How many processes have Poisson counts?

• Published in: Stochastic processes and their applications Vol. 98; no. 2; pp. 331 - 339
• Main Authors: Brown, Timothy C., Xia, Aihua
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01-04-2002; Elsevier Science; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: Distribution theory; Exact sciences and technology; Finite dimensional distribution; Mathematics; Phase transition; Point process; Point process Poisson process Finite dimensional distribution Probability generating function Phase transition; Poisson process; Probability and statistics; Probability generating function; Probability theory and stochastic processes; Sciences and techniques of general use; Stochastic processes; secondary 60G07; Finite dimensional distribution; Point process; Probability generating function; Poisson process; primary 60G55; Phase transition; Completeness; Randomness; Distribution function; Poisson distribution; Joint distribution; Mean estimation; Phase transitions
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/S0304-4149(01)00150-8

The Poisson process has the well-known Poisson count property: the count of points in any subset of the carrier space has a Poisson distribution. To specify the complete distribution of a point process it is necessary and sufficient to specify all of the joint distributions of the counts of points in any (finite disjoint) collection of bounded sets in the carrier space. Suppose that only the Poisson count property is specified for a random collection of points. We reveal the circumstances in which the Poisson count property does indeed determine the distribution. Curiously, there is a ‘phase transition’ in this property with the boundary being mean measures having 2 atoms.