A limit theorem for random coverings of a circle |
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Authors: | Leopold Flatto |
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Institution: | (1) Belfer Graduate School of Science, Yeshiva University, New York, U.S.A. |
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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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Keywords: | |
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