# Thinning of point processes, revisited

Thinning of point processes, revisited LetN,N 1,N 2 be simple point processes on a LCCB space (E,ε) such thatN=N 1+N 2, andp(·) be a measurable function with 0<p(·)<1 on (E,ε). Then any two of the following statements yield another two: (I) N is a Poisson process; (II) N 1 is thep(·)-thinning ofN,N 2 is the (1−p(·))-thinning ofN; (III) N 1 andN 2 are independent; (IV) N 1,N 2 are Poisson processes with respect to a filtration {F(A),A∈g3}, where $$F(A) = \sigma \{ N_1 (B),N_2 (B),B \in \varepsilon ,B \subset A\}$$ , i.e., for each bounded setA∈ε,N 1(A) andN 2(A) are Poisson variables, independent ofF(A c ). Indeed, only the fact, (II)+(III)æ(IV)+(I), is new. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# Thinning of point processes, revisited

, Volume 12 (3) – Jul 14, 2005
6 pages

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Publisher
Springer Journals
Copyright © 1996 by Science Press, Beijing, China and Allerton Press, Inc., New York, U.S.A.
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/BF02011892
Publisher site
See Article on Publisher Site

### Abstract

LetN,N 1,N 2 be simple point processes on a LCCB space (E,ε) such thatN=N 1+N 2, andp(·) be a measurable function with 0<p(·)<1 on (E,ε). Then any two of the following statements yield another two: (I) N is a Poisson process; (II) N 1 is thep(·)-thinning ofN,N 2 is the (1−p(·))-thinning ofN; (III) N 1 andN 2 are independent; (IV) N 1,N 2 are Poisson processes with respect to a filtration {F(A),A∈g3}, where $$F(A) = \sigma \{ N_1 (B),N_2 (B),B \in \varepsilon ,B \subset A\}$$ , i.e., for each bounded setA∈ε,N 1(A) andN 2(A) are Poisson variables, independent ofF(A c ). Indeed, only the fact, (II)+(III)æ(IV)+(I), is new.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jul 14, 2005

### References

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