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 $r_x(\tau) = E[(x(t)-\mu) (x(t+\tau)-\mu)]$ 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 $T \rightarrow \infty$, then the process $x(t)$ is said to be mean-ergodic or mean-square ergodic in the first moment.

Likewise, one can estimate the autocovariance $r_x(\tau)$ by performing a time average:

$\hat{r}_x(\tau) = \frac{1}{2T} \int_{-T}^{T} [x(t+\tau)-\mu] [x(t)-\mu] \, dt.$

If this expression converges in squared mean to the true autocovariance $r_x(\tau) = E[(x(t)-\mu) (x(t+\tau)-\mu)]$, 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.

An important example of an ergodic processes is the stationary Gaussian process with continuous spectrum.

See also

• Poincaré recurrence theorem
• Loschmidt's paradox
• Ergodic theory, a branch of mathematics concerned with a more general formulation of ergodicity
• Ergodic hypothesis
• Ergodicity

References

• Porat, B. (1994). Digital Processing of Random Signals: Theory & Methods. Prentice Hall. pp. 14. ISBN 0130637513. 
• Papoulis, Athanasios (1991). Probability, random variables, and stochastic processes. New York: McGraw-Hill. pp. 427–442. ISBN 0-07-048477-5.