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 . ${\displaystyle {\mathcal {L}}^{p}}$

statement

A dimensionally conserving dynamic system and a measurable function are given . Besides, be ${\displaystyle (X,{\mathcal {A}},\mu ,T)}$ ${\displaystyle f\colon X\to \mathbb {R} }$

${\displaystyle s_{n}:=\sum _{i=0}^{n-1}f\circ T^{i}}$

the sum of the first iterations and ${\displaystyle n}$

${\displaystyle m_{n}:=\max\{0,s_{1},\dots ,s_{n}\}}$

the maximum of these sums. Then applies

${\displaystyle \int _{\{m_{n}>0\}}f\mathrm {d} \mu \geq 0}$

for everyone . ${\displaystyle n\in \mathbb {N} }$

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 ${\displaystyle \tau }$ ${\displaystyle X=(X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X_{0}}$

${\displaystyle S_{n}:=\sum _{i=0}^{n-1}X_{i}}$

and

${\displaystyle M_{n}:=\max\{0,S_{1},\dots ,S_{n}\}}$,

that

${\displaystyle \operatorname {E} (X_{0}\chi _{\{M_{n}>0\}})\geq 0}$

is. To get this, one sets and due to the shift operator then applies . Thus corresponds to that in the formulation above. ${\displaystyle f=X_{0}}$ ${\displaystyle X_{n}=X_{0}(\tau ^{n})}$ ${\displaystyle \tau }$ ${\displaystyle T}$

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 .