the shift on a stationary process
The shift is the dictionary that makes ergodic theory and the theory of stationary processes the same subject. It says that to study a stationary stochastic process you may, with no loss, study one canonical measure-preserving transformation — the shift on sequence space — and read every property of the process off the dynamics. This is why all the ergodic theorems above are simultaneously theorems about time series.
Construction. Given a stationary real-valued process (X_n)_(n in Z) (or n >= 0), let Omega = R^Z be the path space of all sequences, P the law of (X_n) (its joint distribution), and define the shift T by (T omega)_n = omega_(n+1). The coordinate map f(omega) = omega_0 then reproduces the process: X_n = f(T^n omega) (= omega_n) under P. The process is stationary precisely when its law is shift-invariant, i.e. P(T^(-1) A) = P(A) — so stationarity of the process is identical to T being measure-preserving. The system (R^Z, P, T) is the canonical model, and a process is called ergodic / mixing / weakly mixing exactly when its shift is. Through this lens, the invariant sigma-algebra I becomes the shift-invariant events of the process (the tail-like events unchanged by time-translation).
Consequences flow immediately. Birkhoff's theorem becomes the ergodic theorem for stationary sequences: (1/n) sum_(k<n) g(X_k, X_(k+1), ...) -> E[ g(X_0, X_1, ...) given I ] almost surely, the constant E[g(...)] when ergodic — the law of large numbers for dependent stationary data. The ergodic decomposition becomes the statement that every stationary process is a mixture of ergodic stationary processes (the dynamical de Finetti picture). Honest cautions: a process can be stationary without being ergodic (the random-bias coin again), in which case time averages converge but to a random limit; and stationarity is about the whole law being shift-invariant, not merely about constant mean and variance (that weaker condition is wide-sense / second-order stationarity, which is not enough for the ergodic theorems).
A stationary Gaussian process with spectral density vanishing nowhere is ergodic; if moreover its covariance r(k) -> 0 as k -> infinity it is mixing. Then the sample autocovariance (1/n) sum_(k<n) X_k X_(k+h) converges almost surely to the true r(h) — Birkhoff via the shift is exactly why time-series autocovariance estimators are consistent.
Every stationary process is a measure-preserving system in disguise — the shift is the translation key.
Strict stationarity (whole law shift-invariant) is what the ergodic theorems need. Wide-sense stationarity (constant mean and covariance depending only on the lag) is weaker and does not by itself guarantee an ergodic theorem.