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August, 1978 A Martingale Approach to the Poisson Convergence of Simple Point Processes
Tim Brown
Ann. Probab. 6(4): 615-628 (August, 1978). DOI: 10.1214/aop/1176995481


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.


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Tim Brown. "A Martingale Approach to the Poisson Convergence of Simple Point Processes." Ann. Probab. 6 (4) 615 - 628, August, 1978.


Published: August, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0383.60050
MathSciNet: MR482991
Digital Object Identifier: 10.1214/aop/1176995481

Primary: 60G99
Secondary: 60G45

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

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 4 • August, 1978
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