Definition of Ergodic: A stochastic process is ergodic if no sample helps meaningfully to predict values that are very far away in time from that sample. Another way to say that is that the time path of the stochastic process is not sensitive to initial conditions.

Two events A and B (e.g. possible sets of states of the process) are ergodic iff, taking the limit as h goes to infinity:
lim (1/h)SUM_{from i=1}^{to i=h} |Pr(A intersection with L^-iB)-Pr(A)Pr(B)| = 0

Here L is the lag operator. This definition is like that of 'mixing on average'. A stochastic process is ergodic, I believe, if all possible events in it are ergodic by this definition.

If a random process is mixing, it is ergodic.

Priestly, p 340: A process is ergodic iff 'time averages' over a single realization of the process converge in mean square to the corresponding 'ensemble averages' over many realizations.

Example 1: Let x_t (for integer t=0 to infinity) is known to be drawn iid from a standard normal distribution. Then knowing the value of x₁ doesn't help predict the value of x₂, because they are independently drawn. This time series process is ergodic.

Example 2: Suppose the process is x_t=k+sin(t)+e_t where k is unknown and e_t is a white noise error. Then any sample of x_t for a known t gives information about k and that is enough information to make predictions at remote times in the future that are just as good as predictions at nearby times. This process is not ergodic. (Econterms)

Terms related to Ergodic:

• Mixing