Specific Definitions
One can discuss the ergodicity of various properties of a stochastic process. For example, a wide-sense stationary process has mean and autocovariance which do not change with time. One way to estimate the mean is to perform a time average:
If converges in squared mean to as, then the process is said to be mean-ergodic or mean-square ergodic in the first moment.
Likewise, one can estimate the autocovariance by performing a time average:
If this expression converges in squared mean to the true autocovariance, 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.
Read more about this topic: Ergodic Process
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