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A limit theorem for random coverings of a circle

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Abstract

LetN α, m equal the number of randomly placed arcs of length α (0<α<1) required to cover a circleC of unit circumferencem times. We prove that limα→0 P(Nα,m≦(1/α) (log (1/α)+mlog log(1/α)+x)=exp ((−1/(m−1)!) exp (−x)). Using this result for m=1, we obtain another derivation of Steutel's resultE(Nα,1)=(1/α) (log(1/α)+log log(1/α)+γ+o(1)) as α→0, γ denoting Euler's constant.

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Authors and Affiliations

  1. Belfer Graduate School of Science, Yeshiva University, New York, U.S.A.

    Leopold Flatto

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  1. Leopold Flatto
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Flatto, L. A limit theorem for random coverings of a circle. Israel J. Math. 15, 167–184 (1973). https://doi.org/10.1007/BF02764603

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  • DOI: https://doi.org/10.1007/BF02764603

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