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A Characterization Related to the Equilibrium Distribution Associated with a Polynomial Structure

Part of: Special processes Computational aspects and applications

Published online by Cambridge University Press:  14 July 2016

Shaul K. Bar-Lev ,
Onno Boxma  and
Gérard Letac
Shaul K. Bar-Lev*
Affiliation:
University of Haifa
Onno Boxma*
Affiliation:
EURANDOM and Eindhoven University of Technology
Gérard Letac*
Affiliation:
Université Paul Sabatier
*
∗Postal address: Department of Statistics, University of Haifa, Haifa 31905, Israel. Email address: barlev@stat.haifa.ac.il
∗∗Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: boxma@win.tue.nl
∗∗∗Postal address: Laboratoire de Statistique et Probabilité, Université Paul Sabatier, 31062 Toulouse, France. Email address: gerard.letac@alsatis.net
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Abstract

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Let f be a probability density function on (a, b) ⊂ (0, ∞), and consider the class Cf of all probability density functions of the form Pf, where P is a polynomial. Assume that if X has its density in Cf then the equilibrium probability density x ↦ P(X > x) / E(X) also belongs to Cf: this happens, for instance, when f(x) = Ce−λx or f(x) = C(b − x)λ−1. We show in the present paper that these two cases are the only possibilities. This surprising result is achieved with an unusual tool in renewal theory, by using ideals of polynomials.

Keywords

Renewal theoryexcess lifetimepolynomial densityideals of polynomials

MSC classification

Secondary: 60K05: Renewal theory 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 1 , March 2010 , pp. 293 - 299
Copyright
Copyright © Applied Probability Trust 2010 

References

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