Restricted random walk model as a new testing ground for the applicability of q-statistics

Abstract

We present exact results obtained from Master Equations for the probability function P(y,T) of sums $y={\sum }_{t=1}^T{x}_t$ of the positions x_t of a discrete random walker restricted to the set of integers between -L and L. We study the asymptotic properties for large values of L and T. For a set of position dependent transition probabilities the functional form of P(y,T) is with very high precision represented by q-Gaussians when T assumes a certain value $T^{\ast }\propto L^2$. The domain of y values for which the q-Gaussian apply diverges with L. The fit to a q-Gaussian remains of very high quality even when the exponent $a$ of the transition probability g(x)=|x/L|^a+p with 0<p<<1 is different from 1, all though weak, but essential, deviation from the q-Gaussian does occur for $a\ne 1$. To assess the role of correlations we compare the T dependence of P(y,T) for the restricted random walker case with the equivalent dependence for a sum y of uncorrelated variables x each distributed according to 1/g(x).

5 pages, 7 figs, EPL (2011), in press

References in corpus (3)

• Numerical indications of a q-generalised central limit theorem
• A note on q-Gaussians and non-Gaussians in statistical mechanics
• On a representation of the inverse Fq transform

Cited by in corpus (6)

• Reply to Comment on "Towards a large deviation theory for strongly correlated systems"
• Quasi-power laws in multiparticle production processes
• Inter-occurrence times and universal laws in finance, earthquakes and genomes
• Long memory constitutes a unified mesoscopic mechanism consistent with nonextensive statistical mechanics
• Central limit theorem for a class of globally correlated random variables
• On the relevance of q-distribution functions: The return time distribution of restricted random walker