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The moment index of minima

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

D. J. Daley
D. J. Daley*
Affiliation:
The Australian National University
*
1Postal address: Centre for Mathematics and its Applications (SMS), The Australian National University, Canberra, ACT 0200, Australia. Email: daryl@maths.anu.edu.au
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Abstract

For a random variable (RV) X its moment index κ(X) ≡ sup{κ : E(|X|κ) < ∞} lies in 0 ≤ κ (X) ≤∞; it is a critical quantity and finite for heavy-tailed RVs. The paper shows that κ (min(X, Y)) ≥ κ(X) + κ (Y) for independent non-negative RVs X and Y. For independent non-negative ‘excess' RVs Xs and Ys whose distributions are the integrated tails of X and Y, κ (X) + κ (Y) ≤ κ (min(Xs, Ys)) + 2 ≤ κ (min(X, Y)). An example shows that the inequalities can be strict, though not if the tail of the distribution of either X or Y is a regularly varying function.

Keywords

Moment indexHurst indexregularly varying tail

MSC classification

Primary: 60E05: Distributions
Secondary: 60E15: Inequalities; stochastic orderings
Type
Probability theory
Information
Journal of Applied Probability , Volume 38 , Issue A: Probability, Statistics and Seismology , 2001 , pp. 33 - 36
Copyright
Copyright © Applied Probability Trust 2001 

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References

Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
Daley, D. J., Rolski, T. and Vesilo, R. (2000). Long-range dependent point processes and their Palm-Khinchin distributions. Adv. Appl. Probab. 32, 1051–1063.Google Scholar