Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-27T23:42:53.900Z Has data issue: false hasContentIssue false
  • English
  • Français

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: jose.blanchet@columbia.edu
Carlos G. Pacheco-González
Affiliation:
Departamento de Matematicas, CINVESTAV-IPN, A. Postal 14-740, Mexico D.F. 07000, Mexico E-mail: cpacheco@math.cinvestav.mx
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Chakrabarty, A. (2010). Central limit theorem for truncated heavy tailed Banach valued random vectors. Electronic Communication in Probability 15: 346–364.Google Scholar
2.Feller, W. (1968). An introduction to probability theory and its applications, vol. I, 3rd ed.New York: Wiley.Google Scholar
3.Feller, W. (1971). An introduction to probability theory and its applications, vol. II, 2nd ed.New York: Wiley.Google Scholar
4.Nadarajah, S. & Kotz, S. (2006). A truncated Cauchy distribution. International Journal of Mathematica Education in Science and Technology 37(5): 605–608.Google Scholar
5.Resnick, S.I. (1992). Adventures in stochastic processes. Boston: Birkhäuser.Google Scholar
6.Ross, S.M. (2006). A first course in probability, 7th ed.Upper Saddlet River, NJ: Pearson Prentice Hall.Google Scholar
7.Kim, P. & Song, R. (2007). Potential theory of truncated stable processes. Mathematische Zeitschrift 256(1): 139–173.Google Scholar
8.Applebaum, D. (2004). Lévy processes and stochastic calculus. Cambrigde, UK: Cambridge University Press.CrossRefGoogle Scholar
9.Whitt, W. (2002). Stochastic-processes limits. New York: Springer.Google Scholar