4 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
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
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