In econometrics and signal processing, a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long, random sample of the process. The reasoning is that any collection of random samples from a process must represent the average statistical properties of the entire process. In other words, regardless of what the individual samples are, a birds-eye view of the collection of samples must represent the whole process. Conversely, a process that is not ergodic is a process that changes erratically at an inconsistent rate.
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 process 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 ergodic in mean if
converges in squared mean to the ensemble average , as .
Examples of non-ergodic random processes
- An unbiased random walk  is non-ergodic. Its expectation value is zero at all times, whereas its time average is a random variable with divergent variance.
- Suppose that we have two coins: one coin is fair and the other has two heads. We choose (at random) one of the coins first, and then perform a sequence of independent tosses of our selected coin. Let X[n] denote the outcome of the nth toss, with 1 for heads and 0 for tails. Then the ensemble average is ½ (½ + 1) = ¾; yet the long-term average is ½ for the fair coin and 1 for the two-headed coin. So the long term time-average is either 1/2 or 1. Hence, this random process is not ergodic in mean.
- Ergodic hypothesis
- Ergodic theory, a branch of mathematics concerned with a more general formulation of ergodicity
- Loschmidt's paradox
- Poincaré recurrence theorem