Abstract
Denote byA(x) = {a: |aτx| ≦h} a circle zone on the three-dimensional sphere surface for each givenh> 0. For a given integerm, we investigate how many zones chosen randomly are needed to contain at least one of the points on the sphere surfacemtimes. 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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