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2009 Density Formula and Concentration Inequalities with Malliavin Calculus
Ivan Nourdin, Frederi Viens
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Electron. J. Probab. 14: 2287-2309 (2009). DOI: 10.1214/EJP.v14-707

Abstract

We show how to use the Malliavin calculus to obtain a new exact formula for the density $\rho$ 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 $\delta$ (dual of the Malliavin derivative $D$). The formula is based on an auxilliary random variable $G:= < DZ,-DL^{-1}Z >_H$, where $L$ is the generator of the so-called Ornstein-Uhlenbeck semigroup. The use of $G$ was first discovered by Nourdin and Peccati (PTRF 145 75-118 2009 <a href="http://www.ams.org/mathscinet-getitem?mr=2520122">MR-2520122</a>), 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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Ivan Nourdin. Frederi Viens. "Density Formula and Concentration Inequalities with Malliavin Calculus." Electron. J. Probab. 14 2287 - 2309, 2009. https://doi.org/10.1214/EJP.v14-707

Information

Accepted: 29 September 2009; Published: 2009
First available in Project Euclid: 1 June 2016

zbMATH: 1192.60066
MathSciNet: MR2556018
Digital Object Identifier: 10.1214/EJP.v14-707

Subjects:
Primary: 60G15
Secondary: 60H07

Keywords: concentration inequality , Density , fractional Brownian motion , Malliavin calculus , suprema of Gaussian processes

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