Mourier's theorem

The set of Mourier is a theorem of probability theory , one of the branches of mathematics . It goes back to the French mathematician Édith Mourier and formulates a sufficient condition for the existence of the strong law of large numbers for certain sequences of random elements in a separable Banach space over the field of real numbers . The theorem can be understood as a generalization of Kolmogorov's second law of large numbers .

Formulation of the sentence

The sentence can be stated as follows:

Let a probability space , a separable -additional space and a sequence be given ${\displaystyle (\Omega ,{\mathcal {A}},\operatorname {P} )}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle (X,\|{\cdot }\|)}$

${\displaystyle {\xi }_{n}\colon (\Omega ,{\mathcal {A}},\operatorname {P} )\to (X,\|{\cdot }\|)\;(n\in \mathbb {N} )}$

of random elements in . ${\displaystyle X}$

The sequence is stochastically independent and its terms are distributed identically . ${\displaystyle {\xi }_{n}\;(n\in \mathbb {N} )}$

This applies

${\displaystyle \operatorname {E} {\bigl (}\|{\xi }_{1}\|{\bigr )}<{\infty }}$   .

Then - almost certainly the convergence ${\displaystyle \operatorname {P} }$

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{j=1}^{n}\xi _{j}=\operatorname {E} {\bigl (}{\xi }_{1}{\bigr )}}$   .

Explanations

• A Borel - measurable random variable with values in a topological space is generally called random element referred to. ${\displaystyle \xi \colon (\Omega ,{\mathcal {A}},\operatorname {P} )\to (X,{\mathcal {B}}(X))}$ ${\displaystyle X}$
• In a random element with values in a separable normalized - vector space is with the always expected value referred to, if it is defined. He is at least always defined when for the Pettis integral exists. If this is the case, the expected value is equal to the Pettis integral. The expected value is characterized by the fact that it always holds for continuous linear forms . ${\displaystyle \xi }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle (X,\|{\cdot }\|)}$ ${\displaystyle \operatorname {E} {\bigl (}\xi {\bigr )}}$ ${\displaystyle \xi }$ ${\displaystyle \operatorname {E} {\bigl (}\xi {\bigr )}}$ ${\displaystyle f\colon (X,\|{\cdot }\|)\to \mathbb {R} }$ ${\displaystyle \operatorname {E} {\bigl (}f\circ \xi {\bigr )}=f(\operatorname {E} {\bigl (}\xi {\bigr )})}$
• For a random element with values ​​in a separable -additional space there is always a nonnegative real random variable for which the expected value always exists. If there is even then the expected value also exists . ${\displaystyle \xi }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle (X,\|{\cdot }\|)}$ ${\displaystyle \|\xi \|\colon (\Omega ,{\mathcal {A}},\operatorname {P} )\to \mathbb {R} \;,\;\omega \mapsto {\|\xi (\omega )\|}\;(\omega \in \Omega )}$ ${\displaystyle \operatorname {E} {\bigl (}\|\xi \|{\bigr )}\in [0\;,\;{\infty }]}$ ${\displaystyle \operatorname {E} {\bigl (}\|\xi \|{\bigr )}<{\infty }}$ ${\displaystyle \operatorname {E} {\bigl (}\xi {\bigr )}}$

Related result in connection with Kolmogorov's first law of large numbers

Based on Mourier's theorem, the question arises whether and to what extent Kolmogorov's first law of large numbers should also be extended to sequences of random elements in standardized vector spaces. As can be shown, this expansion is at least always possible in the case of the separable Hilbert spaces . The following theorem applies:

Let a probability space , a separable -Hilbert space and a sequence be given ${\displaystyle (\Omega ,{\mathcal {A}},\operatorname {P} )}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle H=(H,\langle \cdot ,\cdot \rangle ,{\|{\cdot }\|})}$

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

of Pettis-integrable random elements in . ${\displaystyle H}$

The consequence is stochastically independent and it applies

${\displaystyle \sum _{n=1}^{\infty }\;{\frac {\operatorname {E} {\bigl (}{\|{\xi }_{n}-\operatorname {E} ({\xi }_{n})\|}^{2}{\bigr )}}{n^{2}}}<{\infty }}$   .

Then the consequence of the condition is sufficient

${\displaystyle \operatorname {P} \left(\lim _{n\to \infty }{\frac {\|\sum _{j=1}^{n}{{\bigl (}{\xi }_{j}-{\operatorname {E} ({\xi }_{j})}{\bigr )}\|}}{n}}=0\right)=1}$

and with it the strong law of large numbers.

Sources and background literature

• P. Gänssler , W. Stute : Probability Theory (=  university text . Volume 91 ). Springer Verlag, Berlin, Heidelberg, New York 1977, ISBN 3-540-08418-5 . MR0501219
• 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
• 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
• Édith Mourier: Eléments aléatoires dans un espace de Banach . In: Annales de l'Institut Henri Poincaré . tape 13 , 1953, pp. 161-244 . MR0064339
• Pál Révész : The Laws of Large Numbers (=  textbooks and monographs from the field of exact sciences: Mathematical series . Volume 35 ). Birkhäuser Verlag , Basel, Stuttgart 1968. MR0245080
• NN Vakhania , VI Tarieladze , SA Chobanyan : Probability Distributions on Banach Spaces (=  Mathematics and its Applications (Soviet Series) . Volume 14 ). D. Reidel Publishing Company , Dordrecht, Boston, Lancaster, Tokyo 1987, ISBN 90-277-2496-2 .

References and footnotes

1. ↑ P. Gänssler, W. Stute: Probability Theory. 1977, pp. 337-338
2. ^ RG Laha, VK Rohatgi: Probability Theory. 1979, pp. 452-454
3. ^ Pál Révész: The Laws of Large Numbers. 1968, pp. 146-147
4. ↑ P. Gänssler, W. Stute: Probability Theory. 1977, p. 335
5. ^ RG Laha, VK Rohatgi: Probability Theory. 1979, p. 447
6. ↑ P. Gänssler, W. Stute: Probability Theory. 1977, p. 336
7. ^ RG Laha, VK Rohatgi: Probability Theory. 1979, p. 455
8. ↑ Here, on the Hilbert space by the inner product generated standard . ${\displaystyle h\mapsto {\|h\|}={\sqrt {\langle h,h\rangle }}}$
9. ↑ The last mentioned condition corresponds to the variance condition known from the case of real random variables.