On LØvy measures for innitely divisible natural exponential families CØlestin C. Kokonendji a; , a University of Pau - LMA & IUT STID, Pau, France Mohamed Khoudar b b University of Pau - IUT STID, Pau, France Abstract It has appeared that innitely divisible distributions are increasingly dened in terms of their LØvy measures. Let be an innitely divisible positive measure on R (not necessarily a probability). In terms of variance function, we simply characterize the natural exponential family G generated by a modied LØvy measure = () from the one F generated by . This connection points out, in particular, that if F admits a polynomial variance function of degree p (p 2), then the variance function of G has always a quadratic form. For -stable processes with < 1 or p-power variance function with p> 1, the corresponding measures are always gamma distributions. Some other examples related to compound Poisson processes of Hinde-DemØtrio and leading to as negative binomial families are given as their new characterizations. Key words: Compound Poisson process, gamma distribution, LØvy process, negative binomial distribution, stable process, variance function. 1991 MSC: Primary 60E07; Secondary 60G51, 62E10. Abbreviated title: On LØvy measures. Address for correspondence: C.C. Kokonendji. UniversitØ de Pau et des Pays de lAdour. Laboratoire de MathØmatiques AppliquØes - CNRS UMR 5142. DØparte- ment STID. Avenue de lUniversitØ. 64000 Pau, France. Phone: +33 559 407 145. Fax: +33 559 407 140. Email address: celestin.kokonendji@univ-pau.fr (CØlestin C. Kokonendji). Preprint submitted to LMA: Technical Report No. 0504 25 February 2005