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Geometric ergodicity and reversibility give exponential rho-mixing

Statement

Assume AC. Let $P$ be a probability kernel on $(X, X)$ with invariant probability $π$ , and consider its stationary output process $X$ . Assume it is $ϕ$ -irreducible (Phi-irreducibility for a general-state Markov kernel), aperiodic (Aperiodicity for a general-state Markov kernel), and Harris recurrent relative to its nonzero $σ$ -finite irreducibility measure $ϕ$ (Harris recurrence relative to an irreducibility measure). Suppose it is geometrically ergodic in the sense of Geometric and uniform ergodicity for MCMC CLTs and reversible with respect to $π$ in the symmetric-flow sense of General-state reversibility by symmetric flow measure. Then there are $C < \infty$ and $r \in (0, 1)$ such that its maximal-correlation mixing coefficient satisfies $ρ_{X} (n) \leq C r^{n} (n \geq 1) .$

Facts & Assumptions

Given: AC, the stationary $P$ -chain and all the hypotheses in the Statement.

[A1]

AC covers the conditional-expectation and path-law interfaces used below. (The Axiom of Choice)

[F1]

Geometric ergodicity supplies a measurable finite $M \geq 0$ and a common $r \in (0, 1)$ with $∥ P^{k} (x, \cdot) - π ∥_{T V} \leq M (x) r^{k}$ for every $x$ and $k \geq 1$ . (Geometric and uniform ergodicity for MCMC CLTs)

[F2]

Reversibility means that the probability flow measure $π (d x) P (x, d y)$ is invariant under coordinate interchange. (General-state reversibility by symmetric flow measure)

[F3]

The square-integrable kernel conditional-expectation identity holds with zero filling on the marginal-null set where an integral is not finite. (Kernel conditional expectations for square-integrable Markov observables)

[F4]

For square-integrable real variables, $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥_{2} ∥ v ∥_{2}$ . (Cauchy-Schwarz inequality for $L^{2}$ )

[F5]

Dominated convergence passes almost-everywhere limits through integrals under an integrable dominating function. (Dominated convergence)

[F6]

A bounded measurable future-path functional has conditional expectation, given the chain's past through its initial time, equal to a bounded measurable function of the state at that time. (Markov property for bounded future path functionals)

[F7]

For bounded measurable $g$ , $E [g (X_{j + n}) ∣ F_{j}] = P^{n} g (X_{j})$ , where $F_{j} = σ (X_{0}, \dots, X_{j})$ . (Chapman-Kolmogorov equations)

[F8]

Nested conditional expectations satisfy the tower property, and a known factor can be taken out when the products are integrable. (Tower property of conditional expectation, Taking out what is known)

[F9]

Conditional Jensen gives $∣ E [V ∣ G] ∣^{2} \leq E [V^{2} ∣ G]$ . (Conditional jensen inequality)

[F10]

The coefficient $ρ_{X} (n)$ is the supremum of absolute correlations over square-integrable past and separated future variables of positive variance. (Strong, uniform, and maximal-correlation mixing coefficients)

Proof

1.1A1F2F3F4F5F7F9given

On real $L^{2} (π)$ put $T f = P f$ and $H = {f : π (f) = 0}$ . By [F3], conditional Jensen and stationarity give $∥ T f ∥_{2}^{2} \leq π (f^{2})$ and $π (T f) = π (f)$ . These identities also show that the operator is well defined on almost-everywhere classes. Thus $T$ is a contraction preserving $H$ , and $T 1 = 1$ . For bounded $f, g$ , [F2] gives $⟨ f, T g ⟩ = ⟨ T f, g ⟩$ by integrating $f (x) g (y)$ against the flow measure. Bounded truncation converges in $L^{2}$ by [F5], so contraction and [F4] extend this symmetry to all $L^{2}$ functions. Iterated kernel integration agrees with $T^{k}$ ; for bounded functions this is [F7], and contraction extends the identity to $L^{2}$ classes.

1.2F1constructalgebra

For probability measures $μ, ν$ and $0 \leq a \leq 1$ , approximate $a$ uniformly by $2^{- q} \sum_{i = 1}^{2^{q}} 1_{{a \geq i 2^{- q}}}$ . Each indicator integral difference is bounded by the event-supremum TV distance, so $∣ μ (a) - ν (a) ∣ \leq ∥ μ - ν ∥_{T V}$ on taking limits. Applying this to $a = (h + ∥ h ∥_{\infty}) / (2 ∥ h ∥_{\infty})$ for nonzero bounded measurable $h$ , and separating $h = 0$ , gives $∣ P^{k} h (x) - π (h) ∣ \leq 2 ∥ h ∥_{\infty} M (x) r^{k}$ .

2.1F1step 1.1step 1.2algebra

Put $C_{m} = {M \leq m}$ for positive integers $m$ . If bounded $h$ vanishes off $C_{m}$ and $f = h - π (h)$ , symmetry and invariance give $∥ T^{k} f ∥_{2}^{2} = ⟨ f, T^{2 k} f ⟩ = \int h (x) (P^{2 k} h (x) - π (h)) π (d x)$ for $k \geq 1$ . Step 1.2 therefore gives $0 \leq ∥ T^{k} f ∥_{2}^{2} \leq 2 m ∥ h ∥_{\infty}^{2} r^{2 k}$ . Write $A = \sqrt{2 m} ∥ h ∥_{\infty} < \infty$ , so $∥ T^{k} f ∥_{2} \leq A r^{k}$ . No integral of $M$ over the whole state space has been used.

3.1F4step 1.1step 2.1algebra

For $u \in H$ and $a \geq 1$ , symmetry and [F4] give $∥ T^{a} u ∥_{2}^{2} = ⟨ u, T^{2 a} u ⟩ \leq ∥ u ∥_{2} ∥ T^{2 a} u ∥_{2}$ . Repeating with $a = 1, 2, 4, \dots, 2^{ℓ - 1}$ yields $∥ T u ∥_{2} \leq ∥ u ∥_{2}^{1 - 2^{- ℓ}} ∥ T^{2^{ℓ}} u ∥_{2}^{2^{- ℓ}}$ . For the $f$ in step 2.1, if $f = 0$ the bound $∥ T f ∥_{2} \leq r ∥ f ∥_{2}$ is immediate; otherwise $A > 0$ and the last display is at most $r A^{2^{- ℓ}} ∥ f ∥_{2}^{1 - 2^{- ℓ}}$ , tending to $r ∥ f ∥_{2}$ . Hence that bound holds for every such $f$ .

4.1F1F4F5step 1.1step 3.1construct

For $u \in H$ choose a finite measurable representative and put $h_{m} = 1_{C_{m}} \max (- m, \min (u, m))$ . Finiteness of $M$ gives $h_{m} \to u$ almost everywhere and $∣ h_{m} - u ∣ \leq ∣ u ∣$ , so [F5] gives $h_{m} \to u$ in $L^{2}$ . By [F4], $π (h_{m}) \to 0$ , and therefore $h_{m} - π (h_{m}) \to u$ in $L^{2}$ . Applying step 3.1 to these centered functions and taking limits through the contraction $T$ gives $∥ T u ∥_{2} \leq r ∥ u ∥_{2}$ . Iteration on $H$ proves $∥ T^{n} u ∥_{2} \leq r^{n} ∥ u ∥_{2}$ for all $n \geq 1$ . This also covers $H = {0}$ .

5.1A1F4F6F7F8F9step 4.1given

Fix $j \geq 0$ and bounded centered real $U, V$ measurable respectively for $F_{j}$ and $σ (X_{j + n}, X_{j + n + 1}, \dots)$ . By [F6] there is bounded measurable $g$ such that $E [V ∣ F_{j + n}] = g (X_{j + n})$ . Stationarity and [F9] give $π (g) = 0$ and $∥ g ∥_{2} \leq ∥ V ∥_{2}$ . The tower property and [F7] give $E [V ∣ F_{j}] = P^{n} g (X_{j})$ . Taking out $U$ , applying [F4] and stationarity, and then step 4.1, yields $∣ E [U V] ∣ = ∣ E [U P^{n} g (X_{j})] ∣ \leq ∥ U ∥_{2} ∥ T^{n} g ∥_{2} \leq r^{n} ∥ U ∥_{2} ∥ V ∥_{2}$ .

6.1F4F5F10step 5.1∎

For general centered square-integrable $U, V$ in the same sigma-algebras, truncate each at levels $\pm m$ and subtract its expectation. By [F5] and [F4], these bounded centered variables converge in $L^{2}$ ; [F4] passes their covariances and norms to the limit in step 5.1. Dividing by the positive standard deviations and taking the suprema in [F10] proves $ρ_{X} (n) \leq r^{n}$ . If no positive-variance variables exist, the coefficient is zero by the nonnegative empty-supremum convention. Thus the stated conclusion holds with $C = 1$ .

Source notes

Jones, Theorem 2(3), supplies the stated implication. Roberts and Rosenthal, Geometric ergodicity and hybrid Markov chains, Theorem 2 of the author-hosted preprint (published Theorem 2.1), proves the reversible spectral-gap route; its full proof is on PDF pp. 7–10 at https://www.probability.ca/jeff/ftpdir/hybrid.pdf . The proof here instead uses envelope level sets and repeated Cauchy–Schwarz to obtain the operator bound locally. It needs neither an integrable envelope nor a compact or general spectral theorem. The irreducibility, recurrence and aperiodicity hypotheses are retained from the source statement; the explicit common geometric rate and reversibility suffice for this argument.

Depends on

Used by

Sufficient MCMC CLT regimes Theorem

Dependency tree · two levels

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Sources

Galin L. Jones, On the Markov Chain Central Limit Theorem, Theorem 2, part 3 (standard reference, not scraped)
Gareth O. Roberts and Jeffrey S. Rosenthal, Geometric ergodicity and hybrid Markov chains, Theorem 2 (published Theorem 2.1) (standard reference, not scraped)