DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Flegal–Jones overlapping-batch size conditions

Definition

Let $δ > 0$ and let $(b_{n})_{n \geq 2}$ be positive integers. Say that the sequence satisfies the Flegal–Jones OBM batch-size conditions for exponent $δ$ , with the integer monotonicity correction described below, when $b_{n} < n$ for all sufficiently large $n$ , $b_{n} \to \infty$ , $n / b_{n} \to \infty$ , and there is a constant $c \geq 1$ such that $\sum_{n = 2}^{\infty} (b_{n} / n)^{c} < \infty,$ while $\frac{b_{n}}{n} \log n \to 0, \frac{n^{2 / (2 + δ)} (\log n)^{3}}{b_{n}} \to 0,$ there are $n_{0} \geq 2$ and $c_{1} < \infty$ with $\log n / b_{n} \leq c_{1}$ for all $n \geq n_{0}$ , and $b_{n}^{2} n^{- 2} \log \log n \to 0, b_{n}^{4} n^{- 3} \log \log n \to 0.$

These are all the numerical rate, summability and endpoint hypotheses of Flegal–Jones Theorem 2. The source additionally asks that both $b_{n}$ and $n / b_{n}$ be nondecreasing. Those two requirements are omitted here: for an integer sequence with $b_{n} < n$ eventually they are incompatible with $b_{n} \to \infty$ . Indeed, ratio monotonicity would imply $n (b_{n + 1} - b_{n}) \leq b_{n} < n$ , while integer monotonicity would force the nonnegative integer $b_{n + 1} - b_{n}$ to be zero. Thus the literal source conditions would make $b_{n}$ eventually constant. The corrected interface preserves the intended floor-power batch sequences, and the Brownian quadratic-form proof of OBM consistency needs neither monotonicity assumption.

This is a sufficient-condition interface; failure of a condition does not imply inconsistency of the estimator.

For the floor-power sequence $b_{n} = ⌊ n^{ν} ⌋$ , the intended source range $δ > 2 / 3$ and $2 / (2 + δ) < ν < 3 / 4$ satisfies this corrected contract. Indeed, eventually $n^{ν} / 2 \leq b_{n} \leq n^{ν}$ . Choose $c > \max {1, 1 / (1 - ν)}$ to bound the summability series by $\sum_{n} n^{- c (1 - ν)} < \infty$ . The other displayed quantities are bounded respectively by $n^{ν - 1} \log n$ , $2 n^{2 / (2 + δ) - ν} (\log n)^{3}$ , $2 n^{- ν} \log n$ , $n^{2 ν - 2} \log \log n$ and $n^{4 ν - 3} \log \log n$ , all tending to zero. Also $b_{n} \to \infty$ , $n / b_{n} \to \infty$ and $b_{n} < n$ eventually. These checks establish the numerical contract without asserting the impossible simultaneous monotonicity.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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