Ottaviani-Skorokhod inequality

The inequality of Ottaviani-Skorokhod is a stochastic inequality within the area of the calculation of probabilities , which at the two mathematician Giuseppe Ottaviani and Anatoly Skorokhod back. It refers to finite families of stochastically independent real random variables and is a useful tool for proofs related to the strong law of large numbers .

Formulation of the inequality

Following the presentation by Heinz Bauer , the inequality can be stated as follows:

A probability space and a finite number of independent random variables are given ${\displaystyle (\Omega ,{\mathcal {A}},\operatorname {P} )}$ ${\displaystyle X_{1},\ldots ,X_{n}\colon (\Omega ,{\mathcal {A}},\operatorname {P} )\to \mathbb {R} \;(n\in \mathbb {N} )}$

Be here for ${\displaystyle i=1,\ldots ,n}$

${\displaystyle S_{i}=\sum _{j=1}^{i}{X_{j}}}$

set.

Then for each index and for two real numbers and ${\displaystyle m=1,\ldots ,n}$ ${\displaystyle \epsilon >0}$ ${\displaystyle \eta >0}$

the inequality

${\displaystyle \operatorname {P} {\bigl (}\max _{k=m,\ldots ,n}{|S_{k}|}>\epsilon +\eta {\bigr )}\cdot \min _{k=m,\ldots ,n}{\operatorname {P} {\bigl (}|S_{n}-S_{k}|\leq \epsilon {\bigr )}}\;\leq \;\operatorname {P} {\bigl (}|S_{n}|>\eta {\bigr )}}$

Fulfills.

Corollaries: A Lévy Theorem and other corollaries

With the inequality of Ottaviani-Skorokhod the following theorem of the French mathematician Paul Lévy can be derived and some corollaries can be derived.

Levy's theorem says:

For every independent sequence of real random variables , the stochastic convergence of the series results in   the almost certain convergence of this series. ${\displaystyle {\bigl (}X_{n}{\bigr )}_{n\in \mathbb {N} }}$ ${\displaystyle \sum _{n=1}^{\infty }{X_{n}}}$

This gives the following corollary:

Is an independent sequence of real random variables with ${\displaystyle {\bigl (}X_{n}{\bigr )}_{n\in \mathbb {N} }}$

(1) ${\displaystyle \sum _{n=1}^{\infty }{\operatorname {E} {\bigl (}{X_{n}}^{2}{\bigr )}}<\infty }$

(2) ${\displaystyle \operatorname {E} (X_{n})=0\;\;(n\in \mathbb {N} )}$

so the series is almost certainly convergent. ${\displaystyle \sum _{n=1}^{\infty }{X_{n}}}$

From this corollary one can then directly obtain Kolmogoroff's criterion for the strong law of large numbers using Kronecker's lemma :

Is an independent sequence of integrable real random variables with ${\displaystyle {\bigl (}X_{n}{\bigr )}_{n\in \mathbb {N} }}$

(*) ${\displaystyle \sum _{n=1}^{\infty }{\frac {\operatorname {Var} {\bigl (}X_{n}{\bigr )}}{n^{2}}}<\infty }$

so the sequence satisfies the strong law of large numbers.

Remarks

1. The inequality of Ottaviani Skorokhod (and variations thereof) connect some authors only the name of Giuseppe Ottaviani and refer to them as inequality of Ottaviani or as ottavianische inequality ( English Ottaviani's inequality ). Often the case is dealt with on its own . ${\displaystyle \epsilon =\eta }$
2. In the university text by Peter Gänssler and Winfried Stute , the inequality appears (in a different and even more general version) as the Skorokhod inequality .
3. The above representation of the inequality, which is based on independent real random variables, can also be formulated in a corresponding manner (for example) for independent Borel measurable random variables with values in a separable Banach space . The norm of the Banach space takes the place of the above absolute value function .

literature

Original work

• Nasrollah Etemadi : Maximal inequalities for partial sums of independent random vectors with multi-dimensional time parameters . In: Communications in Statistics. Theory and Methods . tape 20 , 1991, p. 3909-3923 ( MR1158554 ).
• G. Ottaviani : Sulla teoria astratta del calcolo delle probabilità proposita dal Cantelli . In: Giornale dell'Istituto Italiano degli Attuari . tape 10 , 1939, pp. 10-40 .

Monographs

• Heinz Bauer: Probability Theory (=  De Gruyter textbook ). 5th, revised and improved edition. de Gruyter, Berlin / New York 2002, ISBN 3-11-017236-4 ( MR1902050 ).
• P. Gänssler, W. Stute: Probability Theory (=  university text . Volume 91 ). Springer Verlag, Berlin / Heidelberg / New York 1977, ISBN 3-540-08418-5 ( MR0501219 ).
• J. Hoffmann-Jørgensen: Probability with a View toward Statistics . Volume I (=  Chapman & Hall Probability Series . Volume 91 ). Chapman & Hall, New York 1994, ISBN 0-412-05221-0 ( MR1278485 ).
• Oleg Klesov: Limit Theorems for Multi-Indexed Sums of Random Variables (=  Probability Theory and Stochastic Modeling ). Springer Verlag, Heidelberg / New York / Dordrecht / London 2014, ISBN 978-3-662-44387-3 , doi : 10.1007 / 978-3-662-44388-0 ( MR3244237 ).
• 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 .
• Michel Ledoux , Michel Talagrand : Probability in Banach Spaces . Isoperimetry and Processes (=  results of mathematics and its border areas (3rd part) . Volume 23 ). Springer Verlag, Berlin (inter alia) 1991, ISBN 3-540-52013-9 ( MR1102015 ).
• AN Širjaev : Probability (=  university books for mathematics . Volume 91 ). VEB Deutscher Verlag der Wissenschaften, Berlin 1988, ISBN 3-326-00195-9 ( MR0967761 ).

References and comments

1. ↑ ^{a b c} Heinz Bauer: Probability Theory. 2002, pp. 107-113
2. ↑ The real amount function is denoted by. ${\displaystyle |\cdot |}$
3. ↑ Kolmogoroff's criterion is often referred to as Kolmogoroff's First Law of Large Numbers . Cf. Norbert Kusolitsch: Measure and probability theory: An introduction. 2014, p. 251!
4. ^ J. Hoffmann-Jørgensen: Probability with a View toward Statistics. 1994, pp. 472-473
5. ↑ Oleg Klesov: Limit Theorems for Multi-Indexed Sums of Random Variables. 2014, pp. 30–31
6. ↑ TO Širjaev: Probability. 1988, p. 491
7. ↑ P. Gänssler, W. Stute: Probability Theory. 1977, p. 101
8. ↑ Michel Ledoux, Michel Talagrand: Probability in Banach Spaces. 1991, pp. 151-152