Individual ergodic set
The individual ergodic set is an important set of ergodic theory , a branch of mathematics in the border area between stochastics and the theory of dynamic systems . Alternatively, the individual ergodic set is also called the Birkhoff ergodic set or point-wise ergodic set . It provides a form of the strong law of large numbers for dependent random variables and provides the mathematical basis of the ergodic hypothesis of statistical physics . The theorem was proven in 1931 by George David Birkhoff , after whom it is also named. A compact proof is possible using Hopf's maximal ergodic lemma . In addition, the can -Ergodensatz be derived without much effort from the individual ergodic theorem.
statement
Let it be an integrable random variable (i.e. it has a finite expectation value ) and a measure-preserving transformation on the underlying probability space (i.e. for all in ). Then the means converge
for almost certain against a random variable .
can be chosen measurably with respect to the σ-algebra generated by the -invariant sets (i.e. ) and can be represented as a conditional expected value .
If is ergodic , then is almost certainly constant equal to the expected value of .
The example of a stationary process
The random variables ( ) form a stationary stochastic process , i. H. is distributed as . Conversely, any stationary stochastic process can be represented in this way if one assumes that and is of the form . (If this is not the case, can the pictorial space with the size of place of and consider.) Here , and the left shift , the on displayed is the maßerhaltende transformation.
If they have a finite expectation, it converges according to the Ergod theorem
for almost certain against a random variable . This is everyone's conditional expectation . When ergodicity is present, it is almost certainly constant; H.
- almost certain ( any).
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 .
Web links
- DV Anosov: Birkhoff ergodic theorem . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- Eric W. Weisstein : Birkhoff's Ergodic Theorem . In: MathWorld (English).
- Vitaly Bergelson: History of the Ergodic Theorem
Individual evidence
- ^ GD Birkhoff : Proof of the ergodic theorem , (1931), Proc Natl Acad Sci USA, 17 pp. 656-660. pdf. At: PNAS.org