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Some remarks on conditional distributions for point processes. (English) Zbl 0712.60055
Let N be a point process on a Polish space E, and let \(C'\) be the compound Campbell measure of N. For A a subset of E, let \(N_ A()=N(\cdot \cap A)\) denote the restriction of N to A. Using a standard representation of certain conditional distributions, the author obtains several theorems concerning distributions of point processes. These include a consistency theorem for conditional distributions \(P\{\) \(N\in (\cdot) | N_{B\cup C}\}\), where B and C are bounded sets, an expression for conditional probabilities \(P\{N_ a\in (\cdot) | N_{B^ c}\}\) in terms of the Gibbs measure of N and some properties of thinned point processes and Cox processes.
The results are very similar to those in O. Kallenberg, Random measures. 3 rd ed. (1983; Zbl 0544.60053)].
Reviewer: A.F.Karr

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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[1] Georgii, H.O., Canonical Gibbs measures, () · Zbl 0409.60094
[2] Georgii, H.O., Equilibria for particle motions: conditionally balanced point random fields, () · Zbl 0495.60054
[3] Kallenberg, O., Random measures, (1983), Akademie-Verlag Berlin · Zbl 0288.60053
[4] Kallianpur, G.; Striebel, C., Estimation or stochastic systems: arbitrary system processes with additive white noise observation errors white noise observation errors, Ann. math. statist., 39, 785-801, (1968) · Zbl 0174.22102
[5] Karr, A.F., State estimation for Cox processes with unknown probability law, Stochastic process. appl., 20, 115-131, (1985) · Zbl 0578.60049
[6] Karr, A.F., Point processes and their statistical inference, (1986), Dekker New York and Basel · Zbl 0601.62120
[7] Matthes, K.; Kerstan, J.; Mecke, J., Infinitely divisible point processes, (1978), Wiley New York · Zbl 0383.60001
[8] Matthes, K.; Warmuth, W.; Mecke, J., Bemerkungen zu einer arbeit von nguyen xuan xanh und Hans zessin, Math. nachr., 88, 117-127, (1979) · Zbl 0417.60062
[9] Mecke, J., Stationäre zufällige maβe auf lokalkompakten abelschen gruppen, Z. wahrsch. verw. gebiete, 9, 36-58, (1967) · Zbl 0164.46601
[10] Nguyen, X.X.; Zessin, H., Integral and differential characterizations of the Gibbs process, Math. nachr., 88, 105-115, (1979) · Zbl 0444.60040
[11] Papangelou, F., The conditional intensity of general point processes and an application to line processes, Z. wahrsch. verw. gebiete, 28, 207-226, (1974) · Zbl 0265.60047
[12] Preston, C., Random fields, () · Zbl 0357.60052
[13] Ryll-Nardzewski, C., Remarks on processes of calls, (), 455-465
[14] Wakolbinger, A.; Eder, G., A condition ∑^{c}λ for point processes, Math. nachr., 116, 209-232, (1984) · Zbl 0583.60043
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