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Moment measures of mixed empirical random point processes and marked point processes in compact metric spaces. I. (English. Ukrainian original) Zbl 1323.60067
Theory Probab. Math. Stat. 88, 161-174 (2014); translation from Teor. Jmovirn. Mat. Stat. 88, 144-156 (2013).
Summary: Moment measures of mixed empirical random point processes and marked point processes are investigated by using probability generating functions of random counting measures.

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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[1] N. G. Semeĭko, Yu. I. Petunin, and V. P. Yatsenko, Studying the morphometric characteristics of nuclear pore complexes of a sensory neuron using methods of spherical stochastic geometry, Kibernetika and Sistemnyi Analiz 42 (2006), no. 6, 175-182; English transl. in Cybernetics and Systems Analysis 42 (2006), no. 6, 917-922.
[2] Adrian Baddeley and Eva B. Vedel Jensen, Stereology for statisticians, Monographs on Statistics and Applied Probability, vol. 103, Chapman & Hall/CRC, Boca Raton, FL, 2005. · Zbl 1086.62108
[3] Geoffrey S. Watson, Mathematical morphology, A survey of statistical design and linear models (Proc. Internat. Sympos., Colorado State Univ., Ft. Collins, Colo., 1973) North-Holland, Amsterdam, 1975, pp. 547 – 553. · Zbl 0303.62051
[4] Alan F. Karr, Point processes and their statistical inference, 2nd ed., Probability: Pure and Applied, vol. 7, Marcel Dekker, Inc., New York, 1991. · Zbl 0733.62088
[5] M. Csörgő and P. Révész, Strong approximations in probability and statistics, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. · Zbl 0539.60029
[6] P. Gaenssler, Empirical Processes: On Some Basic Results from the Probabilistic Point of View, Institute of Mathematical Statistics, Hayward, CA, 1984.
[7] P. Gaenssler and W. Stute, On uniform convergence of measures with applications to uniform convergence of empirical distributions, Empirical distributions and processes (Selected Papers, Meeting on Math. Stochastics, Oberwolfach, 1976) Springer, Berlin, 1976, pp. 45 – 56. Lecture Notes in Math., Vol. 566. · Zbl 0349.60005
[8] David Pollard, Convergence of stochastic processes, Springer Series in Statistics, Springer-Verlag, New York, 1984. · Zbl 0544.60045
[9] Robert J. Serfling, Approximation theorems of mathematical statistics, John Wiley & Sons, Inc., New York, 1980. Wiley Series in Probability and Mathematical Statistics. · Zbl 0538.62002
[10] Yu. Ī. Petunīn and M. G. Semeĭko, Mixed empirical random point processes in compact metric spaces. I, Teor. Ĭmovīr. Mat. Stat. 74 (2006), 98 – 107 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 74 (2007), 113 – 123. · Zbl 1150.60381
[11] Yu. Ī. Petunīn and M. G. Semeĭko, Mixed empirical random point processes in compact metric spaces. II, Teor. Ĭmovīr. Mat. Stat. 75 (2006), 121 – 126 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 75 (2007), 139 – 145.
[12] N. G. Semeĭko, Mixed empirical Poisson random spherical-cap process, Kibernetika ta sistemnyi analiz (2011), no. 5, 119-130; English transl. in Cybernetics and Systems Analysis 47 (2011), no. 5, 773-782.
[13] J. E. Moyal, The general theory of stochastic population processes, Acta Math. 108 (1962), 1 – 31. · Zbl 0128.40302 · doi:10.1007/BF02545761 · doi.org
[14] B. D. Ripley, Locally finite random sets: foundations for point process theory, Ann. Probability 4 (1976), no. 6, 983 – 994. · Zbl 0359.60066
[15] Klaus Matthes, Johannes Kerstan, and Joseph Mecke, Infinitely divisible point processes, John Wiley & Sons, Chichester-New York-Brisbane, 1978. Translated from the German by B. Simon; Wiley Series in Probability and Mathematical Statistics. · Zbl 0383.60001
[16] Olav Kallenberg, Random measures, Akademie-Verlag, Berlin, 1975. Schriftenreihe des Zentralinstituts für Mathematik und Mechanik bei der Akademie der Wissenschaften der DDR, Heft 23. · Zbl 0345.60031
[17] Daryl J. Daley, Various concepts of orderliness for point-processes, Stochastic geometry (a tribute to the memory of Rollo Davidson), Wiley, London, 1974, pp. 148 – 161. · Zbl 0285.60039
[18] Garrett Birkhoff, Lattice Theory, American Mathematical Society Colloquium Publications, vol. 25, revised edition, American Mathematical Society, New York, N. Y., 1948. · Zbl 0033.10103
[19] Yu. I. Petunin and N. G. Semeĭko, A random process of segments on a two-dimensional Euclidean sphere. I, Teor. Veroyatnost. i Mat. Statist. 39 (1988), 107 – 113, 128 (Russian); English transl., Theory Probab. Math. Statist. 39 (1989), 129 – 135. · Zbl 0665.60019 · doi:10.1090/s0094-9000-07-00701-6 · doi.org
[20] Peter Gänssler, Empirical processes, Institute of Mathematical Statistics Lecture Notes — Monograph Series, vol. 3, Institute of Mathematical Statistics, Hayward, CA, 1983. · Zbl 1356.60003
[21] D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes, Springer Series in Statistics, Springer-Verlag, New York, 1988. · Zbl 0657.60069
[22] Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. · Zbl 0984.53001
[23] Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. · Zbl 0083.28204
[24] Справочник по теории вероятностей и математической статистике, ”Наукова Думка”, Киев, 1978 (Руссиан).
[25] William Feller, An Introduction to Probability Theory and Its Applications. Vol. I, John Wiley & Sons, Inc., New York, N.Y., 1950. · Zbl 0077.12201
[26] N. A. J. Hastings and J. B. Peacock, Statistical distributions, Halsted Press [John Wiley & Sons], New York-Toronto, Ont., 1975. A handbook for students and practitioners. · Zbl 0960.62001
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