DefinitionDefinition: 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.

Nested rare-event levels and conditional laws

Definition

Let $(Ω, F, P)$ be a probability space as in Probability measures and probability spaces. A finite nested event sequence of length $L \geq 1$ is a sequence of measurable sets $Ω = A_{0} \supseteq A_{1} \supseteq \dots \supseteq A_{L} = A$ such that $P (A_{ℓ}) > 0$ for $0 \leq ℓ \leq L$ . Its conditional level probabilities and level laws are $p_{ℓ} = \frac{P (A_{ℓ})}{P (A_{ℓ - 1})} \in (0, 1], π_{ℓ} (B) = \frac{P (B \cap A_{ℓ})}{P (A_{ℓ})} (B \in F) .$ The target probability is $p = P (A_{L})$ . Independence across levels means that the complete sample arrays used at distinct levels are mutually independent; correct marginal laws alone do not imply this condition.

Depends on

Probability measures and probability spaces

Used by

Full-resampling fixed-level SIR splitting with invariant mutations Algorithm
Independent fixed-level splitting Algorithm
Conditional future means alone do not bound finite-particle SIR error Counterexample
Correct level marginals do not justify a dependent product Counterexample
A zero-particle prefix does not prove a zero event probability Example
Nested-event probabilities factor into conditional levels Lemma
Fixed-level full-resampling SIR splitting is unbiased Theorem
Fixed-level SIR has an absolute O(L/N) mean-square bound Theorem
Independent splitting estimator moments and work Theorem

Dependency tree · two levels

3 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

Gino Biondini, An Introduction to Rare Event Simulation and Importance Sampling, Handbook of Statistics 33, Chapter 2, §8 (standard reference, not scraped)
Adam M. Johansen, Pierre Del Moral, and Arnaud Doucet, Sequential Monte Carlo Samplers for Rare Events, ``2–3 (standard reference, not scraped)