Ergodic process

In econometrics and 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 $m_x(t)= E[x(t)]$ and autocovariance $r_x(\tau) = E[(x(t)-m_x(t)) (x(t+\tau)-m_x(t+\tau))]$ which do not change with time. One way to estimate the mean is to perform a time average:

$\hat{m}_x(t)_{T} = \frac{1}{2T} \int_{-T}^{T} x(t) \, dt.$

If $\hat{m}_x(t)_{T}$ converges in squared mean to $m_x(t)$ 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:

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

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

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