A stopping time concerning sphere data and its applications

Denote by A( x ) = { a : | a τ x | ≦ 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...

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Bibliographic Details
Published in:Advances in applied probability 1996-09, Vol.28 (3), p.674-686
Main Authors: Zhu, Li-Xing, Cheng, Ping, Wei, Gang, Shi, Pei-De
Format: Article
Language:English
Subjects:
Applied mathematics
Distribution theory
Exact sciences and technology
Gaussian distributions
Integers
Logarithms
Mathematical analysis
Mathematics
Monte Carlo methods
Multivariate analysis
Probability
Probability and statistics
Probability theory and stochastic processes
Sample mean
Sciences and techniques of general use
Sine function
Standard deviation
Statistics
Stochastic Geometry and Statistical Applications
Stochastic processes
Stopping distances
Systems science
Time & motion studies
Useful life
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Summary:Denote by A( x ) = { a : | a τ x | ≦ 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.
ISSN:0001-8678
1475-6064
DOI:10.2307/1428176