Convergent Multiplicative Processes Repelled from Zero: Power Laws and Truncated Power Laws

Levy and Solomon have found that random multiplicative processes w t =12 ::: t (with j ? 0) lead, in the presence of a boundary constraint, to a distribution P (w t ) in the form of a power law w \Γ(1+¯) t . 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 ! 1) distribution of w t and should be distinguished from the central limit theorem which is a statement on the asymptotic distribution of the reduced variable 1 p t (log w t \Γ hlog w t i); 2) the two necessary and sufficient conditions for P (w t ) to be a power law are that hlog j i ! 0 (corresponding to a drift w t ! 0) and that w t not be allowed to become too small. We discuss several models, previously thought unrelated, showing the common underlying mechanism for the generation of power laws by multiplicative processes: the variable log w t undergoes a random walk repelled from \Γ...