The scalar Poisson count law

Depends on

• Probability measures and probability spaces
• The natural numbers $\mathbb{N}$ (von Neumann)
• The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$
• The real exponential function and the number $e$ by a power series

Used by

• Budgeted uniformization accumulation with time splitting Algorithm
• Explicit Poisson tau-leaping with an invalid-state status Algorithm
• Exposure-parameterized Poisson law Definition
• Poisson and Cox point processes Definition
• A binomial guard prevents overdraw but changes Poisson noise Example
• A CLE can assign negative density near an empty population Example
• A finite-window Poisson process that terminates at time one Example
• A minimum-intensity coupling reduces a Poisson count correction Example
• A Poisson leap can make a pure-death population negative Example
• Exact and pseudo likelihoods for finite-window Poisson and Strauss models Example
• Homogeneous Poisson likelihood and a Hawkes intensity specification Example
• Poisson and Cox counts with the same mean on a finite window Example
• Three-state birth-death chain with a certified Poisson tail Example
• Two-state gating through uniformization Example
• A geometric upper bound for a Poisson right tail Lemma
• Conditional moments of one Poisson tau-leap increment Lemma
• Finite-window homogeneous Poisson construction and ordered event-time density Lemma
• First and second moments of a Poisson random integral Lemma
• Independent Brownian motion straightens a Poisson clock Lemma
• Mecke identity for a finite-intensity Poisson point process Lemma
• Normalization, moments, and generating function of the Poisson count law Lemma
• One-step weak defect of finite-state Poisson tau-leaping Lemma
• Shared-minimum coupling of two Poisson counts Lemma
• Campbell, Laplace, Mecke, and Cox moment formulas Theorem
• Compensators, brackets, and jump-integral isometry for reaction counts Theorem
• Dominating thinning has the prescribed reaction law Theorem
• Existence of a locally finite Poisson point process Theorem
• Finite master equation and uniformization formula Theorem
• Finite-intensity Poisson-process existence and mixed-binomial law Theorem
• Mecke identity for locally finite Poisson point processes Theorem
• Strong invariance principle for geometrically ergodic chains Theorem
• Uniformization truncation, mass loss, and arithmetic-error ledger Theorem

Dependency tree · two levels

29 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

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