Multiplicative strong unimodality for positive stable laws

It is known that real Non-Gaussian stable distributions are unimodal, not additive strongly unimodal, and multiplicative strongly unimodal in the symmetric case. By a theorem of Cuculescu-Theodorescu [5], the only remaining relevant situation for the multiplicative strong unimodality of stable laws is the one-sided. In this paper, we show that positive α−stable laws are multiplicative strongly unimodal iff α ≤ 1/2. 1. The MSU property and stable laws A real random variable X is said to be unimodal (or quasi-concave) if there exists a ∈ R such that the functions P[X ≤ x] and P[X > x] are convex respectively in (−∞, a) and (a,+∞). If X is absolutely continuous, this means that its density increases on (−∞, a] and decreases on [a,+∞). The number a is called a mode of X , and might not be unique. A well-known example due to Chung shows that unimodality is not stable under convolution, and for this reason the notion of strong unimodality had been introduced in [8]: a real variable X is said to be strongly unimodal if the independent sum X + Y is unimodal for any unimodal variable Y (in particular X itself is unimodal, choosing Y degenerated at zero). In [8] Ibragimov also obtained the celebrated criterion that X is strongly unimodal iff it is absolutely continuous with a log-concave density. Proving unimodality or strong unimodality properties is simple for variables with given densities, but the problem might turn out complicated when these densities are not explicit. In this paper we will deal with real (strictly) stable variables, where very few closed formulae (given e.g. in [14] p. 66) are known for the densities. A classical theorem of Yamazato shows that they are all unimodal with a unique mode see Lemma 1 in [4] for the previously shown one-sided case and Theorem 53.1 in [12] for the general result, but except in the Gaussian situation it is easy to see that stable laws are not strongly unimodal, because their heavy tails prevent the densities from being everywhere log-concave see Remark 52.8 in [12]. Having infinitely divisible distributions, stable variables appear naturally in additive identities, a framework where they hence may not preserve unimodality. Stable variables also occur in multiplicative factorizations as a by-product of the so-called M-scheme (or Minfinite divisibility), a feature which has been studied extensively by Zolotarev see Chapter 3 in [14] and also [11] for the one-sided case. Another concrete example of a multiplicative identity involving a stable law is the following. Suppose that X, Y are two positive variables 2000 Mathematics Subject Classification. 60E07, 60E15.