52,716 research outputs found
Thin-shell concentration for random vectors in Orlicz balls via moderate deviations and Gibbs measures
In this paper, we study the asymptotic thin-shell width concentration for
random vectors uniformly distributed in Orlicz balls. We provide both
asymptotic upper and lower bounds on the probability of such a random vector
being in a thin shell of radius times the asymptotic value of
(as
), showing that in certain ranges our estimates are optimal. In
particular, our estimates significantly improve upon the currently best known
general Lee-Vempala bound when the deviation parameter goes down to
zero as the dimension of the ambient space increases. We shall also
determine in this work the precise asymptotic value of the isotropic constant
for Orlicz balls. Our approach is based on moderate deviation principles and a
connection between the uniform distribution on Orlicz balls and Gibbs measures
at certain critical inverse temperatures with potentials given by Orlicz
functions, an idea recently presented by Kabluchko and Prochno in [The maximum
entropy principle and volumetric properties of Orlicz balls, J. Math. Anal.
Appl. {\bf 495}(1) 2021, 1--19].Comment: 27 page
Thin-shell concentration for random vectors in Orlicz balls via moderate deviations and Gibbs measures
In this paper, we study the asymptotic thin-shell width concentration for random vectors uniformly distributed in Orlicz balls. We provide both asymptotic upper and lower bounds on the probability of such a random vector being in a thin shell of radius times the asymptotic value of n^{-1/2}\left(\E\left[\Vert X_n\Vert_2^2\right]\right)^{1/2} (as ), showing that in certain ranges our estimates are optimal. In particular, our estimates significantly improve upon the currently best known general Lee-Vempala bound when the deviation parameter goes down to zero as the dimension of the ambient space increases. We shall also determine in this work the precise asymptotic value of the isotropic constant for Orlicz balls. Our approach is based on moderate deviation principles and a connection between the uniform distribution on Orlicz balls and Gibbs measures at certain critical inverse temperatures with potentials given by Orlicz functions, an idea recently presented by Kabluchko and Prochno in [The maximum entropy principle and volumetric properties of Orlicz balls, J. Math. Anal. Appl. {\bf 495}(1) 2021, 1--19]
Large Deviation Principles and Complete Equivalence and Nonequivalence Results for Pure and Mixed Ensembles
We consider a general class of statistical mechanical models of coherent
structures in turbulence, which includes models of two-dimensional fluid
motion, quasi-geostrophic flows, and dispersive waves. First, large deviation
principles are proved for the canonical ensemble and the microcanonical
ensemble. For each ensemble the set of equilibrium macrostates is defined as
the set on which the corresponding rate function attains its minimum of 0. We
then present complete equivalence and nonequivalence results at the level of
equilibrium macrostates for the two ensembles.Comment: 57 page
The large deviation approach to statistical mechanics
The theory of large deviations is concerned with the exponential decay of
probabilities of large fluctuations in random systems. These probabilities are
important in many fields of study, including statistics, finance, and
engineering, as they often yield valuable information about the large
fluctuations of a random system around its most probable state or trajectory.
In the context of equilibrium statistical mechanics, the theory of large
deviations provides exponential-order estimates of probabilities that refine
and generalize Einstein's theory of fluctuations. This review explores this and
other connections between large deviation theory and statistical mechanics, in
an effort to show that the mathematical language of statistical mechanics is
the language of large deviation theory. The first part of the review presents
the basics of large deviation theory, and works out many of its classical
applications related to sums of random variables and Markov processes. The
second part goes through many problems and results of statistical mechanics,
and shows how these can be formulated and derived within the context of large
deviation theory. The problems and results treated cover a wide range of
physical systems, including equilibrium many-particle systems, noise-perturbed
dynamics, nonequilibrium systems, as well as multifractals, disordered systems,
and chaotic systems. This review also covers many fundamental aspects of
statistical mechanics, such as the derivation of variational principles
characterizing equilibrium and nonequilibrium states, the breaking of the
Legendre transform for nonconcave entropies, and the characterization of
nonequilibrium fluctuations through fluctuation relations.Comment: v1: 89 pages, 18 figures, pdflatex. v2: 95 pages, 20 figures, text,
figures and appendices added, many references cut, close to published versio
Maximum Entropy Production Principle for Stock Returns
In our previous studies we have investigated the structural complexity of
time series describing stock returns on New York's and Warsaw's stock
exchanges, by employing two estimators of Shannon's entropy rate based on
Lempel-Ziv and Context Tree Weighting algorithms, which were originally used
for data compression. Such structural complexity of the time series describing
logarithmic stock returns can be used as a measure of the inherent (model-free)
predictability of the underlying price formation processes, testing the
Efficient-Market Hypothesis in practice. We have also correlated the estimated
predictability with the profitability of standard trading algorithms, and found
that these do not use the structure inherent in the stock returns to any
significant degree. To find a way to use the structural complexity of the stock
returns for the purpose of predictions we propose the Maximum Entropy
Production Principle as applied to stock returns, and test it on the two
mentioned markets, inquiring into whether it is possible to enhance prediction
of stock returns based on the structural complexity of these and the mentioned
principle.Comment: 14 pages, 5 figure
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