Ergodic process

In signal processing, a stochastic process is said to be ergodic if its statistical properties (such as its mean and variance) can be deduced from a single, sufficiently long sample (realization) of the process.

Specific definitions

One can discuss the ergodicity of various properties of a stochastic process. For example, a wide-sense stationary process $x(t)$ has mean $mu\; =\; E\; [x(t)]$ and autocovariance $r\_x(\; au)\; =\; E\; [(x(t)-mu)\; (x(t+\; au)-mu)]$ which do not change with time. One way to estimate the mean is to perform a time average:

:$hat\{mu\}\_T\; =\; frac\{1\}\{2T\}\; int\_\{-T\}^\{T\}\; x(t)\; ,\; dt.$

If $hat\{mu\}\_T$ converges in squared mean to $mu$ as $T\; ightarrow\; infty$, then the process $x(t)$ is said to be mean-ergodic [Papoulis, p.428] or mean-square ergodic in the first moment.Porat, p.14]

Likewise, one can estimate the autocovariance $r\_x(\; au)$ by performing a time average:

:$hat\{r\}\_x(\; au)\; =\; frac\{1\}\{2T\}\; int\_\{-T\}^\{T\}\; [x(t+\; au)-mu]\; [x(t)-mu]\; ,\; dt.$

If this expression converges in squared mean to the true autocovariance $r\_x(\; au)\; =\; E\; [(x(t)-mu)\; (x(t+\; au)-mu)]$, then the process is said to be autocovariance-ergodic or mean-square ergodic in the second moment.

A process which is ergodic in the first and second moments is sometimes called ergodic in the wide sense.

See also

* Ergodic theory, a branch of mathematics concerned with a more general formulation of ergodicity
* Ergodic hypothesis
* Ergodic (adjective)

Notes

References

* cite book
last = Porat
first = B.
title = Digital Processing of Random Signals: Theory & Methods
date = 1994
publisher = Prentice Hall
isbn = 0130637513
pages = 14
* cite book
author=Papoulis, Athanasios
title=Probability, random variables, and stochastic processes
publisher=McGraw-Hill
location=New York
year=1991
pages=427-442
isbn=0-07-048477-5
oclc=
doi=
accessdate=