Abstract.
It is well-known that the distribution of a point process Ξ defined on a carrier space Γ is uniquely characterised by its finite dimensional joint distributions of counts on disjoint subsets of Γ. In this note, we investigate the common structure of point processes whose distributions are specified by their one dimensional distributions. We also show that, if Ξ is such a point process, then a sequence of point processes {Ξ n } converges in distribution to Ξ if and only if {Ξ n (B)} converges in distribution to Ξ(B) for a suitably rich class of sets B⊂Γ.
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Supported by ARC Discovery project number DP0209179
Mathmatics Subject Classification (2000): Primary 60G55; Secondary 60E05, 60B10
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Xia, A. Point processes characterized by their one dimensional distributions. Probab. Theory Relat. Fields 128, 467–474 (2004). https://doi.org/10.1007/s00440-003-0314-y
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DOI: https://doi.org/10.1007/s00440-003-0314-y
Keywords
- Point process
- Simple point process
- Ordinary point process
- Weakly ordinary point process
- Poisson process
- Finite dimensional distribution
- Probability generating function