Thinning of point processes,revisited |
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| Authors: | Shengwu He Jiagang Wang |
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| Institution: | 1. Department of Mathematical Statistics, East China Normal University, 200062, Shanghai, China
2. Institute of Applied Mathematics, East China University of Chemical Technology, 200237, Shanghai, China
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| 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: - N is a Poisson process;
- N 1 is thep(·)-thinning ofN,N 2 is the (1?p(·))-thinning ofN;
- N 1 andN 2 are independent;
- 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. |
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| Keywords: | Point process Poisson process thinning of point process Laplace functional |
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