Ergodic theorems are laws of large numbers for dependent observations generated by a stationary dynamical system. The limit is not automatically a constant: it is the conditional expectation given invariant information. Ergodicity is exactly the condition that removes this residual random environment.
Measure-preserving dynamics and stationarity
Definition. A measurable map \(T:\Omega\to\Omega\) is measure preserving if \[ P(T^{-1}A)=P(A) \qquad(A\in\mathcal F). \] Its invariant sigma-field is \[ \mathcal I =\{A\in\mathcal F:T^{-1}A=A\text{ modulo }P\}. \]
Proposition. If \(T\) is measure preserving and \(f\) is measurable, then \[ X_n=f\circ T^n \] is strictly stationary. Conversely, every one-sided stationary process can be realized on its canonical path space with \(T\) equal to the left shift.
Proof. For any bounded measurable \(g\) on a finite product, \[ \mathbb Eg(X_h,\ldots,X_{h+k}) =\int g(f\circ T^h,\ldots,f\circ T^{h+k})\,\mathrm{d}P. \] Measure preservation removes the common \(T^h\), yielding the law at times \(0,\ldots,k\). Conversely, stationarity says the path law is invariant under the shift.
The mean ergodic theorem
Theorem (von Neumann). Let \(T\) be measure preserving and define the isometry \(Uf=f\circ T\) on \(L^2(P)\). Then \[ A_nf:=\frac1n\sum_{k=0}^{n-1}U^kf \longrightarrow P_{\mathrm{Fix}(U)}f =\mathbb E[f\mid\mathcal I] \quad\text{in }L^2. \]
Proof. In the Hilbert space \(L^2\), \[ \overline{\operatorname{Ran}(I-U)} =\mathrm{Fix}(U^*)^\perp. \] Since \(U\) is an isometry induced by a measure-preserving map, its fixed vectors are precisely the \(\mathcal I\)-measurable functions, and the relevant orthogonal decomposition is \[ L^2=\mathrm{Fix}(U)\oplus \overline{\operatorname{Ran}(I-U)}. \] If \(f\in\mathrm{Fix}(U)\), then \(A_nf=f\). If \(f=(I-U)g\), telescoping gives \[ A_nf=\frac{g-U^ng}{n}, \qquad \|A_nf\|_2\le\frac{2\|g\|_2}{n}\to0. \] Uniform boundedness \(\|A_n\|_{2\to2}\le1\) extends convergence to the closure. Orthogonal projection onto \(L^2(\mathcal I)\) is conditional expectation.
Birkhoff's pointwise theorem
Lemma (maximal ergodic lemma). For \(f\in L^1\), let \(S_nf=\sum_{k=0}^{n-1}f\circ T^k\). Then \[ \int_{\{\max_{1\le k\le n}S_kf>0\}}f\,\mathrm{d}P\ge0. \]
Proof outline. On the set where a partial sum first becomes positive, group the orbit into consecutive blocks ending at first positive partial sums. Measure preservation lets one shift and sum the contributions of all blocks without changing their integrals. Every completed block has positive sum; the uncompleted terminal block contributes a remainder whose negative part vanishes under truncation. This yields the displayed integral inequality.
Theorem (Birkhoff). If \(T\) is measure preserving and \(f\in L^1(P)\), then \[ A_nf=\frac1n\sum_{k=0}^{n-1}f\circ T^k \longrightarrow \mathbb E[f\mid\mathcal I] \] almost surely and in \(L^1\).
Proof outline. Apply the maximal ergodic lemma to \(f-c\) and to \(-f+c\), first for bounded \(f\). The resulting inequalities control the sets on which \(\limsup A_nf>c\) and \(\liminf A_nf<c\). Varying \(c\) over the rationals forces the upper and lower limits to coincide almost surely. The limit is invariant because \[ A_nf\circ T-A_nf =\frac{f\circ T^n-f}{n}\to0 \] in probability after truncation. Integrating over every invariant set identifies it with \(\mathbb E[f\mid\mathcal I]\). For general \(f\in L^1\), truncate and use the maximal lemma to control the discarded tails. Finally, choose a convex superlinear \(\Phi\) with \(\mathbb E\Phi(|f|)<\infty\). Jensen and stationarity give \[ \sup_n\mathbb E\Phi(|A_nf|) \le\mathbb E\Phi(|f|), \] so the averages are uniformly integrable. Almost-sure convergence therefore upgrades to \(L^1\).
Ergodicity and invariant information
Definition. The transformation \(T\) is ergodic if every \(A\in\mathcal I\) has probability zero or one.
Theorem. The following are equivalent:
- \(T\) is ergodic;
- every \(T\)-invariant integrable function is almost surely constant;
- for every \(f\in L^1\), \[ A_nf\to\mathbb Ef\quad\text{almost surely}; \]
- for all \(f,g\in L^2\), \[ \frac1n\sum_{k=0}^{n-1} \mathbb E[f(g\circ T^k)] \to\mathbb Ef\,\mathbb Eg. \]
Proof. Indicators of invariant events establish the equivalence of (1) and (2). Birkhoff identifies the limit as \(\mathbb E[f\mid\mathcal I]\), giving (2)\(\Leftrightarrow\)(3). For (4), the mean ergodic theorem gives \[ \frac1n\sum_{k=0}^{n-1} \langle f,U^kg\rangle \to\langle f,P_{\mathrm{Fix}(U)}g\rangle. \] This equals \(\mathbb Ef\,\mathbb Eg\) for all \(f,g\) exactly when the fixed space consists of constants.
Stationary Markov chains
Corollary. Let \(X_n\) be a positive recurrent irreducible Markov chain with invariant probability \(\pi\). Then, for every \(f\in L^1(\pi)\), \[ \frac1n\sum_{k=0}^{n-1}f(X_k) \longrightarrow\int f\,\mathrm{d}\pi \quad\text{almost surely} \] from every starting state.
Proof outline. Under the stationary initial law \(\pi\), irreducibility makes the shift ergodic, so Birkhoff applies. For a fixed starting state, decompose the path into i.i.d. regeneration cycles between returns to a reference state. The renewal-reward law gives the same ratio of expected cycle reward to expected cycle length, equal to \(\int f\,\mathrm{d}\pi\).
This theorem justifies Monte Carlo averages under dependent sampling. A central limit theorem requires additional mixing, spectral-gap, or regenerative moment assumptions; ergodicity alone supplies consistency, not a \(\sqrt n\) rate.