Mission

The Markov Chain Central Limit TheoremResearch Paper

Operations Research Probability Statistics Stochastic Systems·Captain: Shuze Chen

Markov chain Monte Carlo turns hard integration problems into long simulations: to estimate an expectation $E_\pi f$ one runs a Markov chain with stationary distribution $\pi$ and reports the sample average $\bar f_n$ . The ergodic theorem guarantees $\bar f_n \to E_\pi f$ , but honest error bars require more: a central limit theorem

\sqrt{n}(\bar f_n - E_\pi f) \to_d N(0, \sigma_f^2).

On general state spaces this is famously delicate - a merely ergodic chain with a square-integrable functional can fail the CLT, so the classical theory trades convergence rates (drift, minorization, geometric or polynomial total-variation rates) and mixing conditions ( $\alpha$ -, $\rho$ -, $\varphi$ -mixing) against moment conditions on $f$ . This mission formalizes G. L. Jones's survey "On the Markov chain central limit theorem" (Probability Surveys, 2004): the drift-condition CLTs of Meyn-Tweedie and Jarner-Roberts, the classical mixing CLTs of Ibragimov-Linnik, Doukhan-Massart-Rio and Billingsley, the characterizations via uniform integrability and boundedness in probability, and their assembly into the summary theorem: six practically checkable regimes - from polynomial ergodicity with bounded functionals to uniform ergodicity with second moments - each of which guarantees the CLT for every initial distribution. The stationarity, total-variation and mixing infrastructure is general state space and reusable well beyond this mission.

Goal · The Markov chain CLT: six sufficient conditions (Jones Thm 9, mission goal)

OPEN

theorem MarkovChainCLT.markov_chain_clt {X : Type*} [MeasurableSpace X]
    (P : Kernel X X) [IsMarkovKernel P] (π : Measure X) [IsProbabilityMeasure π]
    (hP : HarrisErgodic P π) (f : X → ℝ) (hf : Measurable f)
    (hcase :
      (∃ m : ℝ, 1 < m ∧ PolynomiallyErgodicL1 P π m ∧
        ∃ B : ℝ, ∀ᵐ x ∂π, |f x| < B) ∨
      (∃ m δ : ℝ, 0 < δ ∧ 2 + δ < m * δ ∧ PolynomiallyErgodicL1 P π m ∧
        Integrable (fun x => |f x| ^ (2 + δ)) π) ∨
      (GeometricallyErgodic P π ∧
        ∃ δ : ℝ, 0 < δ ∧ Integrable (fun x => |f x| ^ (2 + δ)) π) ∨
      (GeometricallyErgodic P π ∧
        Integrable (fun x => f x ^ 2 * Real.posLog |f x|) π) ∨
      (GeometricallyErgodic P π ∧ Kernel.IsReversible P π ∧ MemLp f 2 π) ∨
      (UniformlyErgodic P π ∧ MemLp f 2 π)) :
    SatisfiesCLT P π f := by sorry

Let $X = \{X_n\}_{n \ge 0}$ be a Markov chain with transition kernel $P$ on a state space $\mathsf{X}$ , Harris ergodic with invariant probability distribution $\pi$ , and let $f : \mathsf{X} \to \mathbb{R}$ be measurable. Write $\bar f_n = n^{-1} \sum_{i=1}^{n} f(X_i)$ for the sample average and $E_\pi f = \int f \, d\pi$ . Assume one of the following six conditions:

the chain is polynomially ergodic of order $m > 1$ with $E_\pi M < \infty$ for the rate constant $M$ , and $|f| < B$ $\pi$ -almost surely for some $B$ ;
the chain is polynomially ergodic of order $m$ with $E_\pi M < \infty$ , and $E_\pi |f|^{2+\delta} < \infty$ for some $\delta > 0$ with $m\delta > 2 + \delta$ ;
the chain is geometrically ergodic and $E_\pi |f|^{2+\delta} < \infty$ for some $\delta > 0$ ;
the chain is geometrically ergodic and $E_\pi [f^2 \log^+ |f|] < \infty$ ;
the chain is geometrically ergodic, reversible with respect to $\pi$ (detailed balance), and $E_\pi f^2 < \infty$ ;
the chain is uniformly ergodic and $E_\pi f^2 < \infty$ .

Then the chain satisfies the central limit theorem for $f$ : there is a single asymptotic variance $\sigma_f^2 \ge 0$ such that for every initial distribution of the chain,

\sqrt{n}\,\bigl(\bar f_n - E_\pi f\bigr) \xrightarrow{d} N(0, \sigma_f^2) \qquad (n \to \infty).

This is the summary theorem of the source and the goal of the mission: six practically checkable regimes, each guaranteeing honest error bars for Markov chain Monte Carlo estimates, assembled from the drift, mixing, and moment machinery of the milestones.

Formalization Note "Harris ergodic" is encoded by its total-variation characterization: $\pi$ is invariant for $P$ and $\|P^n(x, \cdot) - \pi\| \to 0$ for every starting point $x$ (equivalent to the classical aperiodic, $\psi$ -irreducible, positive Harris recurrent definition; the "every $x$ " quantifier is exactly the Harris property). Convergence in distribution is weak convergence of laws, and $N(0, 0)$ is read as the point mass at $0$ , which absorbs the source's " $\sigma_f^2 > 0$ " caveat.

Frontier · Open leaf nodes

MarkovChainCLT.clt_of_exp_alpha_of_log_moment
MarkovChainCLT.geometricallyErgodic_integrable_rate
MarkovChainCLT.clt_of_summable_rho
MarkovChainCLT.clt_of_moment_of_alpha_pow_summable
MarkovChainCLT.chain_condExp_future_bridge
MarkovChainCLT.reversible_indicator_decay_centered_L2_operator_model

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