ExampleConstruction: 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.

Fixed-horizon asymptotic variance can grow without mixing

Example

Let $X = {0, 1}$ have the uniform target and proposal at time zero. Use identity mutation, unit incremental weights, and multinomial resampling at every time. For each fixed horizon $t$ , take $ϕ (x_{0 : t}) = 1_{{x_{t} = 1}}$ . Compute the fixed-horizon asymptotic variance and compare its dependence on $t$ with a time-uniform bound.

Facts & Assumptions

Given: The uniform initial law, identity mutation, unit weights, multinomial selection, and terminal indicator in the Example.

[F1]

Finite-horizon SMC CLT classes and variance recursions: With identity mutation and unit weights, $E_{s} h = h$ and $π_{s} = π_{0}$ .

[F2]

Multinomial SMC variance unrolls into local contributions: $V_{t} (ϕ)$ is the sum of the proposal expectations of squared backward images of $ϕ - π_{t} ϕ$ over $k = 0, \dots, t$ .

[F3]

Fixed-horizon SMC central limit theorem with multinomial resampling: For each fixed horizon and the stated class, the weighted estimate has CLT variance $V_{t} (ϕ)$ and the post-resampling estimate has variance $V_{t} (ϕ) + {Var}_{π_{t}} (ϕ)$ .

Verification

Given: The finite-state identity-mutation filter in the Example and any fixed integer $t \geq 0$ .

1.1F1F2givenalgebra

The identity kernel leaves the uniform law invariant and has incremental weight one. Thus [F1] gives $E_{k + 1 : t} (ϕ - π_{t} ϕ) = ϕ - 1 / 2$ for every $k \leq t$ . Since $(ϕ - 1 / 2)^{2} = 1 / 4$ everywhere, each of the $t + 1$ summands in [F2] equals $1 / 4$ . Therefore $V_{t} (ϕ) = (t + 1) / 4$ .

2.1F3step 1.1algebra∎

By [F3], at every fixed $t$ the weighted estimate has a valid CLT with variance $(t + 1) / 4$ , and the post-resampling estimate has variance $(t + 2) / 4$ . Both variance sequences are unbounded as $t$ ranges over the integers. The fixed-horizon theorem therefore supplies no uniform-in-time variance bound or finite-particle accuracy guarantee.

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