Hopf's maximal ergodic lemma
The Hopf'sches maximum Ergodenlemma is a result of ergodic theory , a partial area of mathematics , between measure theory and the dynamic theory of systems is to be settled. Hopf's maximal ergodic lemma can be formulated in two variants, a stochastic and an iterated application of maps. With the exception of the notation, both differ only slightly. The lemma is named after Eberhard Hopf and is an important tool for a compact proof of the individual ergodic set and the ergo set based on it .
statement
A dimensionally conserving dynamic system and a measurable function are given . Besides, be
the sum of the first iterations and
the maximum of these sums. Then applies
for everyone .
Stochastic formulation
The stochastic formulation uses that a stationary stochastic process provided with the shift operator is a dimensionally maintaining dynamic system, cf. this example . Hopf's maximal ergodic lemma then reads as follows: If a real stationary stochastic process is integrable, it follows with
and
- ,
that
is. To get this, one sets and due to the shift operator then applies . Thus corresponds to that in the formulation above.
Web links
- DV Anosov: Maximal ergodic 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 .