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.

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Finite-window Gibbs density relative to a Poisson reference

Definition

Let $W$ and $C (W)$ be as in Finite-window spatial observations and configurations, and let $Q$ be a specified finite-window Poisson probability measure on that space. For each parameter $θ$ , let $S_{θ} \subseteq C (W)$ be a measurable admissible support and let $H_{θ} : S_{θ} \to R$ be measurable. Extend the weight by zero off $S_{θ}$ : $w_{θ} (η) = e^{- H_{θ} (η)}$ on $S_{θ}$ and $0$ elsewhere. Put $Z_{θ} : = \int_{C (W)} w_{θ} (η) Q (d η) .$ Only when $0 < Z_{θ} < \infty$ do these data define the finite-window Gibbs law, with density $\frac{d P_{θ}}{d Q} (η) = Z_{θ}^{- 1} w_{θ} (η) .$ The support and normalizer are part of the model specification. No general existence, mixing, consistency, or asymptotic result is included in this definition.

Depends on

Used by

Dependency tree · two levels

35 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

Adrian Baddeley, Spatial Point Processes and their Applications, complete lecture notes (standard reference, not scraped)
Günter Last and Mathew Penrose, Lectures on the Poisson Process, author-hosted 2017 version (standard reference, not scraped)