Gliwenko-Cantelli's theorem

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

${\displaystyle (\Omega ,{\mathcal {A}},\operatorname {P} )}$

and then a consequence

${\displaystyle X_{n}\colon (\Omega ,{\mathcal {A}},\operatorname {P} )\to \mathbb {R} \;(n\in \mathbb {N} )}$

of stochastically independent and identically distributed random variables with a common distribution function . ${\displaystyle F\colon \mathbb {R} \to \mathbb {R} }$

The the sample size corresponding empirical distribution function is ${\displaystyle n\in \mathbb {N} }$

${\displaystyle F_{n}\colon {\mathbb {R} \times \Omega }\to [0,1]}$

With

${\displaystyle F_{n}(x,\omega )={\frac {1}{n}}\cdot \sum _{k=1}^{n}{\chi _{(-\infty ,x]}(X_{k}(\omega ))}\;\;(x\in \mathbb {R} ,\omega \in \Omega ,)}$   .

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

${\displaystyle D_{n}\colon \Omega \to \mathbb {R} }$

With

${\displaystyle D_{n}(\omega )=\sup _{x\in \mathbb {R} }{\bigl |}F_{n}(x,\omega )-F(x){\bigr |}}$   ,

which indicates the upper limit of all distances of this empirical distribution from the common distribution , taking into account all possible characteristics . ${\displaystyle F}$ ${\displaystyle x\in \mathbb {R} }$

Then:

They converge with probability 1, almost certainly , to zero. ${\displaystyle D_{n}}$

So it applies

${\displaystyle \operatorname {P} {\bigl (}\lim _{n\to \infty }D_{n}=0{\bigr )}=1}$   .

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

• FP Cantelli : Sulla determinazione empirica delle leggi di probabilità . In: Giornale dell'Istituto Italiano degli Attuari . tape 4 , 1933, pp. 421-424 .
• V. Glivenko : Sulla determinazione empirica delle leggi di probabilità . In: Giornale dell'Istituto Italiano degli Attuari . tape 4 , 1933, pp. 92-99 .
• Flemming Topsøe: On the Glivenko-Cantelli theorem . In: Journal of Probability Theory and Related Areas . tape 14 , 1970, pp. 239-250 , doi : 10.1007 / BF01111419 ( MR0292143 ).

Monographs

• Krishna B. Athreya , Soumendra N. Lahiri : Measure Theory and Probability Theory (=  Springer Texts in Statistics ). Springer Verlag , New York 2006, ISBN 978-0-387-32903-1 . MR2247694
• Kai Lai Chung : A Course in Probability Theory . Academic Press, Inc. , San Diego (et al.) 2001, ISBN 0-12-174151-6 . MR1796326
• Marek Fisz : Probability calculation and mathematical statistics (=  university books for mathematics . Volume 40 ). 10th edition. VEB Deutscher Verlag der Wissenschaften , Berlin 1980.
• P. Gänssler , W. Stute : Probability Theory (=  university text . Volume 91 ). Springer Verlag, Berlin, Heidelberg, New York 1977, ISBN 3-540-08418-5 . MR0501219
• Boris Vladimirovich Gnedenko : Textbook of the theory of probability . Verlag Harri Deutsch , Thun, Frankfurt am Main 1997, ISBN 3-8171-1531-8 .
• Achim Klenke : Probability Theory . 3rd, revised and expanded edition. Springer Spectrum , Berlin, Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-6 .
• Norbert Kusolitsch : Measure and probability theory . An introduction (=  Springer textbook ). 2nd, revised and expanded edition. Springer-Verlag, Berlin, Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-322-96418-2 .
• RG Laha , VK Rohatgi : Probability Theory (=  Wiley Series in Probability and Mathematical Statistics ). John Wiley & Sons , New York (et al.) 1979, ISBN 0-471-03262-X . MR0534143
• M. Loève : Probability Theory I (=  Graduate Texts in Mathematics . Volume 45 ). 4th edition. Springer Verlag, Berlin, Heidelberg 1977, ISBN 3-540-90210-4 . MR0651017
• Klaus D. Schmidt : Measure and probability (=  Springer textbook ). Springer Verlag, Berlin, Heidelberg 2009, ISBN 978-3-540-89729-3 .
• Vladimir Spokoiny , Thorsten Dickhaus : Basics of Modern Mathematical Statistics (=  Springer Texts in Statistics ). Springer-Verlag, Heidelberg, New York, Dordrecht, London 2015, ISBN 978-3-642-39908-4 . MR3289985
• Walter Vogel : Probability Theory (=  Studia Mathematica . Volume XXII ). Vandenhoeck & Ruprecht , Göttingen 1970. MR0286145

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. ${\displaystyle \chi }$
9. ↑ It stands for the supremum . ${\displaystyle \sup }$
10. ^ Flemming Topsøe: On the Glivenko-Cantelli theorem. in: Z. Probability Theory and Related Areas 14, pp. 239 ff