Non-extensive random walks

Abstract

The stochastic properties of variables whose addition leads to $q$-Gaussian distributions $G_q(x)=[1+(q-1)x^2{]}_{+}^{1/(1-q)}$ (with $q\in \mathbb{R}$ and where $[f(x){]}_{+}=max\{f(x),0\}$) as limit law for a large number of terms are investigated. These distributions have special relevance within the framework of non-extensive statistical mechanics, a generalization of the standard Boltzmann-Gibbs formalism, introduced by Tsallis over one decade ago. Therefore, the present findings may have important consequences for a deeper understanding of the domain of applicability of such generalization. Basically, it is shown that the random walk sequences, that are relevant to this problem, possess a simple additive-multiplicative structure commonly found in many contexts, thus justifying the ubiquity of those distributions. Furthermore, a connection is established between such sequences and the nonlinear diffusion equation ${\partial }_t\rho ={\partial }_{xx}^2{\rho }^{\nu }$ ($\nu \ne 1$).

5 pages, 2 figures

References in corpus (6)

• Superstatistics
• Option Pricing Formulas based on a non-Gaussian Stock Price Model
• Nonextensive statistical mechanics and economics
• Volatility clustering and scaling for financial time series due to attractor bubbling
• Defect turbulence and generalized statistical mechanics
• Multiplicative noise: A mechanism leading to nonextensive statistical mechanics

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