1996/09/07 by Rama Cont, Didier Sornette · 4 citations
Physics and Astronomy · #cond-mat
published as J. Phys. I France 7, 431-444 (1997) · 19 pages, Latex, 4 figures available on request from [email protected]
arxiv created 1996/09/07 · arxiv updated 2009/11/30
Random multiplicative processes wt =λ1 λ2 ... λt (with < λj > 0 ) lead, in the presence of a boundary constraint, to a distribution P(wt) in the form of a power law wt-(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 → ∞) distribution of wt and should be distinguished from the central limit theorem which is a statement on the asymptotic distribution of the reduced variable 1 \over √(t)(log wt -< log wt >); 2) the necessary and sufficient conditions for P(wt) to be a power law are that <log λj > < 0 (corresponding to a drift wt → 0) and that wt 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 wt 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 ⟨ λμ ⟩ = 1 and is therefore non-universal and depends on the distribution of λ.