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Exact Strong Laws for the Range

Vol. 15 No. 1 (2020) P.53~P.63
 DOI: https://doi.org/10.21915/BIMAS.2020104 10.21915/BIMAS.2020104

ABSTRACT

In this paper we establish exact strong laws of large numbers for the range of a Pareto random variable. The underlying density is $f(x)=x^{-2}I(x\ge 1)$. Neither the first nor second moments of this random variable exist, which makes these theorems unusual. The results are of the form $\sum_{i=1}^n a_i R_i/b_n\to \gamma,$ as $n \to\infty$, where $R_i$ is the range from the $i^{th}$ sample.

KEYWORDS
Almost sure convergence, order statistics, strong laws of large numbers, exact strong laws

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 60F15

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