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.

Finite-budget stopped-event kernel with a cemetery outcome

Definition

Let $(E, E)$ be a standard Borel space, $M$ a probability kernel on it, and $T, R \in E$ disjoint target and return sets. Write $S = E ∖ (T \cup R)$ and let $C = {h i t, r e t u r n, \partial}$ have the discrete sigma-algebra; $\partial$ is the cemetery outcome meaning the budget expired before either set was reached.

For budget $m = 0$ , define $κ_{0} (x, \cdot) = {\begin{cases} δ_{h i t}, & x \in T, \\ δ_{r e t u r n}, & x \in R, \\ δ_{\partial}, & x \in S . \end{cases}$ For $m \geq 0$ , define the finite-budget stopped-status kernel recursively by $κ_{m + 1} (x, \cdot) = {\begin{cases} δ_{h i t}, & x \in T, \\ δ_{r e t u r n}, & x \in R, \\ \int_{E} κ_{m} (y, \cdot) M (x, d y), & x \in S . \end{cases}$ Thus a target hit at the initial state counts at budget zero; each additional budget unit permits one more transition. The exact-hitting status kernel, when used, is the limit law of these statuses and treats nontermination as $\partial$ ; it is an exact sampler for $P_{x} (τ_{T} < τ_{R})$ only when the requested event is the finite first hit represented by that limit. The budgeted kernel is distinct from an exact-hitting oracle.

Depends on

Used by

Dependency tree · two levels

16 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

Adam M. Johansen, Pierre Del Moral, and Arnaud Doucet, Sequential Monte Carlo Samplers for Rare Events, `3 (standard reference, not scraped)
Frédéric Cérou, Pierre Del Moral, and Arnaud Guyader, A nonasymptotic theorem for unnormalized Feynman–Kac particle models, `6 (standard reference, not scraped)