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.
Publié le : 1978-08-14
Classification:
Simple point processes,
local submartingale,
Doob-Meyer increasing process,
Poisson process,
weak convergence,
60G99,
60G45
@article{1176995481,
author = {Brown, Tim},
title = {A Martingale Approach to the Poisson Convergence of Simple Point Processes},
journal = {Ann. Probab.},
volume = {6},
number = {6},
year = {1978},
pages = { 615-628},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995481}
}
Brown, Tim. A Martingale Approach to the Poisson Convergence of Simple Point Processes. Ann. Probab., Tome 6 (1978) no. 6, pp. 615-628. http://gdmltest.u-ga.fr/item/1176995481/