2,020 research outputs found
The LIL for -statistics in Hilbert spaces
We give necessary and sufficient conditions for the (bounded) law of the
iterated logarithm for -statistics in Hilbert spaces. As a tool we also
develop moment and tail estimates for canonical Hilbert-space valued
-statistics of arbitrary order, which are of independent interest
Non-linear Plank Problems and polynomial inequalities
We study lower bounds for the norm of the product of polynomials and their
applications to the so called \emph{plank problem.} We are particularly
interested in polynomials on finite dimensional Banach spaces, in which case
our results improve previous works when the number of polynomials is large.Comment: 19 page
Local properties of Hilbert spaces of Dirichlet series
We show that the asymptotic behavior of the partial sums of a sequence of
positive numbers determine the local behavior of the Hilbert space of Dirichlet
series defined using these as weights. This extends results recently obtained
describing the local behavior of Dirichlet series with square summable
coefficients in terms of local integrability, boundary behavior, Carleson
measures and interpolating sequences. As these spaces can be identified with
functions spaces on the infinite-dimensional polydisk, this gives new results
on the Dirichlet and Bergman spaces on the infinite dimensional polydisk, as
well as the scale of Besov-Sobolev spaces containing the Drury-Arveson space on
the infinite dimensional unit ball. We use both techniques from the theory of
sampling in Paley-Wiener spaces, and classical results from analytic number
theory.Comment: 27 pages, 1 figur
Optimal Uncertainty Quantification
We propose a rigorous framework for Uncertainty Quantification (UQ) in which
the UQ objectives and the assumptions/information set are brought to the forefront.
This framework, which we call Optimal Uncertainty Quantification (OUQ), is based
on the observation that, given a set of assumptions and information about the problem,
there exist optimal bounds on uncertainties: these are obtained as extreme
values of well-defined optimization problems corresponding to extremizing probabilities
of failure, or of deviations, subject to the constraints imposed by the scenarios
compatible with the assumptions and information. In particular, this framework
does not implicitly impose inappropriate assumptions, nor does it repudiate relevant
information.
Although OUQ optimization problems are extremely large, we show that under
general conditions, they have finite-dimensional reductions. As an application,
we develop Optimal Concentration Inequalities (OCI) of Hoeffding and McDiarmid
type. Surprisingly, contrary to the classical sensitivity analysis paradigm, these results
show that uncertainties in input parameters do not necessarily propagate to
output uncertainties.
In addition, a general algorithmic framework is developed for OUQ and is tested
on the Caltech surrogate model for hypervelocity impact, suggesting the feasibility
of the framework for important complex systems
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