Une extension de l'équation fonctionnelle de B. Mandelbrot

Abstract

Let W ≥ 0 be a r.v. of finite mean and c a real number > 1. Let ℒ be the set { f : ℝ + → ℝ : f ( t ) = E ( e − t Y ) . Y r.v. ≥ 0 } . In [1], B. Mandelbrot introduces the equation ( E) : f ( t ) = ( E ( f ( t W ) ) ) c . when c ε , and asks under what conditions (E) has a non-trivial solution in ℒ . When f ∈ ℒ satisfies (E) and 0 < E ( Y ) < ∞ , he also asks conditions under which Y would have moments of order > 1. These questions have given rise to many works [1], [2], [4], [5], [6] and also to the study of the moments of negative orders of Y [7]. When c ∉ . we study (E) in a space containing ℒ and we look into the equivalent problems to the existence of the moments.