Uniform convergence to a law containing Gaussian and Cauchy distributions (with C. Pacheco-Gonzalez).

Summary:

A source of light is placed d inches apart from the center of a detection bar of length 2L with 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 who actually hit the bar. Which law better 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 a particular case the Cauchy and Gaussian distributions. This result is not too surprising if one know the definition of a “truncated stable” distribution, what is somewhat tricky is to get all the parameters in right place to have the uniform convergence.

Bibtex:

@Article{BlaPac11,

author = {J. Blanchet and C. Pacheco-Gonzalez},

title = {Uniform convergence to a law containing Gaussian and Cauchy distributions},

journal = {To appear in Probability in the Engineering and Informational Sciences},

year = {2012},

volume = {},

pages = {}

}