L ^p -Erod set

The L ^p ergo set , also known as the statistical ergodic set , is a central set of ergodic theory , a sub-area of ​​mathematics that lies in the area between measure theory , theory of dynamic systems and probability theory. It deals with the circumstances under which, when a mapping is iterated, the mean values ​​over the iterations agree with the mean values ​​of the function. In contrast to the individual ergodic set , the -ergodic set deals with the convergence in the p-th mean and not with the almost certain convergence . The theorem was proven by John von Neumann in 1930/31 , but was not published until 1932. A compact proof is possible, for example, by means of Hopf's maximal ergodic lemma and the individual ergodic set. The theorem can also be formulated more generally on Hilbert spaces with isometric operators and norm convergence. ${\displaystyle {\mathcal {L}}^{p}}$

statement

A probability space and a measure-preserving mapping as well as the σ-algebra of the T-invariant events are given . Let the space of all -fold Lebesgue-integrable functions (see also L ^p -space ) be denoted by, as well as the conditional expectation of with respect to the σ-algebra . ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle T:\Omega \to \Omega }$ ${\displaystyle {\mathcal {I}}}$ ${\displaystyle {\mathcal {L}}^{p}(\Omega ,{\mathcal {A}},P)}$ ${\displaystyle p}$ ${\displaystyle {\mathcal {L}}^{p}}$ ${\displaystyle \operatorname {E} (g|{\mathcal {S}})}$ ${\displaystyle g}$ ${\displaystyle {\mathcal {S}}}$

Is , then for all that even in lies and ${\displaystyle p\in [1,\infty )}$ ${\displaystyle f\in {\mathcal {L}}^{p}}$ ${\displaystyle \operatorname {E} (f|{\mathcal {I}})}$ ${\displaystyle {\mathcal {L}}^{p}}$

${\displaystyle \lim _{n\to \infty }{\biggl \Vert }{\frac {1}{n}}\sum _{i=0}^{n-1}f\circ T^{i}-\operatorname {E} (f|{\mathcal {I}}){\biggr \Vert }_{{\mathcal {L}}^{p}}=0}$.

Here the L ^p standard denotes . ${\displaystyle \|\cdot \|_{{\mathcal {L}}^{p}}}$

If P-trivial (or, equivalently, an ergodic transformation ), then and accordingly ${\displaystyle {\mathcal {I}}}$ ${\displaystyle T}$ ${\displaystyle \operatorname {E} (f|{\mathcal {I}})=\operatorname {E} (f)}$

${\displaystyle \lim _{n\to \infty }{\biggl \Vert }{\frac {1}{n}}\sum _{i=0}^{n-1}f\circ T^{i}-\operatorname {E} (f){\biggr \Vert }_{{\mathcal {L}}^{p}}=0}$.

The mean values ​​of the iterated maps thus converge in the p-th mean to the (conditional) expected value.

Application in stochastics

The -Ergode theorem can be applied to stochastic processes as follows : To do this, one considers a canonical process on the probability space , where a Polish space is, for example, a finite or countably infinite set or the . The transformation is then defined as the shift that is given by ${\displaystyle {\mathcal {L}}^{p}}$ ${\displaystyle X=(X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (\Omega ,{\mathcal {A}},P)=(E^{\mathbb {N} },{\mathcal {B}}(E)^{\otimes E},P)}$ ${\displaystyle E}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \tau :\Omega \to \Omega }$

${\displaystyle \tau ((\omega _{n})_{n\in \mathbb {N} })=(\omega _{n+1})_{n\in \mathbb {N} }}$.

For the stochastic process the following applies and is a dimensionally maintaining dynamic system if and only if there is a stationary stochastic process . ${\displaystyle X_{n}(\omega )=X_{0}(\tau ^{n}(\omega ))}$ ${\displaystyle (\Omega ,{\mathcal {A}},P,\tau )}$ ${\displaystyle X}$

If one now sets , where should be, as well as , it follows that for stationary processes ${\displaystyle f=X_{0}}$ ${\displaystyle X_{0}\in {\mathcal {L}}^{p}}$ ${\displaystyle \tau =T}$

${\displaystyle \lim _{n\to \infty }{\biggl \Vert }{\frac {1}{n}}\sum _{i=0}^{n-1}X_{i}-\operatorname {E} (X_{0}|{\mathcal {I}}){\biggr \Vert }_{{\mathcal {L}}^{p}}=0}$

applies. If again a P-trivial σ-algebra (or an ergodic transformation or an ergodic stochastic process ), then it follows exactly as above that is. ${\displaystyle {\mathcal {I}}}$ ${\displaystyle \tau }$ ${\displaystyle X}$ ${\displaystyle \operatorname {E} (X_{0}|{\mathcal {I}})=\operatorname {E} (X_{0})}$

Individual evidence

1. ↑ Achim Klenke: Probability Theory. 2013, p. 454

Web links

• DV Anosov: Von Neumann ergodic theorem . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Manfred Einsiedler , Klaus Schmidt : Dynamic Systems . Ergodic theory and topological dynamics. Springer, Basel 2014, ISBN 978-3-0348-0633-6 , doi : 10.1007 / 978-3-0348-0634-3 .
• 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 .