Three-row set
The three-tier set , sometimes called kolmogoroffscher three-tier set ( English Kolmogorov's three-series theorem ) or as a three-tier criterion ( English three-series criterion referred to) is a mathematical theorem in the field of probability theory , which is based on a work of the Russian mathematician Alexander Yakovlevich Khintchine and Andrei Nikolajewitsch Kolmogoroff from 1925. The theorem deals with the question under which conditions a series formed from stochastically independent real random variables almost certainly converges , and traces this question back to the convergence behavior of three associated series of real quantities . It is closely related to the strong law of large numbers .
Formulation of the sentence
The sentence can be stated in modern terms as follows:
- A probability space and a sequence of stochastically independent random variables are given.
- Then:
- Then and only then is the series almost certain to converge,
-
if a real number exists such that the three rows formed for it
- (1)
- (2)
- (3)
-
in converge, wherein the sequence of random variables is formed by for
- is set.
annotation
The three-row theorem can be extended to the case of families of independent Pettis-integrable random variables with values in a separable Hilbert space - like many sentences in the context of the law of large numbers . In this case, the above absolute value function takes the place by the inner product generated the Hilbert space on this standard . Details can be found in the monograph by Vakhania , Tarieladze, and Chobanyan .
application
The convergence of the random harmonic series follows from the three-series theorem .
literature
- 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, San Diego et al. a. 2001, ISBN 0-12-174151-6 ( MR1796326 ).
- Marek Fisz : Probability calculation and mathematical statistics (= university books for mathematics . Volume 40 ). 8th edition. VEB Deutscher Verlag der Wissenschaften, Berlin 1976.
- Achim Klenke : Probability Theory . 3rd, revised and expanded edition. Springer Spectrum, Berlin / Heidelberg 2013, ISBN 978-3-642-36017-6 .
- A. Kolmogoroff : Basic concepts of probability theory . Reprint (= results of mathematics and its border areas . Volume 3 ). Springer Verlag, Berlin / Heidelberg / New York 1973, ISBN 3-540-06110-X ( MR0494348 ).
- A. Kolmogoroff: About the sums by chance of certain independent quantities . In: Mathematical Annals . tape 99 , 1928, pp. 309-319 , doi : 10.1007 / BF01459098 ( MR1512588 ).
- AJ Khintchine, AN Kolmogoroff: On the convergence of series, the terms of which are determined by chance . In: Recueil mathématique de la Société mathématique de Moscou [Matematicheskii Sbornik] . tape 32 , 1925, pp. 668-677 .
- Radha Govinda Laha , Vijay K. Rohatgi : Probability Theory (= Wiley Series in Probability and Mathematical Statistics ). John Wiley & Sons, New York a. a. 1979, ISBN 0-471-03262-X ( MR0534143 ).
- NN Vakhania, VI Tarieladze, SA Chobanyan: Probability Distributions on Banach Spaces (= Mathematics and its Applications ( Soviet Series ) . Volume 14 ). D. Reidel Publishing, Dordrecht / Boston / Lancaster / Tokyo 1987, ISBN 90-277-2496-2 .
References and comments
- ↑ Achim Klenke: Probability Theory. 2013, pp. 332–333.
- ^ Krishna B. Athreya, Soumendra N. Lahiri: Measure Theory and Probability Theory. 2006, p. 249 ff.
- ^ Kai Lai Chung: A Course in Probability Theory. 2001, p. 125 ff.
- ^ Marek Fisz: Probability calculation and mathematical statistics. 1976, p. 294.
- ↑ A. Kolmogoroff: Basic concepts of the calculation of probability. 1973, pp. 59-60.
- ^ RG Laha, VK Rohatgi: Probability Theory. 1979, pp. 88-89.
- ↑ For an integrable real random variable , the expected value of and the variance of is denoted by.
- ↑ NN Vakhania, VI Tarieladze, SA Chobanyan: Probability Distributions on Banach Spaces. 1987, p. 289 ff.