1,126 research outputs found
Central limit theorem for Fourier transforms of stationary processes
We consider asymptotic behavior of Fourier transforms of stationary ergodic
sequences with finite second moments. We establish a central limit theorem
(CLT) for almost all frequencies and also an annealed CLT. The theorems hold
for all regular sequences. Our results shed new light on the foundation of
spectral analysis and on the asymptotic distribution of periodogram, and it
provides a nice blend of harmonic analysis, theory of stationary processes and
theory of martingales.Comment: Published in at http://dx.doi.org/10.1214/10-AOP530 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A generalization of the central limit theorem consistent with nonextensive statistical mechanics
The standard central limit theorem plays a fundamental role in
Boltzmann-Gibbs statistical mechanics. This important physical theory has been
generalized \cite{Tsallis1988} in 1988 by using the entropy (with ) instead of its
particular BG case . The theory which emerges
is usually referred to as {\it nonextensive statistical mechanics} and recovers
the standard theory for . During the last two decades, this
-generalized statistical mechanics has been successfully applied to a
considerable amount of physically interesting complex phenomena. A
conjecture\cite{Tsallis2005} and numerical indications available in the
literature have been, for a few years, suggesting the possibility of
-versions of the standard central limit theorem by allowing the random
variables that are being summed to be strongly correlated in some special
manner, the case corresponding to standard probabilistic independence.
This is what we prove in the present paper for . The attractor, in
the usual sense of a central limit theorem, is given by a distribution of the
form with , and normalizing
constant . These distributions, sometimes referred to as -Gaussians,
are known to make, under appropriate constraints, extremal the functional
(in its continuous version). Their and particular cases recover
respectively Gaussian and Cauchy distributions.Comment: 19 pages (the new version contains further simplifications and
precisions with regard to the previous one
General expression for the component size distribution in infinite configuration networks
In the infinite configuration network the links between nodes are assigned
randomly with the only restriction that the degree distribution has to match a
predefined function. This work presents a simple equation that gives for an
arbitrary degree distribution the corresponding size distribution of connected
components. This equation is suitable for fast and stable numerical
computations up to the machine precision. The analytical analysis reveals that
the asymptote of the component size distribution is completely defined by only
a few parameters of the degree distribution: the first three moments, scale and
exponent (if applicable). When the degree distribution features a heavy tail,
multiple asymptotic modes are observed in the component size distribution that,
in turn, may or may not feature a heavy tail
Functional Limit Theorems for Toeplitz Quadratic Functionals of Continuous time Gaussian Stationary Processes
\noindent The paper establishes weak convergence in of normalized
stochastic processes, generated by Toeplitz type quadratic functionals of a
continuous time Gaussian stationary process, exhibiting long-range dependence.
Both central and non-central functional limit theorems are obtained
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