A characterization related to the equilibrium distribution associated with a polynomial structure

Shaul K. Bar-Lev, Onno Boxma, Gérard Letac

Research output: Contribution to journalArticlepeer-review

Abstract

Let f be a probability density function on (a, b) ⊂ (0,∞), and consider the classCf 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 showin 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.

Original languageEnglish
Pages (from-to)293-299
Number of pages7
JournalJournal of Applied Probability
Volume47
Issue number1
DOIs
StatePublished - 1 Jan 2010

Keywords

  • Excess lifetime
  • Ideals of polynomials
  • Polynomial density
  • Renewal theory

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics (all)
  • Statistics, Probability and Uncertainty

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