21 research outputs found
Strictly and asymptotically scale-invariant probabilistic models of correlated binary random variables having {\em q}--Gaussians as limiting distributions
In order to physically enlighten the relationship between {\it
--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 ,
unifying the Leibnitz triangle () and the case of independent variables
(); (ii) two slightly different discretizations of
--Gaussians; (iii) a special family, characterized by the parameter ,
which generalizes the usual case of independent variables (recovered for
). Models (i) and (iii) are in fact strictly scale-invariant. For
models (i), we analytically show that the probability
distribution is a --Gaussian with . Models (ii) approach
--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} --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)
--independence, which, in turn, mandates --Gaussian attractors.Comment: The present version is accepted for publication in JSTA
A note on q-Gaussians and non-Gaussians in statistical mechanics
The sum of sufficiently strongly correlated random variables will not in
general be Gaussian distributed in the limit N\to\infty. We revisit examples of
sums x that have recently been put forward as instances of variables obeying a
q-Gaussian law, that is, one of type (cst)\times[1-(1-q)x^2]^{1/(1-q)}. We show
by explicit calculation that the probability distributions in the examples are
actually analytically different from q-Gaussians, in spite of numerically
resembling them very closely. Although q-Gaussians exhibit many interesting
properties, the examples investigated do not support the idea that they play a
special role as limit distributions of correlated sums.Comment: 17 pages including 3 figures. Introduction and references expande
Note on a q-modified central limit theorem
A q-modified version of the central limit theorem due to Umarov et al.
affirms that q-Gaussians are attractors under addition and rescaling of certain
classes of strongly correlated random variables. The proof of this theorem
rests on a nonlinear q-modified Fourier transform. By exhibiting an invariance
property we show that this Fourier transform does not have an inverse. As a
consequence, the theorem falls short of achieving its stated goal.Comment: 10 pages, no figure
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
Functional-differential equations for %-transforms of -Gaussians
In the paper the question - Is the q-Fourier transform of a q-Gaussian a
q'-Gaussian (with some q') up to a constant factor? - is studied for the whole
range of . This question is connected with applicability of
the q-Fourier transform in the study of limit processes in nonextensive
statistical mechanics. We prove that the answer is affirmative if and only if q
> 1, excluding two particular cases of q<1, namely, q = 1/2 and q = 2/3, which
are also out of the theory valid for q \ge 1. We also discuss some applications
of the q-Fourier transform to nonlinear partial differential equations such as
the porous medium equation.Comment: 14 pages A new section on a related solution of the porous medium
equation in comparison with the previous version has been introduc
The q-exponential family in statistical physics
The notion of generalised exponential family is considered in the restricted
context of nonextensive statistical physics. Examples are given of models
belonging to this family. In particular, the q-Gaussians are discussed and it
is shown that the configurational probability distributions of the
microcanonical ensemble belong to the q-exponential family.Comment: 18 pages, 4 figures, proceedings of SigmaPhi 200
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
Nonadditive entropy and nonextensive statistical mechanics - Some central concepts and recent applications
We briefly review central concepts concerning nonextensive statistical
mechanics, based on the nonadditive entropy . Among others, we focus on
possible realizations of the -generalized Central Limit Theorem, including
at the edge of chaos of the logistic map, and for quasi-stationary states of
many-body long-range-interacting Hamiltonian systems.Comment: 15 pages, 9 figs., to appear in Journal of Physics: Conf.Series (IOP,
2010
Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example
Extremization of the Boltzmann-Gibbs (BG) entropy under appropriate norm and
width constraints yields the Gaussian distribution. Also, the basic solutions
of the standard Fokker-Planck (FP) equation (related to the Langevin equation
with additive noise), as well as the Central Limit Theorem attractors, are
Gaussians. The simplest stochastic model with such features is N to infinity
independent binary random variables, as first proved by de Moivre and Laplace.
What happens for strongly correlated random variables? Such correlations are
often present in physical situations as e.g. systems with long range
interactions or memory. Frequently q-Gaussians become observed. This is
typically so if the Langevin equation includes multiplicative noise, or the FP
equation to be nonlinear. Scale-invariance, i.e. exchangeable binary stochastic
processes, allow a systematical analysis of the relation between correlations
and non-Gaussian distributions. In particular, a generalized stochastic model
yielding q-Gaussians for all q (including q>1) was missing. This is achieved
here by using the Laplace-de Finetti representation theorem, which embodies
strict scale-invariance of interchangeable random variables. We demonstrate
that strict scale invariance together with q-Gaussianity mandates the
associated extensive entropy to be BG.Comment: 6 pages, 1 fig, to appear in EPJ
Deviation from Gaussianity in the cosmic microwave background temperature fluctuations
Recent measurements of the temperature fluctuations of the cosmic microwave
background (CMB) radiation from the WMAP satellite provide indication of a
non-Gaussian behavior. Although the observed feature is small, it is detectable
and analyzable. Indeed, the temperature distribution P^{CMB}(Delta T) of these
data can be quite well fitted by the anomalous probability distribution
emerging within nonextensive statistical mechanics, based on the entropy S_q =
k (1 - \int dx [P(x)]^q)/(q - 1) (where in the limit case q -> 1 we obtain the
Boltzmann-Gibbs entropy S_1 = - k \int dx P(x) ln[P(x)]). For the CMB
frequencies analysed, \nu= 40.7, 60.8, and 93.5 GHz, P^{CMB}(Delta T) is well
described by P_q(Delta T) \propto 1/[1 + (q-1) B(\nu) (Delta T)^2]^{1/(q-1)},
with q = 1.04 \pm 0.01, the strongest non-Gaussian contribution coming from the
South-East sector of the celestial sphere. Moreover, Monte Carlo simulations
exclude, at the 99% confidence level, P_1(Delta T) \propto e^{- B(\nu) (Delta
T)^2} to fit the three-year WMAP data.Comment: 6 pages, 1 figur