On q-Gaussians and Exchangeability

The q-Gaussians are discussed from the point of view of variance mixtures of normals and exchangeability. For each q< 3, there is a q-Gaussian distribution that maximizes the Tsallis entropy under suitable constraints. This paper shows that q-Gaussian random variables can be represented as variance mixtures of normals. These variance mixtures of normals are the attractors in central limit theorems for sequences of exchangeable random variables; thereby, providing a possible model that has been extensively studied in probability theory. The formulation provided has the additional advantage of yielding process versions which are naturally q-Brownian motions. Explicit mixing distributions for q-Gaussians should facilitate applications to areas such as option pricing. The model might provide insight into the study of superstatistics.Comment: 14 page

Strictly and asymptotically scale-invariant probabilistic models of NN correlated binary random variables having {\em q}--Gaussians as N→∞N\to \infty limiting distributions

In order to physically enlighten the relationship between {\it $q$--independence} and {\it scale-invariance}, we introduce three types of asymptotically scale-invariant probabilistic models with binary random variables, namely (i) a family, characterized by an index $\nu=1,2,3,...$, unifying the Leibnitz triangle ($\nu=1$) and the case of independent variables ($\nu\to\infty$); (ii) two slightly different discretizations of $q$--Gaussians; (iii) a special family, characterized by the parameter $\chi$, which generalizes the usual case of independent variables (recovered for $\chi=1/2$). Models (i) and (iii) are in fact strictly scale-invariant. For models (i), we analytically show that the $N \to\infty$ probability distribution is a $q$--Gaussian with $q=(\nu -2)/(\nu-1)$. Models (ii) approach $q$--Gaussians by construction, and we numerically show that they do so with asymptotic scale-invariance. Models (iii), like two other strictly scale-invariant models recently discussed by Hilhorst and Schehr (2007), approach instead limiting distributions which are {\it not} $q$--Gaussians. The scenario which emerges is that asymptotic (or even strict) scale-invariance is not sufficient but it might be necessary for having strict (or asymptotic) $q$--independence, which, in turn, mandates $q$--Gaussian attractors.Comment: The present version is accepted for publication in JSTA

Probability densities for the sums of iterates of the sine-circle map in the vicinity of the quasi-periodic edge of chaos

We investigate the probability density of rescaled sum of iterates of sine-circle map within quasi-periodic route to chaos. When the dynamical system is strongly mixing (i.e., ergodic), standard Central Limit Theorem (CLT) is expected to be valid, but at the edge of chaos where iterates have strong correlations, the standard CLT is not necessarily to be valid anymore. We discuss here the main characteristics of the central limit behavior of deterministic dynamical systems which exhibit quasi-periodic route to chaos. At the golden-mean onset of chaos for the sine-circle map, we numerically verify that the probability density appears to converge to a q-Gaussian with q<1 as the golden mean value is approached.Comment: 7 pages, 7 figures, 1 tabl