Convergent multiplicative processes repelled from zero: power laws and truncated power laws
| dc.creator | Cont, Rama | |
| dc.creator | Sornette, Didier | |
| dc.date | 1996-09-07 | |
| dc.date.accessioned | 2026-07-07T03:08:52Z | |
| dc.date.available | 2026-07-07T03:08:52Z | |
| dc.description | Random multiplicative processes $w_t =λ_1 λ_2 ... λ_t$ (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 $w_t^{-(1+μ)}$. 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 ${1 \over \sqrt{t}}(log w_t -< log w_t >)$; 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 $λ$, in terms of the Fokker-Planck equation. 3) For all these models, the exponent $μ$ is shown exactly to be the solution of $\langle λ^μ \rangle = 1$ and is therefore non-universal and depends on the distribution of $λ$. | |
| dc.description | 19 pages, Latex, 4 figures available on request from cont@ens.fr | |
| dc.identifier | https://arxiv.org/abs/cond-mat/9609074 | |
| dc.identifier | http://arxiv.org/abs/cond-mat/9609074 | |
| dc.identifier | J. Phys. I France 7, 431-444 (1997) | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/27667 | |
| dc.subject | Condensed Matter | |
| dc.title | Convergent multiplicative processes repelled from zero: power laws and truncated power laws | |
| dc.type | text |