Ergodic process

In econometrics and signal processing, a 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 ${\displaystyle X(t)}$ has constant mean

and

that depends only on the lag ${\displaystyle \tau }$ and not on time ${\displaystyle t}$. The properties ${\displaystyle \mu _{X}}$ and ${\displaystyle r_{X}(\tau )}$ are ensemble averages not time averages.

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