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.
One can discuss the ergodicity of various statistics of a stochastic process. For example, a wide-sense stationary process has constant mean
that depends only on the lag and not on time . The properties and are ensemble averages not time averages.
converges in squared mean to the ensemble average as .
Likewise, the process is said to be autocovariance-ergodic or mean-square ergodic in the second moment if the time average estimate
converges in squared mean to the ensemble average , as . A process which is ergodic in the mean and autocovariance is sometimes called ergodic in the wide sense.
An important example of an ergodic processes is the stationary Gaussian process with continuous spectrum.
Discrete-time random processes
The notion of ergodicity also applies to discrete-time random processes for integer .
A discrete-time random process is erdogic in mean if
converges in squared mean to the ensemble average , as .
Example of non-ergodic random process
Suppose you have two coins, one of which is fair, and the other of which has two heads. You choose (at random) one of the coins and then perform a sequence of independent tosses of that selected coin, and let be 1 for heads and 0 for tails for the th toss. The ensemble average is 1/2 1/2 + 1/2 1 = 3/4, whereas the long-term average is 1/2 for the fair coin and 1 for the two-headed coin. So this random process is not ergodic in mean.
- Poincaré recurrence theorem
- Loschmidt's paradox
- Ergodic theory, a branch of mathematics concerned with a more general formulation of ergodicity
- Ergodic hypothesis
- Porat, B. (1994). Digital Processing of Random Signals: Theory & Methods. Prentice Hall. p. 14. ISBN 0-13-063751-3.
- Papoulis, Athanasios (1991). Probability, random variables, and stochastic processes. New York: McGraw-Hill. pp. 427–442. ISBN 0-07-048477-5.
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