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.

1 Specific definitions
2 See also
3 Notes
4 References

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 $r_x(\tau) = E[(x(t)-\mu) (x(t%2B\tau)-\mu)]$ 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^[1] or mean-square ergodic in the first moment.^[2]

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

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

If this expression converges in squared mean to the true autocovariance $r_x(\tau) = E[(x(t)-\mu) (x(t%2B\tau)-\mu)]$ , then the process is said to be autocovariance-ergodic or mean-square ergodic in the second moment.^[2]

A process which is ergodic in the first and second moments is sometimes called ergodic in the wide sense.^[2]

Notes

^ Papoulis, p.428
^ ^a ^b ^c Porat, p.14

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.

Ergodic process

Contents

Specific definitions

See also

Notes

References