# zbMATH — the first resource for mathematics

Mixed empirical Poisson random spherical-cap process. (English. Russian original) Zbl 1306.60054
Cybern. Syst. Anal. 47, No. 5, 773-782 (2011); translation from Kibern. Sist. Anal. 2011, No. 5, 119-130 (2011).
Summary: A mathematical model of a mixed empirical Poisson random cap process (RCP) on the sphere $$S^2$$ is investigated using the theory of mixed empirical marked point processes. The first-order moment measure of the RCP is proposed for spherical sets of special form.
##### MSC:
 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60D05 Geometric probability and stochastic geometry
Full Text:
##### References:
 [1] M. G. Kendall and P. A. P. Moran, Geometric Probability, Hafner, New York (1963). [2] P. A. P. Moran and S. Fazekas, ”Random circles on a sphere,” Biometrics, 49, No. 3, 4, 389–396 (1962). · Zbl 0108.31602 · doi:10.1093/biomet/49.3-4.389 [3] N. G. Semeiko, 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,” Cybern. Syst. Analysis, 42, No. 6, 917–924 (2006). · Zbl 05206913 · doi:10.1007/s10559-006-0131-2 [4] C. A. Cross and D. L. Fisher, ”The computer simulation of lunar craters,” Month. Not. Roy. Austral. Soc., 139, 261–272 (1968). · doi:10.1093/mnras/139.2.261 [5] A. H. Marcus, ”Some point process models of lunar and planetary surfaces,” in: P. A. W. Lewis (ed.), Stochastic Point Processes: Statistical Analysis, Theory and Applications, New York (1972), pp. 682–699. [6] J. Neyman and R. L. Scott, ”Statistical approach to problems of cosmology,” J. Roy. Stat. Soc., Ser. B, 20, 1–43 (1958). · Zbl 0085.42906 [7] E. N. Gilbert, ”The probability of covering a sphere with N circular caps,” Biometrics, 52, No. 3, 4, 323–330 (1965). · Zbl 0137.36202 · doi:10.1093/biomet/52.3-4.323 [8] E. P. Harding and D. G. Kendall, Stochastic Geometry, Wiley, New York (1974). [9] K. V. Mardia, Statistics of Directional Data, Academ. Press, London (1972). · Zbl 0244.62005 [10] N. Bourbaki, Elements of Mathematics. General Topology, Addison-Wesley (1966). · Zbl 0145.19302 [11] R. E. Miles, ”Random points, set and tessellations on the surface of a sphere,” Sankhya, Ser. A, 33, No. 2, 145–174 (1971). · Zbl 0243.60014 [12] B. D. Ripley, ”Locally finite random sets: Foundations for point process theory,” Ann. Prob., 4, No. 6, 983–994 (1976). · Zbl 0359.60066 · doi:10.1214/aop/1176995941 [13] Yu. I. Petunin and M. G. Semeiko, ”Mixed empirical stochastic point processes in compact metric spaces. I,” Theor. Probability Math. Statist., Issue 74, 113–123 (2007). [14] Yu. I. Petunin and N.G. Semeiko, ”Random cap process on a two-dimensional Euclidean sphere. I,” Teor. Veroyatn. Mat. Statist., Issue 39, 107–113 (1988). [15] Yu. I. Petunin and M. G. Semeiko, ”Mixed empirical stochastic point processes in compact metric spaces. II,” Theor. Probability Math. Statist., Issue 75, 139–145 (2007). · Zbl 1164.60350 [16] M. G. Semeiko, ”Mixed empirical random marked processes in compact metric spaces,” Teor. Veroyatn. Mat. Statist., (2009). [17] M. Loeve, Probability Theory, Van Nostrand, Princeton (1963). [18] Yu. I. Petunin and N.G. Semeiko, ”Random cap process on a two-dimensional Euclidean sphere. II,” Teor. Veroyatn. Mat. Statist., Issue 41, 88–96 (1989).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.