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A stopping time concerning sphere data and its applications

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Li-Xing Zhu ,
Ping Cheng ,
Gang Wei  and
Pei-De Shi
Li-Xing Zhu*
Affiliation:
Institute of Applied Mathematics, Beijing
Ping Cheng*
Affiliation:
Institute of Systems Science, Beijing
Gang Wei*
Affiliation:
Institute of Applied Mathematics, Beijing
Pei-De Shi*
Affiliation:
Institute of Systems Science, Beijing
*
∗ Postal address: Institute of Applied Mathematics, Chinese Academy of Sciences, PO Box 2734, Beijing 100080, People's Republic of China.
∗∗ Postal address: Institute of Systems Science, Chinese Academy of Sciences, Beijing 100080, People's Republic of China.
∗ Postal address: Institute of Applied Mathematics, Chinese Academy of Sciences, PO Box 2734, Beijing 100080, People's Republic of China.
∗∗ Postal address: Institute of Systems Science, Chinese Academy of Sciences, Beijing 100080, People's Republic of China.
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Abstract

Denote by A(x) = {a: |aτx| ≦ h} a circle zone on the three-dimensional sphere surface for each given h > 0. For a given integer m, we investigate how many zones chosen randomly are needed to contain at least one of the points on the sphere surface m times. As an application, the lifetime of a sphere roller is investigated. We present empirical formulas for the mean, standard deviation and distribution of the lifetime of the sphere roller. Furthermore, some limit behaviors of the above stopping time are obtained, such as the limit distribution, the law of the iterated logarithm, and the upper and lower bounds of the tail probability with the same convergent order.

Keywords

STOPPING TIMELIFETIME OF SPHERE ROLLERNUMBER-THEORETIC METHOD

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 28 , Issue 3 , September 1996 , pp. 674 - 686
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

This work is partly supported by the National Natural Science Foundation of China.

References

Alexander, K. S. (1984) Probability inequalities for empirical processes and a law of the iterated logarithm. Ann. Prob. 12, 1041–1067.Google Scholar
Beran, R. and Millar, P. W. (1986) Confidence sets for a multivariate distribution. Ann. Statist. 14, 431–443.Google Scholar
Cheng, P. (1983) An applied mathematical problem. Math. Pract. Theory 2, 79–89.Google Scholar
Cheng, P., Zhu, L. X., Wei, G. and Shi, P. D. (1991) On the life of sphere roller. Acta Math. Sci. 11, 308–316.CrossRefGoogle Scholar
Dudley, R. M. (1978) Central limit theorems for empirical measure. Ann. Prob. 6, 899–929.CrossRefGoogle Scholar
Fang, K. T. Kotz, S. and Ng, K. W. (1990) Symmetric Multivariate and Related Distribution. Chapman and Hall, London.Google Scholar
Fang, K. T. and Wang, Y. (1993) The Applications of Number Theoretic Methods in Statistics . Chapman and Hall, London.Google Scholar
Fang, K. T. and Wei, G. (1992) The distribution of a special first-hit random variable. Acta Appl. Math. Sinica 15, 460–467. (In Chinese).Google Scholar
Hua, L. G. and Wang, Y. (1981) The Applications of Number Theory to Approximate Analysis. Academic Press, Beijing.Google Scholar
Huber, P. (1985) Projection pursuit (with discussion). Ann. Statist. 13, 435–475.Google Scholar
Pollard, D. (1984) Convergence of Stochastic Processes. Springer, New York.Google Scholar
Pollard, D. (1989) Asymptotics via empirical processes (with discussion). Stat. Sci. 4, 341–366.Google Scholar
Pollard, D. (1990) Empirical Processes Theory and Applications. (NSF-CBMS Regional Conf. Series in Probability and Statistics.) Volume 2.CrossRefGoogle Scholar
Stout, W. F. (1974) Almost Sure Convergence. Academic Press, New York.Google Scholar
Wang, Y. and Fang, K. T. (1990) Number theoretic methods in applied statistics. Chinese Ann. Math. 11, 41–55.Google Scholar
Weber, M. (1989) The supremum of Gaussian processes with a constant variance. Prob. Theory Rel. Fields. 81, 585–592.Google Scholar
Zhu, L. X. and Cheng, P. (1994) The optimal lower bound for the tail probability of the Kolmogorov distance in the worst direction. Sankhya 56, 264–293.Google Scholar