On a functional equation characterizing some probability distributions (Q6585756)

``We find all nonnegative solutions \(f\) of the equation \(f(x)=\prod_{j=1}^n f(s_j x)^{p_j}\), defined in a one-sided vicinity of 0 and having a prescribed asymptotic at 0. The main theorem extends a result obtained by \textit{J. A. Baker} [Proc. Am. Math. Soc. 121, No. 3, 767--773 (1994; Zbl 0808.39010)].''\N\NHere \(n\) is a fixed positive integer, \(s_1,\ldots ,s_n\) are given numbers from (0, 1), and \(p_1,\ldots, p_n\) are given positive real numbers. The functional equation has a probability background (among others in the theory of renewal processes).\N\NThe paper is structured in 4 chapters:\N\N1. Introduction (Theorem B, Baker [loc. cit.]) - 2. Main result (Theorem JJ, extension of Baker's Theorem B.) - 3. Proofs of Theorem JJ and Theorem B (now derived from Theorem JJ) - 4. Proofs of Lemmas 1 and 2 - References (13 references).