J. Phys. I Flaace MARCH 1997, PAGE 431 Convergent Multiplicative Processes Repelled from Zero: Power Laws and libuncated Power Laws Didier Sornette (I,2,') and Raina Cola (I) (I ) Laboratoire de Physique de la Matière Condensée ("), Université des Sciences, BP 70, Parc Valrose, 06108 Nice Cedex 2, France Department of Earth and Space Science. and Institute of Geophysics and Planetary Physics, University of California, Lin Angeles, California 90095, USA (Received 2 September 1996, received in final tom 12 November 1996. accepted 20 November 1996) PACS.05.40.+j - Fluctuation phenomena, random processes, and Brownian motion PACS.64.60.Ht - Dynamic critical phenomena PACS.05.70.Ln - Nonequilibriuin thermodynamics, irreversible processes Abstract. — Levy and Solomon have found that random multiplicative processes we = (with At > lead, in the presence of a boundary constraint, to a distribution P(tur) in the form of a power law wifte" ). We provide a simple exact physically intuitive derivation of this result based on a random walk analogy and show the following: 1) the result applies to the asymptotic (t —› distribution of we and should be distinguished from the central limit theorem which is a statement on the asymptotic distribution of the reduced variable *(log we — (log tve)); 2) the two necessary and sufficient conditions for P(we) to be a power law are that (log ;) < 0 (corresponding to a drift w, —e 0) and that w, not be allowed to become too small. We discuss several models, previously thought unrelated, showing the commun underlying mechanism for the generation of power laws by multiplicative processes: the variable log we undergoes a random walk repelled from —oo, which we describe by a Fokker-Planck equatiou. 3) For all these models, we obtain the exact result that p is solution of (A.) = 1 and thus depends on the distribution of A. 4) For finite t, the power law is cut-off by a log-normal tail, reflecting the fact that the random