CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.

The source function class need not give finite SMC CLT variances

Statement refuted

The source class $Φ_{t}$ in Finite-horizon SMC CLT classes and variance recursions, together with $1 \in Φ_{t}$ , guarantees that every displayed finite-horizon weighted and post-resampling CLT covariance is finite, including $V_{0}$ for a separately weighted initial proposal.

Facts & Assumptions

Given: The recursive class $Φ_{t}$ , target/proposal laws, and variance notation in the cited definition.

[F1]

Finite-horizon SMC CLT classes and variance recursions: $Φ_{0}$ requires a finite proposal second moment; $ϕ \in Φ_{t}$ requires ${\tilde{π}}_{t} (∣ υ_{t} ϕ ∣^{2 + δ}) < \infty$ and $k_{t} (υ_{t} ϕ) \in Φ_{t - 1}$ ; the post-multinomial covariance adds ${Var}_{π_{t}} (ϕ)$ .

[F2]

Finite-horizon SMC CLT classes and variance recursions: For an initially weighted proposal, $V_{0} (ϕ) = {Cov}_{{\tilde{π}}_{0}} (υ_{0} (ϕ - π_{0} ϕ))$ ; finiteness requires its displayed second moment to exist.

Counterexample

Given: Let $Θ_{0} = Θ_{1} = N_{0}$ with its power-set sigma-algebra and proposal $p_{0} = 1 / 2$ , $p_{n} = 2^{- (n + 1)}$ for $n \geq 1$ . At time zero set ${\tilde{π}}_{0} = π_{0} = p$ and $υ_{0} = 1$ . Put $w_{n} = 2^{- n^{2}}$ for $n \geq 1$ , $s = \sum_{n \geq 1} p_{n} w_{n}$ , and $w_{0} = (1 - s) / p_{0}$ . Let $k_{1} (n, \cdot) = δ_{n}$ , ${\tilde{π}}_{1} = p$ , $υ_{1} (n) = w_{n}$ , and $π_{1} (n) = p_{n} w_{n}$ . Define $ϕ (0) = 0$ and $ϕ (n) = w_{n}^{- 3 / 4}$ for $n \geq 1$ .

1.1givenalgebra

The geometric proposal masses sum to one. Since $0 < s < 1 / 2$ , $w_{0}$ is finite and positive, and $\sum_{n \geq 0} p_{n} w_{n} = 1$ by construction; hence $π_{1}$ is a probability law with density $w$ relative to $p$ . The identity kernel gives ${\tilde{π}}_{1} = p$ .

1.2F1F2givenalgebra

For a separate initial-proposal witness, let $Θ_{0} = N_{\geq 1}$ , ${\tilde{π}}_{0} {n} = 2^{- n}$ , $c = (\sum_{n \geq 1} 2^{- n / 2})^{- 1}$ , and $υ_{0} (n) = c 2^{n / 2}$ . Put $π_{0} {n} = 2^{- n} υ_{0} (n)$ and $ϕ (n) = 1_{{n even}}$ . Then $\sum_{n} {\tilde{π}}_{0} {n} υ_{0} (n) = 1$ , so $π_{0}$ is a probability law and $υ_{0} = d π_{0} / d {\tilde{π}}_{0}$ . Both $ϕ$ and $1$ are bounded, hence belong to $Φ_{0}$ .

2.1F1step 1.1

Choose $δ = 1$ . For $n \geq 1$ , $w_{n} ϕ (n) = w_{n}^{1 / 4}$ , so $\sum p_{n} ∣ w_{n} ϕ (n) ∣^{3} = \sum_{n \geq 1} p_{n} w_{n}^{3 / 4} < \infty$ and $\sum p_{n} ∣ w_{n} ϕ (n) ∣^{2} = \sum_{n \geq 1} p_{n} w_{n}^{1 / 2} < \infty$ . Thus $k_{1} (w ϕ) = w ϕ \in Φ_{0}$ and $ϕ \in Φ_{1}$ . Also $\sum p_{n} w_{n}^{3} < \infty$ and $k_{1} w = w \in Φ_{0}$ , so $1 \in Φ_{1}$ .

2.2step 1.1algebra

The target mean is $π_{1} (ϕ) = \sum_{n \geq 1} p_{n} w_{n}^{1 / 4} < \infty$ , but $π_{1} (ϕ^{2}) = \sum_{n \geq 1} p_{n} w_{n}^{- 1 / 2} = \sum_{n \geq 1} 2^{- (n + 1)} 2^{n^{2} / 2} = \infty,$ since the summands do not tend to zero. Therefore ${Var}_{π_{1}} (ϕ) = \infty$ .

3.1F1step 2.1step 2.2algebra

The weighted covariance $V_{1} (ϕ)$ is finite: the identity mutation has zero conditional variance, $w (ϕ - π_{1} ϕ) \in L^{2} (p)$ by step 2.1 and the finite mean in step 2.2, and the initial multinomial covariance at time zero is finite because $υ_{0} = 1$ and $g = w (ϕ - π_{1} ϕ) \in L^{2} (p)$ . Yet the post-multinomial formula in [F1] adds the infinite target variance from step 2.2. Thus the source class and $1 \in Φ_{1}$ do not suffice for the claimed finite post-resampling covariance.

4.1F2step 1.2algebra∎

Let $q = π_{0} (ϕ)$ . Positive proposal and target mass occur on both parities, so $0 < q < 1$ . For every even $n$ , ${\tilde{π}}_{0} {n} υ_{0} (n)^{2} (ϕ (n) - q)^{2} = c^{2} (1 - q)^{2} > 0$ , so the sum over even $n$ diverges. Thus ${\tilde{π}}_{0} (υ_{0}^{2} (ϕ - q)^{2}) = \infty$ and [F2] gives $V_{0} (ϕ) = \infty$ , despite $ϕ, 1 \in Φ_{0}$ . This independently refutes the claimed initial weighted-estimator covariance finiteness.

Depends on

Finite-horizon SMC CLT classes and variance recursions

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Nicolas Chopin, Central Limit Theorem for Sequential Monte Carlo Methods and its Application to Bayesian Inference, Theorem 1 and Appendix A.3 (standard reference, not scraped)