Math. Review MR2520122</a>),
in the context of rates of convergence in law. Here, thanks to <i>G</i>,
density lower bounds can be obtained in some instances. Among several
examples, we provide an application to the (centered) maximum of a general
Gaussian process. We also explain how to derive concentration inequalities
for <i>Z</i> in our framework.">
Density formula and concentration inequalities with Malliavin calculus
Ivan Nourdin, Université Paris 6
Frederi G Viens, Purdue University
Abstract
We show how to use the Malliavin calculus to obtain a new exact formula for
the density ρ of the law of any random variable Z which is measurable
and differentiable with respect to a given isonormal Gaussian process. The
main advantage of this formula is that it does not refer to the divergence
operator δ (dual of the Malliavin derivative D). The formula is based
on an auxilliary random variable G:=<DZ,-DL-1Z>H,
where L is the
generator of the so-called Ornstein-Uhlenbeck semigroup. The use of G was
first discovered by Nourdin and Peccati (Probab. Theory Relat. Fields 145, 75-118,
2009,
Math. Review MR2520122),
in the context of rates of convergence in law. Here, thanks to G,
density lower bounds can be obtained in some instances. Among several
examples, we provide an application to the (centered) maximum of a general
Gaussian process. We also explain how to derive concentration inequalities
for Z in our framework.
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