Countable random sets: uniqueness in law and constructiveness. (English) Zbl 1277.60086
Summary: The first part of this article deals with theorems on uniqueness in law for \(\sigma \)-finite and constructive countable random sets, which in contrast to the usual assumptions may have points of accumulation. We discuss and compare two approaches on uniqueness theorems: first, the study of generators for \(\sigma \)-fields used in this context and, secondly, the analysis of hitting functions.
The last section of this paper deals with the notion of constructiveness. We prove a measurable selection theorem and a decomposition theorem for constructive countable random sets, and study constructive countable random sets with independent increments.
The last section of this paper deals with the notion of constructiveness. We prove a measurable selection theorem and a decomposition theorem for constructive countable random sets, and study constructive countable random sets with independent increments.
MSC:
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 60D05 | Geometric probability and stochastic geometry |
| 28B20 | Set-valued set functions and measures; integration of set-valued functions; measurable selections |
Keywords:
constructive countability; constructiveness; countable random sets; decomposition; generators; hitting functions; independent increments; measurable selections; point processes; Poisson processes; Rényi; uniqueness in lawReferences:
| [1] | Blackwell, D., Dubins, L.E.: On existence and non-existence of proper, regular, conditional distributions. Ann. Probab. 3, 741-752 (1975) · Zbl 0348.60003 · doi:10.1214/aop/1176996261 |
| [2] | Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Springer, New York (1988) · Zbl 0657.60069 |
| [3] | Dravecký, J.: Spaces with measurable diagonal. Math. Slovaca 25(1), 3-9 (1975). http://dml.cz/dmlcz/127041 · Zbl 0295.28002 |
| [4] | Halmos, P.R.: Measure Theory. Van Nostrand, New York (1951) · Zbl 0045.05702 |
| [5] | Halmos, P.R.: Lectures on Boolean Algebras. Van Nostrand, Princeton (1963) · Zbl 0114.01603 |
| [6] | Himmelberg, C.J.: Measurable relations. Fundam. Math. 87, 53-72 (1975) · Zbl 0296.28003 |
| [7] | Himmelberg, C.J., Parthasarathy, T., Van Vleck, F.S.: On measurable relations. Fundam. Math. 111, 161-167 (1981) · Zbl 0465.28002 |
| [8] | Kallenberg, O.: Random Measures. Akademie Verlag, Berlin (1976) · Zbl 0345.60032 |
| [9] | Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2002) · Zbl 0996.60001 |
| [10] | Kendall, W.S.: Stationary countable dense random sets. Adv. Appl. Probab. 32, 86-100 (2000) · Zbl 0999.60012 · doi:10.1239/aap/1013540024 |
| [11] | Kingman, J.F.C.: Poisson Processes. Clarendon, Oxford (1993) · Zbl 0771.60001 |
| [12] | Leadbetter, M. R., On basic results of point process theory, No. 3, 449-462 (1972) · Zbl 0274.60039 |
| [13] | Matheron, G.: Random Sets and Integral Geometry. John Wiley, Chichester (1975) · Zbl 0321.60009 |
| [14] | Mönch, G.: Verallgemeinerung eines Satzes von A. Rényi. Studia Sci. Math. Hung. 6, 81-90 (1971) · Zbl 0308.60030 |
| [15] | Rényi, A.: Remarks on the Poisson process. Studia Sci. Math. Hung. 2, 119-123 (1967) · Zbl 0149.12904 |
| [16] | Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987) · Zbl 0925.00005 |
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