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Overlap Problems on the Circle

Part of: Limit theorems Probabilistic methods, simulation and stochastic differential equations Combinatorial probability

Published online by Cambridge University Press:  04 January 2016

S. Juneja  and
M. Mandjes
S. Juneja*
Affiliation:
Tata Institute for Fundamental Research
M. Mandjes*
Affiliation:
University of Amsterdam, EURANDOM and Eindhoven University of Technology
*
∗ Postal address: Tata Institute for Fundamental Research, Homi Bhabha Road, Colaba Mumbai, India. Email address: [email protected]
∗∗ Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, Amsterdam 1098 XH, The Netherlands. Email address: [email protected]
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Abstract

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Consider a circle with perimeter N > 1 on which k < N segments of length 1 are sampled in an independent and identically distributed manner. In this paper we study the probability π (k,N) that these k segments do not overlap; the density φ(·) of the position of the disks on the circle is arbitrary (that is, it is not necessarily assumed uniform). Two scaling regimes are considered. In the first we set k≡ a√N, and it turns out that the probability of interest converges (N→ ∞) to an explicitly given positive constant that reflects the impact of the density φ(·). In the other regime k scales as aN, and the nonoverlap probability decays essentially exponentially; we give the associated decay rate as the solution to a variational problem. Several additional ramifications are presented.

Keywords

Overlap probabilityasymptoticsnonuniformity

MSC classification

Primary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 60F10: Large deviations 65C05: Monte Carlo methods
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 3 , September 2013 , pp. 773 - 790
Copyright
© Applied Probability Trust 

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