ErgodicTheorem

For stationary Gaussian random functions

The General Ergodic Theorem for stationary Gaussian random functions states that for a wide-sense stationary (WSS) Gaussian process $X(t)$, under certain conditions, time averages can be replaced by ensemble averages. This is often expressed as:

$$\lim_{{T \to \infty}} \frac{1}{2T} \int_{{-T}}^{{T}} X(t) dt = \mathbb{E}[X(t)]$$

This holds almost surely, meaning with probability 1. The conditions generally require the process to be ergodic in the mean and second moment, which is commonly true for WSS Gaussian processes.

The theorem allows us to infer long-term statistical behavior of the system from a single, sufficiently long, realization.