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Strong Brownian approximation implies overlapping-batch consistency

Statement

Assume AC. Let $Y_{1}, Y_{2}, \dots$ be real random variables and $S_{k} = \sum_{i = 1}^{k} Y_{i}$ , with $S_{0} = 0$ . Suppose on a probability-space extension with the original process as marginal there are a standard Brownian motion $B$ , constants $σ \geq 0$ and $0 < α < 1 / 2$ , and finite random constants $K, N$ such that almost surely $∣ S_{k} - σ B (k) ∣ \leq K k^{α} \log k (k \geq N) .$ Let $(b_{n})$ be deterministic positive integers with $b_{n} < n$ eventually and $\frac{b_{n}}{n} \to 0, \sum_{n = 2}^{\infty} (b_{n} / n)^{c} < \infty for some real c \geq 1,$ $\frac{b_{n} \log n}{n} \to 0, \frac{n^{2 α} (\log n)^{2}}{b_{n}} \to 0.$ For the exact estimator of Overlapping batch means for MCMC standard errors, applied to $Y_{1}, \dots, Y_{n}$ , ${\hat{σ}}_{O B M, n}^{2} = \frac{n b_{n}}{(n - b_{n}) (n - b_{n} + 1)} \sum_{j = 0}^{n - b_{n}} {(\frac{S_{j + b_{n}} - S_{j}}{b_{n}} - \frac{S_{n}}{n})}^{2} ⟶ σ^{2} almost surely .$ No stationarity, independence of $B$ from $(Y_{i})$ , or monotonicity of $b_{n}$ or $n / b_{n}$ is required. The conclusion includes $σ = 0$ .

Facts & Assumptions

Given: AC, the original process and its Brownian approximation on the stated extension, the constants $σ, α$ , and the deterministic batch sequence with all displayed numerical hypotheses.

[A1]

AC is assumed as in The Axiom of Choice and the Brownian and Gaussian interfaces.

[F1]

Brownian motion starts at zero and has independent normal increments with variance equal to interval length. Brownian motion

[F2]

A centered Gaussian quadratic form satisfies the dimension-free even-moment bound for every positive integer $r$ . Centered Gaussian quadratic forms have dimension-free even-moment bounds

[F3]

For a nonnegative random variable $X$ and $a > 0$ , $\Pr (X \geq a) \leq E X / a$ . Markov's inequality for random variables

[F4]

Events with summable probabilities occur only finitely often almost surely. The first Borel-Cantelli lemma for measures

[F5]

The standard-normal Mills upper bound $\Pr (Z > x) \leq e^{- x^{2} / 2} / (x \sqrt{2 π})$ , together with symmetry and the normal density, gives $\Pr (∣ Z ∣ \geq x) \leq 2 e^{- x^{2} / 2}$ for $x \geq 1$ . Two-sided Mills bounds for the standard normal tail

[F6]

The Euclidean norm is a norm and satisfies $∣ ∥ x ∥_{2} - ∥ y ∥_{2} ∣ \leq ∥ x - y ∥_{2}$ . Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation, The reverse triangle inequality in a normed space

[F7]

The finite OBM algorithm uses the displayed denominator and overlapping batch means. Overlapping batch means for MCMC standard errors

Proof

1.1A1F1givenconstructalgebra

Work first on the extension. For each sufficiently large $n$ write $b = b_{n}$ , $m = n - b + 1$ , $U_{i} = B (i) - B (i - 1)$ for $1 \leq i \leq n$ , and $D_{j} = B (j + b) - B (j)$ for $0 \leq j \leq n - b$ . By [F1], the $U_{i}$ are independent standard normals. If $h_{j}$ indicates coordinates $j + 1, \dots, j + b$ , then $Q_{n} = (n b)^{- 1} \sum_{j} D_{j}^{2} = U^{T} A_{n} U$ where $A_{n} = (n b)^{- 1} \sum_{j} h_{j} h_{j}^{T}$ . At most $b$ windows contain a fixed pair of coordinates, and none does if their separation is at least $b$ . Hence $0 \leq (A_{n})_{i k} \leq 1 / n$ , $(A_{n})_{i k} = 0$ when $∣ i - k ∣ \geq b$ , $tr A_{n} = m / n$ , and $\sum_{i, k} (A_{n})_{i k}^{2} \leq (2 b - 1) / n \leq 2 b / n$ .

2.1F2F3F4step 1.1givenchoosealgebra

Choose an integer $r \geq c$ . By [F2] and [F3], for every $η > 0$ , $\Pr (∣ Q_{n} - m / n ∣ > η) \leq C_{r} η^{- 2 r} (2 b_{n} / n)^{r}$ . These probabilities are summable since $b_{n} / n < 1$ eventually, $r \geq c$ and $\sum_{n} (b_{n} / n)^{c} < \infty$ . Applying [F4] for each $η = 1, 1 / 2, 1 / 3, \dots$ and intersecting the resulting countable family of full-probability events gives $Q_{n} - m / n \to 0$ almost surely. Since $m / n \to 1$ , $Q_{n} \to 1$ almost surely. No independence between different $n$ is used.

3.1F1F4F5F6step 1.1step 2.1givenalgebra

By [F1] and [F5], $\Pr (∣ B (n) ∣ > \sqrt{6 n \log n}) \leq 2 n^{- 3}$ for $n \geq 2$ . Thus [F4] gives $B (n)^{2} / n \leq 6 \log n$ eventually almost surely. Put $V_{j} = D_{j} - (b / n) B (n)$ and $W_{n} = (n b)^{- 1} \sum_{j} V_{j}^{2}$ . Applying [F6] to the vectors $(V_{j}) / \sqrt{n b}$ and $(D_{j}) / \sqrt{n b}$ gives $∣ \sqrt{W_{n}} - \sqrt{Q_{n}} ∣ \leq [(m / n) (b / n) (B (n)^{2} / n)]^{1 / 2} \to 0$ , using $(b_{n} / n) \log n \to 0$ . Consequently $W_{n} \to 1$ almost surely.

4.1step 3.1givenconstructalgebra

Set $R_{k} = S_{k} - σ B (k)$ and $R_{0} = 0$ . On the full event of the given approximation, enlarge its finite random constant to absorb all finitely many early errors. Since $k^{α} \log k$ is increasing for integers $k \geq 2$ , this gives a finite random $K^{'}$ with $\max_{0 \leq k \leq n} ∣ R_{k} ∣ \leq K^{'} n^{α} \log n$ for every sufficiently large $n$ . For every window, $S_{j + b} - S_{j} - (b / n) S_{n} = σ V_{j} + E_{n, j}$ where $E_{n, j} = R_{j + b} - R_{j} - (b / n) R_{n}$ . Therefore $∣ E_{n, j} ∣ \leq 3 K^{'} n^{α} \log n$ simultaneously for all windows, and $H_{n} = (n b)^{- 1} \sum_{j} E_{n, j}^{2} \leq 9 (K^{'})^{2} (m / n) n^{2 α} (\log n)^{2} / b_{n} \to 0$ .

5.1F6step 3.1step 4.1givenalgebra

Put $T_{n} = (n b)^{- 1} \sum_{j} (S_{j + b} - S_{j} - (b / n) S_{n})^{2}$ . By [F6] and $σ \geq 0$ , $∣ \sqrt{T_{n}} - σ \sqrt{W_{n}} ∣ \leq \sqrt{H_{n}} \to 0$ . Hence $T_{n} \to σ^{2}$ almost surely. This argument divides neither by $σ$ nor by $T_{n}$ , so it also proves the zero-variance case.

6.1F7step 5.1givenalgebra∎

By [F7], ${\hat{σ}}_{O B M, n}^{2} = n^{2} T_{n} / [(n - b_{n}) (n - b_{n} + 1)]$ . This deterministic multiplier tends to one because $b_{n} / n \to 0$ , and step 5.1 proves the desired limit on the extension. The event that this sequence of finite measurable functions of $(Y_{i})$ converges to $σ^{2}$ depends only on the original process. Its probability under the extension equals its probability under the original marginal, so that probability is one for the original process as well. Arbitrarily defining the estimator at any finitely many indices with inadmissible batches does not affect its limit.

Depends on

Used by

Strong consistency of overlapping batch means Theorem

Dependency tree · two levels

46 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

James M. Flegal and Galin L. Jones, Batch means and spectral variance estimators in Markov chain Monte Carlo, Appendix B.3 and Lemma 4 (standard reference, not scraped)