Scorochod representation

The Skorochod representation , also called Skorochod coupling or referred to as the representation theorem of Skorochod , is a statement of stochastics about the convergence in distribution or the weak convergence of probability measures and their link to almost certain convergence . It is named after Anatoly Skorochod , but due to the different transcriptions of his name into different languages ​​it can also be found in the literature in the spelling Skorokhod or Skorohod . The proof of the representation theorem is a classic example of a coupling argument .

statement

Given are random variables with values ​​in a Polish space , provided with Borel's σ-algebra. A typical case would be, for example . Furthermore applies ${\displaystyle X,X_{1},X_{2},X_{3},\dots }$ ${\displaystyle (E,{\mathcal {B}}(E))}$ ${\displaystyle E=\mathbb {R} }$

${\displaystyle X_{n}{\stackrel {\mathcal {D}}{\rightarrow }}X}$,

the random variables thus converge in distribution.

Then the following applies: There is a probability space and random variables ${\displaystyle (\Omega ,{\mathcal {A}},P)}$

${\displaystyle Y,Y_{i}:(\Omega ,{\mathcal {A}},P)\to (E,{\mathcal {B}}(E))}$,

so that

1. the distributions match and ${\displaystyle X_{i}\sim Y_{i}}$ ${\displaystyle Y\sim X}$
2. which almost certainly converge against . ${\displaystyle Y_{i}}$ ${\displaystyle Y}$

variants

The sentence is formulated in different variants. Partly only for real random variables, whereby the convergence in distribution is then defined via the distribution functions. In part, the convergence in distribution is also formalized as a weak convergence of probability measures in the image space.

Web links

• D. Nualart: Skorokhod theorem . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .
• Geoffrey Grimmett, David Stirzaker: Probability and Random Processes . 3. Edition. Oxford University Press, Oxford New York 2001.

Individual evidence

1. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 176.
2. ↑ Klenke: Probability Theory. 2013, p. 392.
3. ↑ Kusolitsch: Measure and probability theory. 2014, p. 289.
4. ↑ Grimmet, Stirzaker: Probability and Random Processes 2001, p. 314