Berry–Esseen theorem

The set of Berry - Esseen tells about the quality of the convergence in the central limit theorem of probability theory. Here, both the convergence rate and a numerical estimate of the approximation to the normal distribution are given. The sentence has been independently by the mathematician Andrew C. Berry ( 1941) and Carl- Gustav Esseen (1942, published 1944) proved.

Set of Berry - Esseen

It is a sequence of independent and identically distributed random variables on a probability space. The expected values ​​and the variances may exist. Then converge on the central limit theorem, the distribution functions

The standardized sums against the normal distribution.

If the third absolute moment of the random variables exists, then for a constant C

Comments

  • For the validity of the principle of Berry - Esseen ( existence of expectation and variance ) is additionally required the existence of the third moment except for the conditions for the central limit theorem. Therefore, the record does not provide for all cases where the central limit theorem applies, a statement about the quality of the convergence to the normal distribution.
  • The set of Berry - Esseen are as qualitative information on the rate of convergence in the central limit theorem with the order. Without further assumptions on the distribution of the random variables, this is the best possible order of magnitude as the special case of the Bernoulli distribution shows with.
  • The set provides a quantitative estimate of the approximation to the normal distribution. The constant C is a " universal constant ", which does not depend on the characteristics of the random variables.

The Berry - Esseen constant

The constant C, which is for the quantitative estimation of the convergence of meaning is referred to in the literature as the Berry - Esseen constant ( engl. Berry - Esseen bound).

In the original work of Carl Gustav Esseen C is given as 7.59. Since then, she has been continuously improved. The best known today is C = 0.7655, which was given in 1985 by Shiganov. Other hand, it follows from the above-mentioned special case of the Bernoulli distribution that C is greater than.

Carl-Gustav Esseen Moment (mathematics)
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