Poisson and Point Processes
A point process is a random scattering of points in time or space; the Poisson process, in which disjoint regions contain independent Poisson-distributed counts, is its fundamental example.
Definition
A point process is a random measure that places a discrete set of points in a space, and the Poisson process is the point process in which the number of points in any region is Poisson distributed with mean given by an intensity measure and counts in disjoint regions are independent.
Scope
This area covers the homogeneous Poisson process and its characterisations through independent exponential interarrival times and independent increments, inhomogeneous and compound Poisson processes, the general theory of point processes as random counting measures, intensity and marks, operations such as superposition, thinning, and mapping, and spatial point patterns.
Sub-topics
Core questions
- What defines a Poisson process and which equivalent characterisations describe it?
- How do independent increments and exponential interarrival times arise?
- How are point processes formalised as random counting measures?
- How do thinning, superposition, and mapping transform Poisson processes?
Key theories
- Characterisations of the Poisson process
- The homogeneous Poisson process is equivalently described by Poisson counts with independent increments, by independent and identically distributed exponential interarrival times, and as the unique simple point process with stationary independent increments and no fixed atoms.
- Mapping, thinning, and superposition theorems
- Independently displacing, randomly deleting, or merging points of Poisson processes again yields Poisson processes with transformed intensity measures, a robustness that makes the Poisson process the canonical model for completely random points.
Clinical relevance
Point processes model the arrival of customers, telephone calls, radioactive decays, insurance claims, neuronal spikes, and the spatial locations of trees, galaxies, or disease cases; the Poisson process serves as the baseline of complete spatial randomness against which clustering or regularity is judged.
History
The Poisson distribution arose in Poisson's 1837 work on judgments, the process was used by Erlang from 1909 to model telephone traffic and by Bateman and Rutherford for radioactive decay, and the modern measure-theoretic theory of point processes was consolidated in the later twentieth century by Kingman, Daley, and Vere-Jones.
Key figures
- Simeon Denis Poisson
- Agner Krarup Erlang
- John Kingman
Related topics
Seminal works
- kingman1993
Frequently asked questions
- What is a Poisson process?
- It is a model for points scattered completely at random in time or space, in which the number of points in any region follows a Poisson distribution and counts in non-overlapping regions are independent.
- Why is the Poisson process so widely used?
- It is the natural model of complete randomness, is preserved under thinning, superposition, and mapping, and arises as a limit whenever many rare independent events accumulate, making it a flexible and tractable baseline.