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.
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Specific definitions
One can discuss the ergodicity of various statistics of a stochastic process. For example, a wide-sense stationary process
and autocovariance
that depends only on the lag
The process
converges in squared mean to the ensemble average
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
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
A discrete-time random process
converges in squared mean to the ensemble average
Example of a non-ergodic random process
Suppose that we have two coins: one coin is fair and the other has two heads. We choose (at random) one of the coins, 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. Hence, this random process is not ergodic in mean.