Convergent Multiplicative Processes Repelled from Zero: Power Laws and Truncated Power Laws

Didier Sornette , Rama Cont

Levy and Solomon have found that random multiplicative processes wt =λ1λ2...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 exact 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 $\frac{1}{\sqrt t}$(log wt - 〈log wt〉); 2) the two necessary and sufficient conditions for P(wt) to be a power law are that $\langle {\rm log}\lambda_j\rangle < 0$ (corresponding to a drift wt →0) and that wt not be allowed to become too small. We discuss several models, previously thought unrelated, showing the common underlying mechanism for the generation of power laws by multiplicative processes: the variable log wt undergoes a random walk repelled from -∞, which we describe by a Fokker-Planck equation. 3) For all these models, we obtain the exact result that μ is solution of 〈λμ〉= 1 and thus depends on the distribution of λ. 4) For finite t, the power law is cut-off by a log-normal tail, reflecting the fact that the random walk has not the time to scatter off the repulsive force to diffusively transport the information far in the tail.

Domaines: Articles anciens

Didier Sornette, Rama Cont. Convergent Multiplicative Processes Repelled from Zero: Power Laws and Truncated Power Laws. Journal de Physique I, 1997, 7 (3), pp.431-444. ⟨10.1051/jp1:1997169⟩. ⟨jpa-00247337⟩