par Patie, Pierre
Référence Bernoulli, 17, 2, page (814-826)
Publication Publié, 2010
Article révisé par les pairs
Résumé : Let ζ be a (possibly killed) subordinator with Laplace exponent f and denote by Iφ =∫∞0 e-ζs ds, the socalled exponential functional. Consider the positive random variable I ψ1 whose law, according to Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95-106], is determined by its negative entire moments as follows:E[I-nψ1 ] =IIn k=1φ(k), n = 1, 2, ⋯ . In this note, we show that I ψ1 is a positive self-decomposable random variable whenever the Lévy measure of ζ is absolutely continuous with a monotone decreasing density. In fact, I?1 is identified as the exponential functional of a spectrally negative (sn, for short) Lévy process. We deduce from Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95-106] the following factorization of the exponential law e:Iφ/Iψ1(d)= e,where Iψ1 is taken to be independent of Iφ. We proceed by showing an identity in distribution between the entrance law of an sn self-similar positive Feller process and the reciprocal of the exponential functional of sn Lévy processes. As a by-product, we obtain some new examples of the law of the exponential functionals, a new factorization of the exponential law and some interesting distributional properties of some random variables. For instance, we obtain that S(α)α is a self-decomposable random variable, where S(a) is a positive stable random variable of index α ε (0, 1) © 2011 ISI/BS.