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

Then:

Then and only then is the series   almost certain to converge, ${\displaystyle \sum _{n=1}^{\infty }{X_{n}}}$

if a   real number   exists such that the three rows formed for it ${\displaystyle K>0}$

(1) ${\displaystyle \sum _{n=1}^{\infty }{\operatorname {P} {\bigl (}|X_{n}|>K{\bigr )}}}$

(2) ${\displaystyle \sum _{n=1}^{\infty }{\operatorname {E} (Y_{n})}}$

(3) ${\displaystyle \sum _{n=1}^{\infty }{\operatorname {Var} (Y_{n})}}$

in converge, wherein the sequence of random variables is formed by for   ${\displaystyle \mathbb {R} }$ ${\displaystyle Y_{n}\colon (\Omega ,{\mathcal {A}},\operatorname {P} )\to \mathbb {R} \;(n\in \mathbb {N} )}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle Y_{n}(\omega )={\begin{cases}X_{n}(\omega ),&{\text{falls }}|X_{n}(\omega )|\leq K,\\0&{\text{falls }}X_{n}(\omega )>K\end{cases}}}$

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

1. ↑ Achim Klenke: Probability Theory. 2013, pp. 332–333.
2. ^ Krishna B. Athreya, Soumendra N. Lahiri: Measure Theory and Probability Theory. 2006, p. 249 ff.
3. ^ Kai Lai Chung: A Course in Probability Theory. 2001, p. 125 ff.
4. ^ Marek Fisz: Probability calculation and mathematical statistics. 1976, p. 294.
5. ↑ A. Kolmogoroff: Basic concepts of the calculation of probability. 1973, pp. 59-60.
6. ^ RG Laha, VK Rohatgi: Probability Theory. 1979, pp. 88-89.
7. ↑ For an integrable real random variable , the expected value of and the variance of is denoted by. ${\displaystyle X}$ ${\displaystyle \operatorname {E} (X)}$ ${\displaystyle X}$ ${\displaystyle \operatorname {Var} (X)}$ ${\displaystyle X}$
8. ↑ NN Vakhania, VI Tarieladze, SA Chobanyan: Probability Distributions on Banach Spaces. 1987, p. 289 ff.