AN EXTENSION OF MANDELBROTS FUNCTIONAL-EQ UATION

Let W greater than or equal to 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(-tY)), Y r.v. greater than or equal to 0}. In [1], B. Mandelbrot introduces the equa tion (E) : f(t) = (E(f(tW)))(c), when c is an element of N, and asks u nder what conditions (E) has a non-trivial solution in L. When f is an element of L satisfies (E) and 0 < E(Y) < infinity; he also asks cond itions 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 is not an ele ment of N, we study (E) in a space containing L and we look into the e quivalent problems to the existence of the moments. (C) Academie des S ciences/Elsevier, Paris.