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Strong consistency of overlapping batch means

Statement

Let $P$ be a geometrically ergodic Markov chain with invariant probability $π$ , and let the initial distribution be arbitrary. Let $g$ be real Borel and suppose $π (∣ g ∣^{2 + δ + ε}) < \infty$ for some $δ, ε > 0$ . Assume the state-dependent one-step minorization State-dependent one-step minorization: there are measurable $s : X \to [0, 1]$ and a probability measure $Q$ such that $P (x, A) \geq s (x) Q (A)$ for all $x, A$ .

Let $(b_{n})$ be positive integer batch sizes with $1 \leq b_{n} < n$ for all sufficiently large $n$ , $b_{n} \to \infty$ , and $n / b_{n} \to \infty$ . Suppose there is $c \geq 1$ such that $\sum_{n = 1}^{\infty} (b_{n} / n)^{c} < \infty,$ and suppose $\frac{b_{n}}{n} \log n \to 0, \frac{n^{2 / (2 + δ)} (\log n)^{3}}{b_{n}} \to 0,$ there are $n_{0}$ and $c_{1} < \infty$ with $\log n / b_{n} \leq c_{1}$ for every $n \geq n_{0}$ , and $b_{n}^{2} n^{- 2} \log \log n \to 0, b_{n}^{4} n^{- 3} \log \log n \to 0.$

For ${\overset{ˉ}{g}}_{n} = n^{- 1} \sum_{i = 1}^{n} g (X_{i})$ and ${\overset{ˉ}{g}}_{n, j} (b_{n}) = b_{n}^{- 1} \sum_{i = j}^{j + b_{n} - 1} g (X_{i})$ , define ${\hat{σ}}_{O B M, n}^{2} = \frac{n b_{n}}{(n - b_{n}) (n - b_{n} + 1)} \sum_{j = 1}^{n - b_{n} + 1} ({\overset{ˉ}{g}}_{n, j} (b_{n}) - {\overset{ˉ}{g}}_{n})^{2} .$ The strong-consistency conclusion is ${\hat{σ}}_{O B M, n}^{2} ⟶ σ_{g}^{2} almost surely,$ where $σ_{g}^{2}$ is the covariance long-run variance defined by Long-run variance for a stationary Markov chain. The batch sequence satisfies Flegal–Jones overlapping-batch size conditions for the displayed exponent $δ$ ; these conditions are written out above. This corrects the source's incompatible requirement that both integer $b_{n}$ and $n / b_{n}$ be nondecreasing, retaining every numerical batch-size hypothesis.

As in the intended source corollary, if $δ > 2 / 3$ and $(1 + δ / 2)^{- 1} < ν < 3 / 4$ , then $b_{n} = ⌊ n^{ν} ⌋$ satisfies these corrected batch-size hypotheses and has the same strong-consistency conclusion. No monotonicity of $n / ⌊ n^{ν} ⌋$ is asserted.

Facts & Assumptions

Given: The geometrically ergodic chain, its arbitrary initial law, the real measurable observable $g$ , the moment exponents $δ, ε > 0$ , the supplied one-step minorization, and every displayed numerical batch-size hypothesis.

[A1]

AC is assumed through the chain, Brownian, mixing and strong-approximation interfaces. The Axiom of Choice

[F1]

The canonical chain exists for each initial law, and choosing its invariant initial law $π$ makes it strictly stationary. Canonical Markov chain on path space, Invariant initial law makes a Markov chain stationary

[F2]

An invariant kernel with an everywhere finite measurable common geometric total-variation bound admits a geometric envelope integrable under $π$ , on the original arbitrary measurable space. Harris geometric ergodicity admits an integrable total-variation envelope

[F3]

An integrable geometric total-variation envelope gives exponentially decaying stationary strong-mixing coefficients. A measurable observation preserves stationarity and does not increase those coefficients. A total-variation rate bounds strong-mixing coefficients, A measurable observation does not increase stationary mixing coefficients

[F4]

For a centered stationary sequence with a finite $(2 + η)$ th absolute moment, $η > 0$ , summability of $α (n)^{η / (2 + η)}$ gives absolute covariance summability, a finite nonnegative covariance-sum variance, and a CLT with that variance, including zero. Strong-mixing CLTs under bounded, moment, and logarithmic-moment conditions

[F5]

The covariance long-run variance is the stationary covariance sum when absolute summability and the CLT have been established; equality to the CLT variance is a theorem obligation, not a consequence of the definition alone. Long-run variance for a stationary Markov chain

[F6]

Under the stated geometric-ergodicity, minorization and $(2 + δ + ε)$ th moment hypotheses, with positive finite asymptotic standard deviation, the strong invariance principle supplies a Brownian approximation with error $O (n^{1 / (2 + δ)} \log n)$ on an extension of the original chain. Strong invariance principle for geometrically ergodic chains

[F7]

Under the everywhere common geometric total-variation bound, a finite centered $f \in L^{q} (π)$ with zero covariance sum has, for every $2 < p < q$ , the maximal partial-sum rate $o (n^{1 / p})$ under every initial law, without a moment condition on that law. Zero-variance geometric Markov sums have a uniform strong rate

[F8]

A partial-sum approximation by $σ B (n)$ with error $O (n^{α} \log n)$ , $σ \geq 0$ and $0 < α < 1 / 2$ , implies consistency of the exact OBM estimator when $b_{n} / n \to 0$ , $\sum_{n} (b_{n} / n)^{c} < \infty$ , $(b_{n} / n) \log n \to 0$ , and $n^{2 α} (\log n)^{2} / b_{n} \to 0$ . It requires neither stationarity nor batch monotonicity and includes zero variance. Strong Brownian approximation implies overlapping-batch consistency

[F9]

The corrected Flegal–Jones contract retains all numerical rate, summability and endpoint conditions. For $δ > 2 / 3$ and $2 / (2 + δ) < ν < 3 / 4$ , the integer sequence $⌊ n^{ν} ⌋$ satisfies that contract; the definition supplies the floor bounds and each numerical limit. Flegal–Jones overlapping-batch size conditions

[F10]

A standard scalar Brownian motion exists, and the product of its probability space with the original probability space gives an extension preserving the original marginal. Existence and scaling of $d$ -dimensional Brownian motion, For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique

Proof

1.1A1F1F2F3givenconstructalgebra

Put $q = 2 + δ + ε$ , $μ = π (g)$ and $f = g - μ$ . The moment hypothesis gives $π ∣ g ∣ < \infty$ , since $∣ g ∣ \leq 1 + ∣ g ∣^{q}$ , so $μ$ is finite. The elementary bound $∣ g - μ ∣^{q} \leq 2^{q} (∣ g ∣^{q} + ∣ μ ∣^{q})$ gives $π ∣ f ∣^{q} < \infty$ , and $π f = 0$ . For variance identification consider the canonical chain with initial law $π$ , which is stationary by [F1]. The given geometric bound has an everywhere finite measurable common envelope, so the stronger arbitrary-measurable-space clause of [F2] supplies an integrable such envelope. By [F3], this stationary chain and its observation $f$ have a strong-mixing bound $α_{f} (n) \leq C r^{n}$ with $C < \infty$ and $r \in (0, 1)$ . This construction is only for identifying the stationary covariance parameter; the observed initial law remains arbitrary.

2.1F4F5step 1.1givenalgebra

Take $η = q - 2 = δ + ε > 0$ . The geometric bound in step 1.1 makes $\sum_{n} α_{f} (n)^{η / (2 + η)}$ finite. Apply [F4] to the stationary observed sequence. Its covariance series is absolutely convergent, its sum $v = π (f^{2}) + 2 \sum_{k \geq 1} {Cov}_{π} (f (X_{0}), f (X_{k}))$ is finite and nonnegative, and its CLT has precisely variance $v$ . Thus [F5] identifies $v = σ_{g}^{2}$ , without using the stationary Harris-only formulation of a separate MCMC regime theorem. Write $σ = \sqrt{v} \geq 0$ and $α = 1 / (2 + δ) \in (0, 1 / 2)$ .

3.1F6step 1.1step 2.1given

If $v > 0$ , the original $g$ , the given $δ, ε$ , geometric ergodicity and supplied minorization meet [F6] with the finite positive standard deviation from step 2.1. Hence, for the original arbitrary initial law, on an extension there are a standard Brownian motion $B$ and finite random constants for which $∣ \sum_{i = 1}^{n} f (X_{i}) - σ B (n) ∣ = O (n^{α} \log n)$ almost surely. The same moment exponents and the same covariance variance are used; no stronger moment assumption is introduced.

4.1F7F10step 1.1step 2.1chooseconstructalgebra

If $v = 0$ , choose a real $p$ with $2 + δ < p < q$ , possible because $ε > 0$ . Apply the all-start geometric-envelope clause of [F7] to $f$ ; its moment and zero covariance hypotheses were established in steps 1.1, 2.1. Use the chain indexed from $X_{1}$ as its time-zero chain, whose initial law is still arbitrary. Then $\max_{j \leq n} ∣ \sum_{i = 1}^{j} f (X_{i}) ∣ = o (n^{1 / p})$ almost surely. Since $1 / p < α$ , this implies $∣ \sum_{i = 1}^{n} f (X_{i}) ∣ = O (n^{α} \log n)$ . Adjoin a standard Brownian motion on the product extension supplied by [F10] and set $σ = 0$ . The same approximation as in step 3.1 now holds, with zero Brownian coefficient. No division by $σ$ or implication from a degenerate CLT to an almost-sure rate is used.

5.1F8step 2.1step 3.1step 4.1givenalgebra

In either variance branch the hypotheses of [F8] hold with this $σ, α$ . Indeed $n / b_{n} \to \infty$ gives $b_{n} / n \to 0$ , while the summability and $(b_{n} / n) \log n \to 0$ assumptions are displayed in the Statement. Moreover $n^{2 α} (\log n)^{2} / b_{n} = [n^{2 / (2 + δ)} (\log n)^{3} / b_{n}] / \log n \to 0$ . The other stated numerical endpoint and growth conditions are retained and impose no additional requirement in this reduction. Thus [F8] gives convergence to $σ^{2} = v$ for the exact OBM estimator applied to $Y_{i} = f (X_{i})$ .

6.1F8step 1.1step 2.1step 5.1algebra

Centering the observations by the constant $μ$ does not change this finite estimator: ${\overset{ˉ}{f}}_{n} = {\overset{ˉ}{g}}_{n} - μ$ and ${\overset{ˉ}{f}}_{n, j} (b_{n}) = {\overset{ˉ}{g}}_{n, j} (b_{n}) - μ$ , so their difference equals ${\overset{ˉ}{g}}_{n, j} (b_{n}) - {\overset{ˉ}{g}}_{n}$ . The indexing $j = 0, \dots, n - b_{n}$ in [F8] is exactly $j = 1, \dots, n - b_{n} + 1$ after adding one to the starting index, and its denominator is $(n - b_{n}) (n - b_{n} + 1)$ as here. Therefore its estimator is the displayed ${\hat{σ}}_{O B M, n}^{2}$ , and steps 2.1 and 5.1 prove its almost-sure convergence to $σ_{g}^{2}$ under the original arbitrary initial law.

7.1F7F8F9step 1.1step 2.1step 4.1step 6.1given∎

Finally, the source-corollary range is $2 / (2 + δ) < ν < 3 / 4$ . By [F9], $b_{n} = ⌊ n^{ν} ⌋$ satisfies every corrected numerical batch hypothesis, so steps 1.1, 2.1, 3.1, 4.1, 5.1, 6.1 apply to it as well. This verifies the intended floor-power conclusion without asserting the incompatible ratio monotonicity. The proof includes $v = 0$ , finite early inadmissible batch indices, and arbitrary initial laws without additional moment hypotheses, as allowed by [F7]–[F8].

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Sources

James M. Flegal and Galin L. Jones, Batch means and spectral variance estimators in Markov chain Monte Carlo, Theorem 2 and Appendix B.3 (standard reference, not scraped)
James M. Flegal and Galin L. Jones, Batch means and spectral variance estimators in Markov chain Monte Carlo, Corollary 1 (standard reference, not scraped)