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UNIFORM CONVERGENCE TO A LAW CONTAINING GAUSSIAN AND CAUCHY DISTRIBUTIONS

Published online by Cambridge University Press:  08 June 2012

Jose H. Blanchet  and
Carlos G. Pacheco-González
Jose H. Blanchet
Affiliation:
Industrial Engineering and Operations Research, Columbia University, 1255 Amsterdam Ave. Mail Code 4403, New York, NY E-mail: [email protected]
Carlos G. Pacheco-González
Affiliation:
Departamento de Matematicas, CINVESTAV-IPN, A. Postal 14-740, Mexico D.F. 07000, Mexico E-mail: [email protected]
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Abstract

A source of light is placed d inches apart from the center of a detection bar of length L ≥ d. The source spins very rapidly, while shooting beams of light according to, say, a Poisson process with rate λ. The positions of the beams, relative to the center of the bar, are recorded for those beams that actually hit the bar. Which law best describes the time-average position of the beams that hit the bar given a fixed but long time horizon t? The answer is given in this paper by means of a uniform weak convergence result in L, d as t → ∞. Our approximating law includes as particular cases the Cauchy and Gaussian distributions.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 26 , Issue 3 , July 2012 , pp. 437 - 448
Copyright
Copyright © Cambridge University Press 2012

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