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\$.
%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
}