323 research outputs found
Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry
A Poisson or a binomial process on an abstract state space and a symmetric
function acting on -tuples of its points are considered. They induce a
point process on the target space of . The main result is a functional limit
theorem which provides an upper bound for an optimal transportation distance
between the image process and a Poisson process on the target space. The
technical background are a version of Stein's method for Poisson process
approximation, a Glauber dynamics representation for the Poisson process and
the Malliavin formalism. As applications of the main result, error bounds for
approximations of U-statistics by Poisson, compound Poisson and stable random
variables are derived, and examples from stochastic geometry are investigated.Comment: Published at http://dx.doi.org/10.1214/15-AOP1020 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Eulerian and Lagrangian solutions to the continuity and Euler equations with vorticity
In the first part of this paper we establish a uniqueness result for
continuity equations with velocity field whose derivative can be represented by
a singular integral operator of an function, extending the Lagrangian
theory in \cite{BouchutCrippa13}. The proof is based on a combination of a
stability estimate via optimal transport techniques developed in \cite{Seis16a}
and some tools from harmonic analysis introduced in \cite{BouchutCrippa13}. In
the second part of the paper, we address a question that arose in
\cite{FilhoMazzucatoNussenzveig06}, namely whether 2D Euler solutions obtained
via vanishing viscosity are renormalized (in the sense of DiPerna and Lions)
when the initial data has low integrability. We show that this is the case even
when the initial vorticity is only in~, extending the proof for the
case in \cite{CrippaSpirito15}
Concentration of empirical distribution functions with applications to non-i.i.d. models
The concentration of empirical measures is studied for dependent data, whose
joint distribution satisfies Poincar\'{e}-type or logarithmic Sobolev
inequalities. The general concentration results are then applied to spectral
empirical distribution functions associated with high-dimensional random
matrices.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ254 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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