A random process is said to be ergodic if the time averages of the process tend to the appropriate ensemble averages. This definition implies that with probability 1, any ensemble average of {X(t)} can be determined from a single sample function of {X(t)}. Clearly, for a process to be ergodic, it has to necessarily be stationary. But not all stationary processes are ergodic.

A random process is said to be ergodic if the time averages of the process tend to the appropriate ensemble averages. This definition implies that with probability 1, any ensemble average of {X(t)} can be determined from a single sample function of {X(t)}. Clearly, for a process to be ergodic, it has to necessarily be stationary. But not all stationary processes are ergodic.

I would add to what Raed Ahman has said that the process does not have to be necessarily random. Also, another equivalent characterization of ergodic processes is that the orbits induced by the transformation are dense in the whole space. 

Good afternoon.

This link can be useful

http://economia.unipv.it/pagp/pagine_personali/erossi/rossi_intro_stochastic_process_PhD.pdf

A process in which ensemble average (at a given time) of an observable quantity E=∫O e-βE dp dq / ∫ e-βE dpdq remains equal to the time average t= lim t →∞ (1/t) ∫0t O(t) dt, is an ergodic process.

Notice that ensemble average is a theoretical concept because it depends on the probability distribution function, whereas time average of an observable is directly related to experiments.

It is surprising! Because we are replacing average over time by an instantaneous average over ensemble. This means that with the passage of time the trajectory of the system in phase space should visit all possible points of the ensemble, for the process to be ergodic.

In other words, in a sufficiently long time if all possible points of the ensemble are accessible then (and only then) the time average of a measurable quantity can be replaced by an average over all possible points of the ensemble.

A process in which every sequence or sizable sample is equally representative of the whole (as in regard to a statistical parameter).