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Sequential random displacements of points in an interval

Published online by Cambridge University Press:  14 July 2016

David Mannion*
Affiliation:
Royal Holloway College
*
Postal address: Department of Statistics and Computer Science, Royal Holloway College, University of London, Egham, Surrey, TW20 0EX, U.K.

Abstract

An array of points {Z1, Z2, …, Zn–1} in the interval V = [0, L], is such that 0 ≦ Z1, ≦ Z2 ≦ … ≦ Zn–1, ≦ L. One of the points is chosen at random (Zk, say, with probability pk) and displaced to a new position within the interval [Zk–1, Zk+ 1], the position again chosen at random according to a probability distribution Gk. We derive some results concerning the limiting distribution of the array after a succession of such displacements. If Gk is a uniform distribution, it appears that the number of displacements necessary to open up a gap between at least one pair of adjacent points of size at least γ is O(ρ n), n →∞, where ρ = L/(L – γ).

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1983 

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References

Blaisdell, B. E. and Solomon, H. (1982) Random sequential packing in Euclidean spaces of dimensions three and four and a conjecture of Palásti. J. Appl. Prob. 19, 382390.CrossRefGoogle Scholar
Mannion, D. (1976) Random packing of an interval. Adv. Appl. Prob. 8, 477501.CrossRefGoogle Scholar
Hasegawa, M. and Tanemura, M. (1976) On the pattern of space division by territories. Ann. Inst. Statist. Math. 28, 509519.CrossRefGoogle Scholar