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

André Adler

 Department of Mathematics, Illinois Institute of Technology, Chicago 60616, IL, USA
Received 16 August 2006; Accepted 20 November 2006

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.