### Abstract

Let f be a probability density function on (a, b) C (0, infinity) 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-¿x or f(x) = C(b-x) ¿-1. The present paper shows that actually they are the only possible two cases. This surprising result is achieved with an unusual tool in renewal theory, by using ideals of polynomials.

Original language | English |
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Place of Publication | Eindhoven |

Publisher | Eurandom |

Number of pages | 10 |

Publication status | Published - 2009 |

### Publication series

Name | Report Eurandom |
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Volume | 2009025 |

ISSN (Print) | 1389-2355 |

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## Cite this

Bar-Lev, S. K., Boxma, O. J., & Letac, G. (2009).

*A characterization related to the equilibrium distribution associated with a polynomial structure*. (Report Eurandom; Vol. 2009025). Eurandom.