Random multiplicative processes $w_t =\lambda_1 \lambda_2 ... \lambda_t$
(with 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 ${1 \over \sqrt{t}}(log w_t -)$; 2) the
necessary and sufficient conditions for $P(w_t)$ to be a power law are that
< 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
$\langle \lambda^{\mu} \rangle = 1$ and is therefore non-universal and depends
on the distribution of $\lambda$.Comment: 19 pages, Latex, 4 figures available on request from [email protected]