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A Stopping Time concerning Sphere Data and Its Applications

Li-Xing Zhu, Ping Cheng, Gang Wei and Pei-De Shi
Advances in Applied Probability
Vol. 28, No. 3 (Sep., 1996), pp. 674-686
DOI: 10.2307/1428176
Stable URL: http://www.jstor.org/stable/1428176
Page Count: 13
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
A Stopping Time concerning Sphere Data and Its Applications
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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.

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