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A one dimensional random space-filling problem

Published online by Cambridge University Press:  14 July 2016

John P. Mullooly
John P. Mullooly*
Affiliation:
National Heart Institute, National Institutes of Health, Bethesda, Maryland
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Consider an interval of the real line (0, x), x > 0; and place in it a random subinterval S(x) defined by the random variables Xx and Yx, the position of the center of S(x) and the length of S(x). The set (0, x)– S(x) consists of two intervals of length δ and η. Let a > 0 be a fixed constant. If δ ≦ a, then a random interval S(δ) defined by Xδ, Yδ is placed in the interval of length δ. If δ < a, the placement of the second interval is not made. The same is done for the interval of length η. Continue to place non-intersecting random subintervals in (0, x), and require that the lengths of all the random subintervals be ≦ a. The process terminates after a finite number of steps when all the segments of (0, x) uncovered by random subintervals are of length < a. At this stage, we say that (0, x) is saturated. Define N(a, x) as the number of random subintervals that have been placed when the process terminates. We are interested in the asymptotic behavior of the moments of N(a, x), for large x.

Type
Research Papers
Information
Journal of Applied Probability , Volume 5 , Issue 2 , August 1968 , pp. 427 - 435
Copyright
Copyright © Applied Probability Trust 1968 

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References

[1] Dvoretzky, A. and Robbins, H. (1964) On the “parking problem”. Comm. Math. Res. Inst. Hung. Acad. Sci. 9, 209–225.Google Scholar
[2] Gornick, F. and Jackson, J. L. (1963) Sequence selection problem in the crystallization of polymers. J. Chem. Phys. 39, 1150–1154.CrossRefGoogle Scholar
[3] Ney, P. E. (1962) A random interval filling problem. Ann. Math. Statist. 33, 702–718.Google Scholar
[4] Rényi, A. (1958) A one-dimensional problem of random space filling. Comm. Math. Res. Inst. Hung. Acad. Sci. 3, 109–127.Google Scholar
[5] Widder, D. V. (1946) The Laplace Transform. Princeton Univ. Press.Google Scholar