The Annals of Probability
- Ann. Probab.
- Volume 6, Number 4 (1978), 615-628.
A Martingale Approach to the Poisson Convergence of Simple Point Processes
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.
Ann. Probab., Volume 6, Number 4 (1978), 615-628.
First available in Project Euclid: 19 April 2007
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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