The Annals of Probability

A Martingale Approach to the Poisson Convergence of Simple Point Processes

Tim Brown

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Abstract

The paper concerns the Doob-Meyer increasing processes of simple point processes on the positive half line. It is shown that the weak convergence of such point processes to a simple Poisson process is implied by the pointwise weak convergence of their increasing processes, provided that the increasing processes satisfy a mild regularity condition. Conditions under which the regularity is satisfied are investigated. One condition is that the increasing process is that of the point process with its generated $\sigma$-fields. The Poisson convergence theorem is applied to superpositions of point processes.

Article information

Source
Ann. Probab., Volume 6, Number 4 (1978), 615-628.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995481

Digital Object Identifier
doi:10.1214/aop/1176995481

Mathematical Reviews number (MathSciNet)
MR482991

Zentralblatt MATH identifier
0383.60050

JSTOR
links.jstor.org

Subjects
Primary: 60G99: None of the above, but in this section
Secondary: 60G45

Keywords
Simple point processes local submartingale Doob-Meyer increasing process Poisson process weak convergence

Citation

Brown, Tim. A Martingale Approach to the Poisson Convergence of Simple Point Processes. Ann. Probab. 6 (1978), no. 4, 615--628. doi:10.1214/aop/1176995481. https://projecteuclid.org/euclid.aop/1176995481


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