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Multivariate subexponential distributions and random sums of random vectors

Part of: Real functions Special processes Distribution theory - Probability

Published online by Cambridge University Press:  08 September 2016

E. Omey ,
F. Mallor  and
J. Santos
E. Omey*
Affiliation:
European University College Brussels
F. Mallor*
Affiliation:
Public University of Navarre
J. Santos*
Affiliation:
Public University of Navarre
*
∗ Postal address: Department of Mathematics and Statistics, European University College Brussels, Stormstraat 2, 1000 Brussels, Belgium.
∗∗ Postal address: Department of Statistics and Operations Research, Public University of Navarre, Campus Arrosadia, 31006 Pamplona, Spain. Email address: mallor@unavarra.es
∗∗∗ Current address: Los Alderetes 14, 4° 3, 14004 Cordoba, Spain.
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Abstract

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Let F(x) denote a distribution function in Rd and let F*n(x) denote the nth convolution power of F(x). In this paper we discuss the asymptotic behaviour of 1 - F*n(x) as x tends to ∞ in a certain prescribed way. It turns out that in many cases 1 - F*n(x) ∼ n(1 - F(x)). To obtain results of this type, we introduce and use a form of subexponential behaviour, thereby extending the notion of multivariate regular variation. We also discuss subordination, in which situation the index n is replaced by a random index N.

Keywords

Subexponential distributionregular variationrandom sum

MSC classification

Primary: 26A12: Rate of growth of functions, orders of infinity, slowly varying functions
Secondary: 60E99: None of the above, but in this section 60K99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 4 , December 2006 , pp. 1028 - 1046
Copyright
Copyright © Applied Probability Trust 2006 

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