Lyapunov's inequality

The inequality of Lyapunov is an elementary stochastic inequality , which the Russian mathematician Aleksandr Lyapunov back. It represents an isotonic property of the absolute moments of real random variables and can be derived using the Jensen inequality for expected values .

Formulation of the inequality

Following the presentation by AN Širjaev and Marek Fisz , the Lyapunov inequality can be summarized as follows:

A probability space and a real random variable are given   . ${\displaystyle (\Omega ,{\mathcal {A}},\operatorname {P} )}$ ${\displaystyle X\colon (\Omega ,{\mathcal {A}},\operatorname {P} )\to \mathbb {R} }$

Then   the inequality always holds for two real numbers     and     with   ${\displaystyle s}$ ${\displaystyle t}$ ${\displaystyle 0<s\leq t}$

${\displaystyle {\operatorname {E} {{\bigl (}|X|^{s}{\bigr )}}}^{\frac {1}{s}}\;\leq \;{\operatorname {E} {{\bigl (}|X|^{t}{\bigr )}}}^{\frac {1}{t}}}$  .

In particular, one always has the chain of inequalities

${\displaystyle \operatorname {E} {{\bigl (}|X|{\bigr )}}\;\leq \;{\operatorname {E} {{\bigl (}|X|^{2}{\bigr )}}}^{\frac {1}{2}}\;\leq \;{\operatorname {E} {{\bigl (}|X|^{3}{\bigr )}}}^{\frac {1}{3}}\;\leq \;\cdots \;\leq \;{\operatorname {E} {{\bigl (}|X|^{n}{\bigr )}}}^{\frac {1}{n}}\;(n\in \mathbb {N} )}$  .

Different representation

For the Lyapunov inequality there is also the following more general representation:

For a real random variable of a probability space   . ${\displaystyle X}$ ${\displaystyle (\Omega ,{\mathcal {A}},\operatorname {P} )}$

and for nonnegative real numbers     with     the inequality always holds ${\displaystyle a\;,\;b\;,\;c}$ ${\displaystyle a\geq b\geq c}$

${\displaystyle {\operatorname {E} {{\bigl (}|X|^{b}{\bigr )}}}^{(a-c)}\;\leq \;{\operatorname {E} {{\bigl (}|X|^{c}{\bigr )}}}^{(a-b)}\;\cdot \;{\operatorname {E} {{\bigl (}|X|^{a}{\bigr )}}}^{(b-c)}}$  .

There are also other equivalent versions of this representation.

literature

• Heinz Bauer : Probability Theory (=  De Gruyter textbook ). 5th, revised and improved edition. de Gruyter, Berlin / New York 2002, ISBN 3-11-017236-4 ( MR1902050 ).
• Harald Cramér : Mathematical Methods of Statistics (=  Princeton Mathematical Series ). 11th edition. Princeton University Press, Princeton 1966.
• Marek Fisz: Probability calculation and mathematical statistics (=  university books for mathematics . Volume 40 ). 8th edition. VEB Deutscher Verlag der Wissenschaften, Berlin 1976.
• 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 ).
• AM Liapounoff : Nouvelle forme du théorème sur la limite de probabilité . In: Mémoires de l'Académie Impériale des Sciences de Saint Pétersbourg . tape 12 , no. 5 , 1901.
• AM Liapounoff: Sur une proposition de la theory des probabilités . In: Bulletin de l'Académie impériale des sciences de Saint-Pétersbourg . tape 13 , 1900, p. 359 .
• M. Loève : Probability Theory I (=  Graduate Texts in Mathematics . Volume 45 ). 4th edition. Springer Verlag, Berlin / Heidelberg 1977, ISBN 3-540-90210-4 ( MR0651017 ).
• AN Širjaev: Probability (=  university books for mathematics . Volume 91 ). VEB Deutscher Verlag der Wissenschaften, Berlin 1988, ISBN 3-326-00195-9 ( MR0967761 ).
• JV Uspensky : Introduction to Mathematical Probability . MacGraw-Hill Book Company, New York / London 1937.

Individual evidence

1. ↑ TO Širjaev: Probability. 1988, p. 204
2. ^ Marek Fisz: Probability calculation and mathematical statistics. 1976, pp. 100-101
3. ^ JV Uspensky: Introduction to Mathematical Probability. 1937, p. 265
4. ↑ M. Loève: Probability Theory I. 1977, p. 174
5. ↑ Harald Cramér: Mathematical Methods of Statistics. 1966, p. 255