In physics, statistics, econometrics and signal processing, a stochastic process is said to be in an ergodic regime if an observable's ensemble average equals the time average. In this regime, any collection of random samples from a process must represent the average statistical properties of the entire regime. Conversely, a regime of a process that is not ergodic is said to be in non-ergodic regime. A regime implies a time-window of a process whereby ergodicity measure is applied.

Specific definitions

One can discuss the ergodicity of various statistics of a stochastic process. For example, a wide-sense stationary process X(t) has constant mean and autocovariance that depends only on the lag \tau and not on time t. The properties \mu_X and r_X(\tau) are ensemble averages (calculated over all possible sample functions X), not time averages. The process X(t) is said to be mean-ergodic or mean-square ergodic in the first moment if the time average estimate converges in squared mean to the ensemble average \mu_X as. Likewise, the process is said to be autocovariance-ergodic or d moment if the time average estimate converges in squared mean to the ensemble average r_X(\tau), as. A process which is ergodic in the mean and autocovariance is sometimes called ergodic in the wide sense.

Discrete-time random processes

The notion of ergodicity also applies to discrete-time random processes X[n] for integer n. A discrete-time random process X[n] is ergodic in mean if converges in squared mean to the ensemble average E[X], as.

Examples

Ergodicity means the ensemble average equals the time average. Following are examples to illustrate this principle.

Call centre

Each operator in a call centre spends time alternately speaking and listening on the telephone, as well as taking breaks between calls. Each break and each call are of different length, as are the durations of each 'burst' of speaking and listening, and indeed so is the rapidity of speech at any given moment, which could each be modelled as a random process.

Electronics

Each resistor has an associated thermal noise that depends on the temperature. Take N resistors (N should be very large) and plot the voltage across those resistors for a long period. For each resistor you will have a waveform. Calculate the average value of that waveform; this gives you the time average. There are N waveforms as there are N resistors. These N plots are known as an ensemble. Now take a particular instant of time in all those plots and find the average value of the voltage. That gives you the ensemble average for each plot. If ensemble average and time average are the same then it is ergodic.

Examples of non-ergodic random processes