311 research outputs found
The Stein-Dirichlet-Malliavin method
The Stein's method is a popular method used to derive upper-bounds of
distances between probability distributions. It can be viewed, in certain of
its formulations, as an avatar of the semi-group or of the smart-path method
used commonly in Gaussian analysis. We show how this procedure can be enriched
by Malliavin calculus leading to a functional approach valid in infinite
dimensional spaces.Comment: in ESAIM: Proceedings, EDP Sciences, 2015, pp.1
Quasi-invariance and integration by parts for determinantal and permanental processes
Determinantal and permanental processes are point processes with a
correlation function given by a determinant or a permanent. Their atoms exhibit
mutual attraction of repulsion, thus these processes are very far from the
uncorrelated situation encountered in Poisson models. We establish a
quasi-invariance result : we show that if atoms locations are perturbed along a
vector field, the resulting process is still a determinantal (respectively
permanental) process, the law of which is absolutely continuous with respect to
the original distribution. Based on this formula, following Bismut approach of
Malliavin calculus, we then give an integration by parts formula.Comment: Journal of Functional Analysis (2010) To appea
Malliavin and dirichlet structures for independent random variables
On any denumerable product of probability spaces, we construct a Malliavin
gradient and then a divergence and a number operator. This yields a Dirichlet
structure which can be shown to approach the usual structures for Poisson and
Brownian processes. We obtain versions of almost all the classical functional
inequalities in discrete settings which show that the Efron-Stein inequality
can be interpreted as a Poincar{\'e} inequality or that Hoeffding decomposition
of U-statistics can be interpreted as a chaos decomposition. We obtain a
version of the Lyapounov central limit theorem for independent random variables
without resorting to ad-hoc couplings, thus increasing the scope of the Stein
method
Hitting times for Gaussian processes
We establish a general formula for the Laplace transform of the hitting times
of a Gaussian process. Some consequences are derived, and particular cases like
the fractional Brownian motion are discussed.Comment: Published in at http://dx.doi.org/10.1214/009117907000000132 the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stein's method for Brownian approximations
Motivated by a theorem of Barbour, we revisit some of the classical limit
theorems in probability from the viewpoint of the Stein method. We setup the
framework to bound Wasserstein distances between some distributions on infinite
dimensional spaces. We show that the convergence rate for the Poisson
approximation of the Brownian motion is as expected proportional to
where is the intensity of the Poisson process. We
also exhibit the speed of convergence for the Donsker Theorem and for the
linear interpolation of the Brownian motion. By iterating the procedure, we
give Edgeworth expansions with precise error bounds.Comment: Communications on Stochastic Analysis (2013
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