488 research outputs found
Conditioning moments of singular measures for entropy optimization. I
In order to process a potential moment sequence by the entropy optimization
method one has to be assured that the original measure is absolutely continuous
with respect to Lebesgue measure. We propose a non-linear exponential transform
of the moment sequence of any measure, including singular ones, so that the
entropy optimization method can still be used in the reconstruction or
approximation of the original. The Cauchy transform in one variable, used for
this very purpose in a classical context by A.\ A.\ Markov and followers, is
replaced in higher dimensions by the Fantappi\`{e} transform. Several
algorithms for reconstruction from moments are sketched, while we intend to
provide the numerical experiments and computational aspects in a subsequent
article. The essentials of complex analysis, harmonic analysis, and entropy
optimization are recalled in some detail, with the goal of making the main
results more accessible to non-expert readers.
Keywords: Fantappi\`e transform; entropy optimization; moment problem; tube
domain; exponential transformComment: Submitted to Indagnationes Mathematicae, I. Gohberg Memorial issu
Wigner chaos and the fourth moment
We prove that a normalized sequence of multiple Wigner integrals (in a fixed
order of free Wigner chaos) converges in law to the standard semicircular
distribution if and only if the corresponding sequence of fourth moments
converges to 2, the fourth moment of the semicircular law. This extends to the
free probabilistic, setting some recent results by Nualart and Peccati on
characterizations of central limit theorems in a fixed order of Gaussian Wiener
chaos. Our proof is combinatorial, analyzing the relevant noncrossing
partitions that control the moments of the integrals. We can also use these
techniques to distinguish the first order of chaos from all others in terms of
distributions; we then use tools from the free Malliavin calculus to give
quantitative bounds on a distance between different orders of chaos. When
applied to highly symmetric kernels, our results yield a new transfer
principle, connecting central limit theorems in free Wigner chaos to those in
Gaussian Wiener chaos. We use this to prove a new free version of an important
classical theorem, the Breuer-Major theorem.Comment: Published in at http://dx.doi.org/10.1214/11-AOP657 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
AFIT School of Engineering Contributions to Air Force Research and Technology Calendar Year 1973
This report contains abstracts of Master of Science Theses, Doctoral dissertations, and selected faculty publications completed during the 1973 calendar year at the School of Engineering, Air Force Institute of Technology, at Wright-Patterson Air Force Base, Ohio
AFIT School of Engineering Contributions to Air Force Research and Technology Calendar Year 1973
This report contains abstracts of Master of Science Theses, Doctoral dissertations, and selected faculty publications completed during the 1973 calendar year at the School of Engineering, Air Force Institute of Technology, at Wright-Patterson Air Force Base, Ohio
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