Ergodic Processes
Simple examples for discrete-time ergodic processes are
- i.i.d. random variables,
i.e. sequences of independent and
identically distributed random variables
- finite state Markov processes with irreducible transition matrix
and more simple ones are the processes generated by
- the deterministic logistic map with µ = 4
- the tent map
See Arnold [1,2] and Katok and Hasselblatt [19, p.159f],
listed in the references.
Ergodic sequences possess a canonical representation as an
ergodic dynamical system, i.e. there exists a canonical
probability space (Omega,F,P) where
- Omega is the space of all sequences
- F is the Borel sigma-algebra on Omega
- P is the probability measure on (Omega,F) given by Kolmogorov's
fundamental theorem
such that theta: Omega -> Omega defined by
theta(omega(.)) := omega(.+1) is an ergodic flow.
That is theta is measure-preserving and for each invariant set A
(i.e. theta(A) = A = theta-1(A)) it is P(A) in {0,1}).
The stochastic process thetan, n in Z, is ergodic
and possesses the same distributions than the original process.
For the last two (deterministic) examples, one can also choose
Omega = [0,1]. Then theta is the logistic resp. the tent map.
(For the logistic map, P is equivalent to Lebesgue measure, and
P is the Lebesgue measure for the tent map.)
To the
Introduction
Additive noise case
Multiplicative noise case
Deterministic case
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