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Holst [10] introduced a discrete spacings model that is related to the Bose-Einstein distribution and obtained the distribution of the number of vacant positions in an associated circle covering problem. We correct his expression for its probability mass function, obtain the first two moments, and describe their limiting properties. We then examine the properties of the vacancy statistic when the number of covering arcs in the associated circle covering problem is random. We also discuss applications of our results to a study of contagion in networks.

Properties of the Vacancy Statistic in the Discrete Circle Covering Problem