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

Let f be a probability density function on (a, b) ⊂ (0,∞), and consider the class C_f of all probability density functions of the form Pf , where P is a polynomial. Assume that if X has its density in C _f then the equilibrium probability density x |→ P(X > x)/ E(X) also belongs to C_f : 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.