The set of real numbers left uncovered by random covering intervals

P. J. Fitzsimmons, Bert Fristedt, L. A. Shepp

Research output: Contribution to journalArticlepeer-review

27 Scopus citations


Random covering intervals are placed on the real line in a Poisson manner. Lebesgue measure governs their (random) locations and an arbitrary measure μ governs their (random) lengths. The uncovered set is a regenerative set in the sense of Hoffmann-Jørgensen's generalization of regenerative phenomena introduced by Kingman. Thus, as has previously been obtained by Mandelbrot, it is the closure of the image of a subordinator -one that is identified explicitly. Well-known facts about subordinators give Shepp's necessary and sufficient condition on μ for complete coverage and, when the coverage is not complete, a formula for the Hausdorff dimension of the uncovered set. The method does not seem to be applicable when the covering is not done in a Poisson manner or if the line is replaced by the plane or higher dimensional space.

Original languageEnglish (US)
Pages (from-to)175-189
Number of pages15
JournalZeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
Issue number2
StatePublished - Aug 1985


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