Uniqueness of Unbounded Loeb Measure Using Choquet's Theorem

Multidimensional random point processes are shown to be a special case of Matheron's random closed sets. This theory, and in particular Choquet's theorem, are then used to prove a general result on heterogeneous Poisson processes and a uniqueness property of mixed Poisson processes with gamma mixing distributions.

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ON THE STRUCTURE OF THE SPACE OF GEOMETRIC PRODUCT-FORM MODELS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964802162073 ◽

2002 ◽

Vol 16 (2) ◽

pp. 241-270 ◽

Cited By ~ 2

Author(s):

Nimrod Bayer ◽

Richard J. Boucherie

Keyword(s):

Invariant Measures ◽

Mathematical Object ◽

Product Form ◽

State Transitions ◽

State Spaces ◽

Discrete State ◽

Finite Dimensional ◽

Geometric Product ◽

Choquet's Theorem ◽

Discrete State Space

This article deals with Markovian models defined on a finite-dimensional discrete state space and possess a stationary state distribution of a product-form. We view the space of such models as a mathematical object and explore its structure. We focus on models on an orthant [script Z]+n, which are homogeneous within subsets of [script Z]+n called walls, and permit only state transitions whose ∥ ∥∞-length is 1. The main finding is that the space of such models exhibits a decoupling principle: In order to produce a given product-form distribution, the transition rates on distinct walls of the same dimension can be selected without mutual interference. This principle holds also for state spaces with multiple corners (e.g., bounded boxes in [script Z]+n).In addition, we consider models which are homogeneous throughout a finite-dimensional grid [script Z]n, now without a fixed restriction on the length of the transitions. We characterize the collection of product-form measures which are invariant for a model of this kind. For such models with bounded transitions, we prove, using Choquet's theorem, that the only possible invariant measures are product-form measures and their combinations.

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Multidimensional point processes and random closed sets

Journal of Applied Probability ◽

10.1017/s0021900200024487 ◽

1984 ◽

Vol 21 (01) ◽

pp. 173-178 ◽

Cited By ~ 5

Author(s):

Michel Baudin

Keyword(s):

General Result ◽

Point Processes ◽

Poisson Processes ◽

Random Point ◽

Uniqueness Property ◽

Closed Sets ◽

Random Closed Sets ◽

Mixing Distributions ◽

Choquet's Theorem ◽

Special Case

Multidimensional random point processes are shown to be a special case of Matheron's random closed sets. This theory, and in particular Choquet's theorem, are then used to prove a general result on heterogeneous Poisson processes and a uniqueness property of mixed Poisson processes with gamma mixing distributions.

Download Full-text