EJP Reading Tools

Density Formula and Concentration Inequalities with Malliavin Calculus

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Density Formula and Concentration Inequalities with Malliavin Calculus
2.	Creator	Author's name, affiliation, country	Ivan Nourdin; Université Paris 6; France
2.	Creator	Author's name, affiliation, country	Frederi G Viens; Purdue University; United States
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Malliavin calculus; density; concentration inequality; suprema of Gaussian processes; fractional Brownian motion
3.	Subject	Subject classification	60G15; 60H07
4.	Description	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 MR-2520122), 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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	National Science Foundation, grant numbers DMS 0606615 and DMS 0907321
7.	Date	(YYYY-MM-DD)	2009-09-29
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/707
10.	Identifier	Digital Object Identifier	10.1214/EJP.v14-707
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 14
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
15.	Rights	Copyright and permissions	The Electronic Journal of Probability applies the Creative Commons Attribution License (CCAL) to all articles we publish in this journal. Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles published in EJP, so long as the original authors and source are credited. This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available. Summary of the Creative Commons Attribution License You are free to copy, distribute, display, and perform the work to make derivative works to make commercial use of the work under the following condition of Attribution: others must attribute the work if displayed on the web or stored in any electronic archive by making a link back to the website of EJP via its Digital Object Identifier (DOI), or if published in other media by acknowledging prior publication in this Journal with a precise citation including the DOI. For any further reuse or distribution, the same terms apply. Any of these conditions can be waived by permission of the Corresponding Author.