Convergent multiplicative processes repelled from zero: power laws and truncated power laws

Abstract

Random multiplicative processes $w_t={\lambda }_1{\lambda }_2...{\lambda }_t$ (with < λ_j > 0 ) lead, in the presence of a boundary constraint, to a distribution $P(w_t)$ in the form of a power law $w_t^{-(1+\mu )}$. We provide a simple and physically intuitive derivation of this result based on a random walk analogy and show the following: 1) the result applies to the asymptotic ($t\to \infty$) 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 $\frac{1}{\sqrt{t}}(log{w}_t-<log{w}_t>)$; 2) the necessary and sufficient conditions for $P(w_t)$ to be a power law are that <log λ_j > < 0 (corresponding to a drift $w_t\to 0$) and that $w_t$ not be allowed to become too small. We discuss several models, previously unrelated, showing the common underlying mechanism for the generation of power laws by multiplicative processes: the variable $\log w_t$ undergoes a random walk biased to the left but is bounded by a repulsive ''force''. We give an approximate treatment, which becomes exact for narrow or log-normal distributions of $\lambda$, in terms of the Fokker-Planck equation. 3) For all these models, the exponent $\mu$ is shown exactly to be the solution of $⟨{\lambda }^{\mu }⟩=1$ and is therefore non-universal and depends on the distribution of $\lambda$.

19 pages, Latex, 4 figures available on request from [email protected]