Density Formula and Concentration Inequalities with Malliavin Calculus
By Ivan Nourdin, Frederi G. Viens. 2009
130 citations recorded by the source corpus.
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 $G:= _H$, 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.
Subfields: Applied Mathematics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/7107268cfceef75eeb0d0c33bd58a27c7a0528eb
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
- Victor Chernozhukov
- D. Chetverikov
- Ivan Nourdin
- Kengo Kato
- Giovanni Peccati
- Gesine Reinert
- Sourav Chatterjee
- Matthias Schulte
In Statistics and Probability
- Victor Chernozhukov
- D. Chetverikov
- Kengo Kato
- Alexandre Belloni
- Ivan Nourdin
- Giovanni Peccati
- Christian Hansen
- Whitney K. Newey
Citations in the corpus
- Stein’s method for invariant measures of diffusions via Malliavin calculus☆ (2012, unknown): “Formula (23) is a consequence of the Mehler formula (see e.g. [12]) and it has been proved in [11, Section 3.2.1].”
- The optimal fourth moment theorem (2013, background): “1 has sparkled a great amount of generalisations and applications, ranging from density estimates [18] to entropic CLTs [17], and from estimates for Gaussian polymers [21] to universality results [16].”
- Comparison and anti-concentration bounds for maxima of Gaussian random vectors (2013, methodology): “Our anti-concentration bounds do not require such positivity (or other) assumptions on covariances and hence are not implied by results [17].”