The Annals of Probability

Random Measures with Aftereffects

Larry P. Ammann and Peter F. Thall

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Abstract

A class of $\mathscr{D}$ of random measures, generalizing the class of completely random measures, is developed and shown to contain the class of Poisson cluster point processes. An integral representation is obtained for $\mathscr{D}$, generalizing the Levy-Ito representation for processes with independent increments. A subclass $\mathscr{D}_n \subset \mathscr{D}$ is defined such that for $X \in \mathscr{D}_n$, the distribution of the random vector $X(A_1), \cdots, X(A_m), m > n, A_1, \cdots, A_m$ disjoint, is determined by the distributions of all subvectors $X(A_{i_1}), \cdots, X(A_{i_k}), 1 \leqq k \leqq n$. The class $\mathscr{D}_1$ coincides with the class of completely random measures.

Article information

Source
Ann. Probab., Volume 6, Number 2 (1978), 216-230.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995569

Digital Object Identifier
doi:10.1214/aop/1176995569

Mathematical Reviews number (MathSciNet)
MR474490

Zentralblatt MATH identifier
0374.60046

JSTOR
links.jstor.org

Subjects
Primary: 60G05: Foundations of stochastic processes
Secondary: 60G20: Generalized stochastic processes 60G17: Sample path properties

Keywords
Infinitely divisible stochastic process stochastic point process random measure completely random measure probability generating functional

Citation

Ammann, Larry P.; Thall, Peter F. Random Measures with Aftereffects. Ann. Probab. 6 (1978), no. 2, 216--230. doi:10.1214/aop/1176995569. https://projecteuclid.org/euclid.aop/1176995569


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