For a measure‑preserving dynamical system or a stationary stochastic process that is ergodic, the time average of an integrable observable along almost every single trajectory converges to the ensemble (expected) average; thus long‑time sample means equal probabilistic expectations under the ergodicity conditions.

Principle

Ergodicity allows replacing ensemble averages by time averages for individual realizations: if a process is stationary and ergodic, empirical averages computed along one sufficiently long sample path converge (almost surely) to the theoretical expectation.

Demonstration

Illustrative scenario → A stationary fading channel is modelled as an ergodic process. Recognition → Measure received signal power continuously over a long interval on one link. Action → Compute the time average of the power. Consequence → If the channel is ergodic and observation time is long relative to its mixing time, the time average approximates the ensemble mean used in capacity and performance analysis.

Misapplication

Treating finite or short observation averages as representative when the process is non‑stationary, non‑ergodic, or has very long correlation times; or assuming ergodicity for spatial ensembles without verifying the necessary measure‑preserving properties.

Consequence

Ergodicity justifies empirical estimation of statistical quantities from single long observations and underpins ergodic channel capacity definitions; misuse can lead to biased estimates and incorrect system design when ergodicity assumptions fail.

Reversal

If the process is non‑ergodic (e.g., contains invariant measurable subsets with different statistics), non‑stationary, or observed for times shorter than its mixing scale, time averages do not converge to ensemble averages and ensemble‑based predictions fail.

Boundary

Applies to stationary, measure‑preserving processes or dynamical systems that satisfy ergodicity; does not apply to transient, non‑stationary, multi‑ergodic, or structurally non‑ergodic processes without further qualification.

Semantic Tension

Theoretical identifiability versus practical sampling — ergodic theorems provide asymptotic equivalence, but finite‑time estimation, non‑stationarity and long memory create a tension between mathematical guarantees and empirical feasibility.

Synthesis

Ergodicity links probability to observation: it legitimizes using long single‑trajectory measurements to infer ensemble properties, but only when stationarity and mixing conditions hold and the observation window is adequate relative to the process timescales.