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3 results

  • On the random coverage of the circle

  • L. Holst, J. Hüsler
  • Journal:
    Journal of Applied Probability / Volume 21 / Issue 3 / September 1984
    Published online by Cambridge University Press:
    14 July 2016, pp. 558-566
    Print publication:
    September 1984

  • Place n arcs of equal length a uniformly at random on the circumference of a circle. We discuss the joint limit distributions of the number of gaps, the uncovered proportion of the circle and the lengths of the largest gap and of the smallest gap, depending on how a → 0 as n →∞.

    We show that the results may be proved in a unified and simple way by using a result of Le Cam.

  • On the lengths of the pieces of a stick broken at random

  • Lars Holst
  • Journal:
    Journal of Applied Probability / Volume 17 / Issue 3 / September 1980
    Published online by Cambridge University Press:
    14 July 2016, pp. 623-634
    Print publication:
    September 1980

  • Consider the pieces of a randomly broken stick. How long is the j th longest piece? How many breaks are necessary for getting all pieces less than a given length? These and related questions are studied in particular when the number of pieces is large. Using simple properties of the exponential distribution new proofs are given of old results and new results are obtained.

  • On multiple covering of a circle with random arcs

  • Lars Holst
  • Journal:
    Journal of Applied Probability / Volume 17 / Issue 1 / March 1980
    Published online by Cambridge University Press:
    14 July 2016, pp. 284-290
    Print publication:
    March 1980

  • On a circle of unit circumference arcs of length a are placed at random. Let Nα be equal to the necessary number of arcs to cover at least the length 1 − p, 0 ≦ p < 1, of the circumference at least m (≧1) times. In the present paper limit distributions of Nα are derived when α → 0. Some results for spacings are also obtained.