Gliwenko-Cantelli's theorem

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Empirical distribution function of a standard normally distributed sample of size n = 100

The set of Gliwenko-Cantelli or set of Gliwenko also law of mathematical statistics or fundamental set of statistics called english Central statistical theorem , is a mathematical theorem on the area of the probability calculation , which is based on two work of the two mathematicians Valery Glivenko and Francesco Cantelli dating back to 1933. The sentence shows that when random experiments are carried out independently , the empirical distribution functions of a random variable obtained from the random samples converge uniformly with probability one to its actual distribution function and that this gives the possibility of estimating this distribution function.

Formulation of the sentence in detail

The sentence can be stated as follows:

Let a probability space be given

and then a consequence

of stochastically independent and identically distributed random variables with a common distribution function .

The the sample size corresponding empirical distribution function is

With
  .

For this one has the random variable on the given probability space

With
  ,

which indicates the upper limit of all distances of this empirical distribution from the common distribution , taking into account all possible characteristics .

Then:

They converge with probability 1, almost certainly , to zero.
So it applies
  .

Remarks

  1. The theorem results from the application of Kolmogorov's law of large numbers .
  2. It has been generalized and modified in various directions. The work of the Danish mathematician Flemming Topsøe from 1970 gives an impression of this .

Sources and background literature

Original work

Monographs

Individual evidence

  1. ^ Marek Fisz: Probability calculation and mathematical statistics. 1980, p. 456 ff
  2. P. Gänssler, W. Stute: Probability Theory. 1977, p. 145
  3. BW Gnedenko: Introduction to Probability Theory 1980, pp. 185 ff
  4. Achim Klenke: Probability Theory. 2013, p. 117 ff
  5. Norbert Kusolitsch: Measure and probability theory: An introduction. 2014, p. 262 ff
  6. Klaus D. Schmidt: Measure and probability. 2009, p. 353 ff
  7. Walter Vogel: Probability Theory. 1970, p. 318 ff
  8. With which is characteristic function called.
  9. It stands for the supremum .
  10. ^ Flemming Topsøe: On the Glivenko-Cantelli theorem. in: Z. Probability Theory and Related Areas 14, pp. 239 ff