Ergodic stochastic process

An ergodic stochastic process , short- ergodic process is a special stochastic process of making it possible, terms ergodic theory to the theory of probability to transfer. The time-discrete stochastic process is interpreted as a dynamic system that is created by iterating shift maps and is dimensionally stable under certain conditions .

Ergodic stochastic processes play an important role, since one of them by the individual ergodic theorem and -Ergodensatzes also strong laws of large numbers can derive not only for independent and identically distributed random variables apply. ${\displaystyle {\mathcal {L}}^{p}}$

definition

A canonical process is given on the probability space , with a Polish space such as a finite or countably infinite set or the . The shift is defined by ${\displaystyle X=(X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (\Omega ,{\mathcal {A}},P)=(E^{\mathbb {N} },{\mathcal {B}}(E)^{\otimes \mathbb {N} },P)}$ ${\displaystyle E}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \tau :\Omega \to \Omega }$

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

Thus, a dynamic system applies and is dimensionally accurate 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 an ergodic transformation is now , the σ-algebra of the -invariant events is a P-trivial σ-algebra , then an ergodic stochastic process is called . ${\displaystyle \tau }$ ${\displaystyle \tau }$ ${\displaystyle X}$

example

Independent, identically distributed random variables

Each sequence of independently identically distributed random variables forms an ergodic process. The process is definitely stationary as the distributions are all identical by definition. If the σ-algebra contains the invariant events , then ${\displaystyle A}$ ${\displaystyle {\mathcal {I}}}$

${\displaystyle A=\{\tau ^{n}\in A\}\in \sigma (X_{n},X_{n+1},\dots )}$

and thus is contained in the terminal σ-algebra . However, according to Kolmogorow's zero-one law, this is P-trivial, so P-trivial must also be. The ergodicity of the process follows from this. ${\displaystyle {\mathcal {I}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {I}}}$

Markov chains

Another example of ergodic processes are Markov chains in discrete time and with a countable infinite state space, which start in their invariant distribution and are irreducible and positively recurrent . This is shown by means of the strong Markov property . These Markov chains are thus an example of stochastic processes in which the strong law of large numbers applies due to the ergodic theorems , although stochastic dependencies are present.

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 .
• 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 .