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Home > Central limit theorem

The most important and famous result is simply called The Central Limit Theorem; it is concerned with independent variables with identical distribution whose expected value and variance are finite. Several generalizations exist which do not require identical distribution but incorporate some condition which guarantees that none of the variables exert a much larger influence than the others. Two such conditions are the Lindeberg condition and the Lyapunov condition. Other generalizations even allow some "weak" dependence of the random variables.

The reader may find it helpful to consider this illustration of the central limit theorem.

1 "The" central limit theorem

Let X₁,X₂,X₃,... be a sequence of random variables which are defined on the same probability space, share the same probability distribution D and are independent. Assume that both the expected value μ and the standard deviation σ of D exist and are finite.

Consider the sum :S_n=X₁+...+X_n. Then the expected value of S_n is nμ and its standard deviation is σ n^½. Furthermore, informally speaking, the distribution of S_n approaches the normal distribution N(nμ,σ²n) as n approaches ∞.

In order to clarify the word "approaches" in the last sentence, we standardize S_n by setting

Then the distribution of Z_n converges towards the standard normal distribution N(0,1) as n approaches ∞ (this is convergence in distribution). This means: if Φ(z) is the cumulative distribution function of N(0,1), then for every real number z, we have

or, equivalently,

where

is the "sample mean".

1.1 Proof of the central limit theorem

For a theorem of such fundamental importance to statistics and applied probabilityMuch of the current research involving probability is done under the auspices of applied probability the application of probability theory to other scientific domains. However, while such research is motivated (to some degree) by applied problems, it is u, the central limit theorem has a remarkably simple proof using characteristic functionSome mathematicians use the phrase characteristic function synonymously with " indicator function". The indicator function of a subset A of a set B is the function with domain B whose value is 1 at each point in A and 0 at each point that is in B but nots. It is similar to the proof of a (weak) law of large numbersIn a statistical context, laws of large numbers imply that the average of a random sample from a large population is likely to be close to the mean of the whole population. In probability theory, several laws of large numbers say that the average of a seq. For any random variable, Y, with zero mean and unit variance (var(Y) = 1), the characteristic function of Y is, by Taylor's theoremIn calculus, Taylor's theorem named after the mathematician Brook Taylor, who stated it in 1712, allows the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that p,

Letting Y_i be (X_i − μ)/σ, the standardised value of X_i, it is easy to see that the standardised mean of the observations X₁, X₂, ..., X_n is just

By simple properties of characteristic functions, the characteristic function of Z_n is

But, this limit is just the characteristic function of a standard normal distribution, N(0,1), and the central limit theorem follows from the Lévy continuity theorem , which confirms that the convergence of characteristic functions implies convergence in distribution.

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Topics: Central Limit Theorem Distribution Variables Functions Normal...

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