1,482 research outputs found
Asymptotic Cram\'er type decomposition for Wiener and Wigner integrals
We investigate generalizations of the Cram\'er theorem. This theorem asserts
that a Gaussian random variable can be decomposed into the sum of independent
random variables if and only if they are Gaussian. We prove asymptotic
counterparts of such decomposition results for multiple Wiener integrals and
prove that similar results are true for the (asymptotic) decomposition of the
semicircular distribution into free multiple Wigner integrals
Arnol'd Tongues and Quantum Accelerator Modes
The stable periodic orbits of an area-preserving map on the 2-torus, which is
formally a variant of the Standard Map, have been shown to explain the quantum
accelerator modes that were discovered in experiments with laser-cooled atoms.
We show that their parametric dependence exhibits Arnol'd-like tongues and
perform a perturbative analysis of such structures. We thus explain the
arithmetical organisation of the accelerator modes and discuss experimental
implications thereof.Comment: 20 pages, 6 encapsulated postscript figure
Classical and free Fourth Moment Theorems: universality and thresholds
Let be a centered random variable with unit variance, zero third moment,
and such that . Let denote a normalized
sequence of homogeneous sums of fixed degree , built from independent
copies of . Under these minimal conditions, we prove that converges in
distribution to a standard Gaussian random variable if and only if the
corresponding sequence of fourth moments converges to . The statement is
then extended (mutatis mutandis) to the free probability setting. We shall also
discuss the optimality of our conditions in terms of explicit thresholds, as
well as establish several connections with the so-called universality
phenomenon of probability theory. Both in the classical and free probability
frameworks, our results extend and unify previous Fourth Moment Theorems for
Gaussian and semicircular approximations. Our techniques are based on a fine
combinatorial analysis of higher moments for homogeneous sums.Comment: 26 page
Four moments theorems on Markov chains
We obtain quantitative Four Moments Theorems establishing convergence
of the laws of elements of a Markov chaos to a Pearson distribution,
where the only assumptionwemake on the Pearson distribution is that it admits
four moments. While in general one cannot use moments to establish convergence
to a heavy-tailed distributions, we provide a context in which only the
first four moments suffices. These results are obtained by proving a general
carré du champ bound on the distance between laws of random variables in the
domain of a Markov diffusion generator and invariant measures of diffusions.
For elements of a Markov chaos, this bound can be reduced to just the first four
moments.First author draf
Four moments theorems on Markov chaos
We obtain quantitative Four Moments Theorems establishing convergence of the
laws of elements of a Markov chaos to a Pearson distribution, where the only
assumption we make on the Pearson distribution is that it admits four moments.
While in general one cannot use moments to establish convergence to a
heavy-tailed distributions, we provide a context in which only the first four
moments suffices. These results are obtained by proving a general carr\'e du
champ bound on the distance between laws of random variables in the domain of a
Markov diffusion generator and invariant measures of diffusions. For elements
of a Markov chaos, this bound can be reduced to just the first four moments.Comment: 24 page
For the Jubilee of Vladimir Mikhailovich Chernov
On April 25, 2019, Vladimir Chernov celebrated his 70th birthday, Doctor of Physics and Mathematics, Chief Researcher at the Laboratory of Mathematical Methods of Image Processing of the Image Processing Systems Institute of the Russian Academy of Sciences (IPSI RAS), a branch of the Federal Science Research Center "Crystallography and Photonics RAS and part-Time Professor at the Department of Geoinformatics and Information Security of the Samara National Research University named after academician S.P. Korolev (Samara University). The article briefly describes the scientific and pedagogical achievements of the hero of the day. © Published under licence by IOP Publishing Ltd
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