%0 Generic %1 citeulike:613403 %A Cont, Rama %A Sornette, Didier %D 1996 %K math %T Convergent multiplicative processes repelled from zero: power laws and truncated power laws %U http://arxiv.org/abs/cond-mat/9609074 %X Random multiplicative processes \$w\_t =łambda\_1 łambda\_2 ... łambda\_t\$ (with < łambda\_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 ınfty\$) 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 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 łambda\_j > < 0 (corresponding to a drift \$w\_t 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 \$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 \$łambda\$, in terms of the Fokker-Planck equation. 3) For all these models, the exponent \$\mu\$ is shown exactly to be the solution of \$łambda^\mu = 1\$ and is therefore non-universal and depends on the distribution of \$łambda\$.

@misc{citeulike:613403, abstract = {Random multiplicative processes \$w\_t =\lambda\_1 \lambda\_2 ... \lambda\_t\$ (with \< \lambda\_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 \${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 \lambda\_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 \$\langle \lambda^{\mu} \rangle = 1\$ and is therefore non-universal and depends on the distribution of \$\lambda\$.}, added-at = {2009-08-06T15:16:38.000+0200}, archiveprefix = {arXiv}, author = {Cont, Rama and Sornette, Didier}, biburl = {https://www.bibsonomy.org/bibtex/23c2ca1c1a6bbc1697d20d5e0b601250a/chato}, citeulike-article-id = {613403}, citeulike-linkout-0 = {http://arxiv.org/abs/cond-mat/9609074}, citeulike-linkout-1 = {http://arxiv.org/pdf/cond-mat/9609074}, eprint = {cond-mat/9609074}, interhash = {0e6737d330c2e7c681be3ba006717150}, intrahash = {3c2ca1c1a6bbc1697d20d5e0b601250a}, keywords = {math}, month = Sep, posted-at = {2006-05-04 16:34:55}, priority = {0}, timestamp = {2009-08-06T15:16:52.000+0200}, title = {Convergent multiplicative processes repelled from zero: power laws and truncated power laws}, url = {http://arxiv.org/abs/cond-mat/9609074}, year = 1996 }