Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T17:16:33.887Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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