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One sided limit theorems for the range of a Pareto

Vol. 15 No. 4 (2020) P.321~P.327
 DOI: https://doi.org/10.21915/BIMAS.2020403 10.21915/BIMAS.2020403

ABSTRACT

The limiting behaviour of $\sum_{k=1}^n a_k R_k/b_n$, where $R_k$ is the range from our $k^{th}$ sample of Pareto random variables is explored. Here we show that a weak laws of large numbers holds, while the corresponding strong law of large numbers fails. We use the weak law to show the lower and upper almost sure limits of our normalized weighted sum. The underlying density is $f(x)=x^{-2}I(x\ge 1)$ and the partial sum consists of the weights times the range from our samples. We also show how these weights must be of a certain form.

KEYWORDS
Weak laws of large numbers, exact strong laws, order statistics

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 60F05, 60F15

MILESTONES