus one can interpret the theorem of the central limit like a formulation of the properties of the densities of probability subjected to a convolution: under the conditions established previously, convolée of a certain number of densities of probability tends towards the normal density when their number grows indefinitely.

As the function characteristic of a convolution is the product of the functions characteristic of the variables in question, the theorem of the central limit can be formulated in a different way: under the preceding conditions, the product of the functions characteristic of several densities of probability tends towards the function characteristic of the normal law when the number of variables grows indefinitely.

Products of random variables

The theorem of the central limit says to us with what it is necessary to expect as regards sums of independent random variables; but what is it products? Eh well, the Logarithme of a product (with strictly positive factors) is the sum of the logarithms of the factors, so that the logarithm of a product of random variables (with strictly positive values) tends towards a normal law, which involves a lognormal Loi for the product itself. Good number of physical sizes (in particular the mass and the length, it is a question of dimension, cannot be negative) are the product various factors Aléatoires, so that they follow a lognormal law. The same applies to the Stock exchange price of a risky credit.

Theorems of the central limit

Condition of Lyapounov

That is to say X _N a sequence of variables defined on the same space of probability. Let us suppose that X _N has a finished hope μ _N and a standard deviation finished σ _N . We will define

$s\_n^2\; =\; \backslash \; sum\_\; \{I\; =\; 1\}\; ^n\; \backslash \; sigma\_i^2.$

Let us suppose that the central moments of order 3

$r\_n^3\; =\; \backslash \; sum\_\; \{I\; =\; 1\}\; ^n\; \backslash \; mbox\; \{E\}\; \backslash \; left\; (\{\backslash \; left|\; X\_i\; -\; \backslash \; mu\_i\; \backslash \; right|\}\; ^3\; \backslash \; right)$

are finished for all N and that

$\backslash \; lim\_\; \{N\; \backslash \; to\; \backslash \; infty\}\; \backslash \; frac\; \{r\_n\}\; \{s\_n\}\; =\; 0.$

(It is the condition of Lyapounov).

Let us consider the sum again S_n=X₁+… +X_n. The expectation of S _N is m _N = ∑ _{I =1. N} μ _I and its standard deviation S _N . If we standardize S _N while posing

$Z\_n\; =\; \backslash \; frac\; \{S\_n\; -\; m\_n\}\; \{s\_n\}$

then the law of Z _N converges towards the reduced centered normal law NR (0,1) like above.

Condition of Lindeberg

With the same definitions and the same notations that previously, we can replace the condition of Lyapounov by the following one which is weaker (Lindeberg 1920). For all ε > 0

$$

\ lim_ {N \ to \ infty} \ sum_ {I = 1} ^ {N} \ mbox {E} \ left ( \ frac {(X_i - \ mu_i) ^2} {s_n^2} : \ left| X_i - \ mu_i \ right| > \ epsilon s_n \ right) = 0

where E ( U : V > C ) represents the conditional hope: the hope of U under the condition V > C . Then the law of Z _N converges towards the reduced centered normal law NR (0,1).

Case of the dependant variables

There exist some theorems which treat the case of sums of dependant variables, for example the theorem of the m-dependant central limit , the theorem of the central limit of the martingales and the theorem of the central limit for the processes of mixture.

Interest of this theorem

One can sometimes read in the general press that the bell-shaped curve represents the law of the chance, which does not have great significance. Success without equal of the law of Gauss is the direct consequence of the theorem of the central limit and it is reinforced by the relative convenience of use of this law.

In itself, convergence towards the normal law many sums of random variables when their number tends towards the infinite one interests only the mathematician. For the expert, it is interesting to stop a little before the limit: the sum of a great number of these variables is almost Gaussian, which provides an approximation often more easily usable than the exact law.

While moving away even more theory, one can say that good number of natural phenomena are due to the superposition of many causes, more or less independent. It results from it that the normal law represents them in a reasonably effective way.

Contrary, one can say that no concrete phenomenon is really Gaussian because it cannot exceed some limiting (in particular if it is with positive values).

External bonds

• Central Theorem Limit, Java

• Central Limit Theorem interactive Simulation to make experiments using several parameters.

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