# 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. The reasoning behind this is that any sample from the process must represent the average statistical properties of the entire process, so that regardless what sample you choose, it represents the whole process, and not just that section of the process. A process that changes erratically at an inconsistent rate is not said to be ergodic.[1]

## 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:

$\mu_X(t)= E[X(t)]=\mu_X$,

$r_X(\tau) = E[(X(t)-\mu_X(t)) (X(t+\tau)-\mu_X(t+\tau))]$,

which do not change with time $t$.

One way to estimate the mean is to perform a time average:

$\hat{\mu}_X(t)_{T} = \frac{1}{T} \int_{0}^{T} X(t) \, dt.$

If $\hat{\mu}_X(t)_{T}$ converges in squared mean to the true value $\mu_X(t)$ as $T \rightarrow \infty$, then the process $X(t)$ is said to be mean-ergodic[2] or mean-square ergodic in the first moment.[3]

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

$\hat{r}_X(\tau)_{T} = \frac{1}{T} \int_{0}^{T} [X(t+\tau)-\mu_X(t+\tau)] [X(t)-\mu_x(t)] \, dt.$

Again, if $\hat{r}_X(\tau)_{T}$ converges in squared mean to the true autocovariance $r_X(\tau) = E[(X(t+\tau)-\mu_X(t+\tau)) (X(t)-\mu_X(t))]$, then the process is said to be autocovariance-ergodic or mean-square ergodic in the second moment.[3]

A process which is ergodic in the first and second moments is sometimes called ergodic in the wide sense.[3]

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