Certain statistical applications deal with the extremal distributions of the number of points in a moving interval or window of fixed length. This article gives an approximation that is highly accurate for several of these distributions. Applications include the maximum cluster of points on a line or circle, multiple coverage by subintervals or subarcs of fixed size, the length of the longest success run in Bernoulli trials, and the generalized birthday problem.
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Joseph Naus (1982) studied this question.
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