RemarkRemark: Literature-sourcedProof: Not applicablePipeline-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.

Finite-particle SIR absolute and relative error scope

Remarks

Assuming AC, under positive prefixes and full multinomial resampling, Fixed-level SIR has an absolute O(L/N) mean-square bound proves the absolute bound $E [(\hat{p} - P (A_{L}))^{2}] \leq (L + 1) / (4 N)$ for event potentials $0 \leq G_{ℓ} \leq 1$ . This does not control relative error uniformly in rarity, because relative error divides by $P (A_{L})$ (equivalently, relative variance divides by $P (A_{L})^{2}$ ). The published Time-uniform SMC asymptotic variance under uniform weight and operator bounds concerns fixed-horizon CLT variances and does not establish a finite-particle bound. Cérou–Del Moral–Guyader's Theorem 1.5 is a distinct finite-particle result with additional potential-ratio and iterated-kernel domination hypotheses.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

Frédéric Cérou, Pierre Del Moral, and Arnaud Guyader, A nonasymptotic theorem for unnormalized Feynman–Kac particle models, Theorem 1.5 and §5 (standard reference, not scraped)