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Regular variation of the tail of a subordinated probability distribution

Published online by Cambridge University Press:  01 July 2016

A. J. Stam
A. J. Stam*
Affiliation:
State University of Groningen
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Abstract

Let F be a probability measure on the real line and G = Σ C(k)Fk∗ the probability measure subordinate to F with subordinator C restricted to the nonnegative integers. Let V(x) vary regularly of order p for x→ ∞ and either (1) V(x) F[x, ∞)→ α ≧ 0 or (2) V(x) C[x, ∞)→ γ ≧ 0. If ρ > 1 and F(–∞, 0) = 0, necessary and sufficient in order that V(x) G[x, ∞)→b, is that both (1) and (2) hold for suitable α and γ. For 0 ≦ ρ ≦ 1 the conditions are of different type. For two-sided F a different situation arises and only sufficient conditions are found. An application to renewal moments of negative order is given.

Keywords

SUBORDINATIONREGULAR VARIATIONCONVOLUTIONTAIL BEHAVIOURRENEWAL THEORYSTABLE LAWSATTRACTION
Type
Research Article
Information
Advances in Applied Probability , Volume 5 , Issue 2 , August 1973 , pp. 308 - 327
Copyright
Copyright © Applied Probability Trust 1973 

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