Journal of Applied Mathematics and Stochastic Analysis
Volume 2007 (2007), Article ID 56924, 6 pages
doi:10.1155/2007/56924
Research Article
Laws of Large Numbers for Asymmetrical Cauchy Random Variables
Department of Mathematics, Illinois Institute of Technology, Chicago 60616, IL, USA
Received 16 August 2006; Accepted 20 November 2006
Copyright © 2007 André Adler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We generalize the Cauchy distribution so that we can have asymmetrical
tails. This allows us to obtain unusual laws of large numbers involving weighted sums of these random variables. Unusual in the sense that even though in every case E|X|=∞, we can still obtain a nonzero limit for these weighted sums.