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Unusual limit theorems for the difference of order statistics from a Pareto
by André Adler  

Vol. 16 No. 2 (2021) P.177~P.192
DOI: https://doi.org/10.21915/BIMAS.2021204
  10.21915/BIMAS.2021204

ABSTRACT

In this paper we establish various limit theorems for the difference in order statistics from a sample from the Pareto distribution. The underlying density is $f(x)=x^{-2}I(x\ge 1)$. We look at both fixed and slowly increasing samples sizes. For our strong and weak laws of large numbers the first moment will be infinite and for our central limit theorem the second moment will be infinite. These theorems are quite unusual since the usual moment conditions do not hold. In order to achieve these results we must attach weights to these random variables and find these appropriate weights and norming sequences in order to establish our results.


KEYWORDS
Almost sure convergence, order statistics, strong laws of large numbers, exact strong laws, weak law of large numbers, central limit theorems.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 60F05, 60F15.

MILESTONES

Received: 2021-03-21
Revised :
Accepted: 2021-06-27


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