International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 21, Pages 3427-3441

Limit theorems for randomly selected adjacent order statistics from a Pareto distribution

André Adler

Department Applied of Mathematics, College of Science and Letters, Illinois Institute of Technology, Chicago 60616, IL, USA

Received 28 May 2005; Revised 23 September 2005

Copyright © 2005 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.


Consider independent and identically distributed random variables {Xnk,  1km, n1} from the Pareto distribution. We randomly select two adjacent order statistics from each row, Xn(i) and Xn(i+1), where 1im1. Then, we test to see whether or not strong and weak laws of large numbers with nonzero limits for weighted sums of the random variables Xn(i+1)/Xn(i) exist, where we place a prior distribution on the selection of each of these possible pairs of order statistics.