Ergodic theory

Ergodicity is a term used within the mathematical subdivision of stochastics. The statistics of a process is described by a model function.

Preparations

This is called a probability space is a measurable mapping T maßerhaltend, if the size of P under T P is again, that is, for all sets A of the σ - algebra. Furthermore ie a set A T- invariant if.

Definition

A maßerhaltende transformation is now called ergodic if for all T- invariant sets A that.

Applications

Mathematically, the Birkhoff ergodic theorem for ergodic Maßtransformationen a variant of the strong law of large numbers dar. This also dependent random variables may well be considered.

Ergodicity in time series analysis

For statistical inference in time-series assumptions must be made, because in practice usually there is only one realization of the time series generating process. The assumption of ergodicity means that sample moments, which are obtained from a finite time series converge for almost against the moments of the population. For constant and: mean- ergodic:

Variance ergodic: These properties dependent random variables can not be proven empirically and must therefore be assumed. Thus, a stochastic process can be ergodic, it must be located in a statistical equilibrium, ie it must be stationary.

Ergodicity in the ergodic hypothesis

A statistical system selects a defined number space of random numbers. The system thereby achieves all of the available number space and perform the probability is for any number of equal size.