Ergodic Theorem

Let $\{X_n\}$ be a stationary and ergodic stochastic process, and let $f$ be an integrable function of the process's state. The (Birkhoff) ergodic theorem states that the time average converges almost surely to the space (ensemble) average: $\frac{1}{n}{\sum }_{k=1}^{n}f(X_k)\to \mathbb{E}[f(X_0)]$ as $n\to \infty$, almost surely. Stationarity alone (an unchanging joint distribution over time) is not sufficient for this conclusion, ergodicity, informally, the impossibility of splitting the process into distinct, statistically-invariant sub-behaviors, is the additional condition that guarantees the time average converges to a single, non-random constant rather than to a random quantity that depends on which 'regime' the trajectory happened to be in. For an ergodic Markov chain, this general theorem specializes to the guarantee that a single chain's time-average converges to the expectation under its stationary distribution $\pi$.

• 'Ergodic' is often loosely used to mean 'well-behaved' or 'mixes nicely,' but its precise technical meaning, the impossibility of decomposing the process into separate invariant components, is exactly the condition that rules out a time average converging to different values depending on which invariant sub-behavior a trajectory happens to land in.
• This theorem is the deep theoretical justification for treating a single Monte Carlo simulation, a single MCMC chain, or a single long observed time series as informative about a process's underlying probabilities, without it, running one long simulation would tell you nothing reliable about the population of possible outcomes.
• The ergodic theorem for Markov chains (a corollary of the general Birkhoff theorem, applied to an ergodic Markov chain's stationary distribution $\pi$) is the precise mathematical guarantee that justifies estimating posterior expectations from a single MCMC chain's output.