Malliavin calculus, Stein’s lemma, and densities and tails of random variables.

Frederi Viens (University Purdue, USA)
vendredi 28 mai 2010

Résumé : Ivan Nourdin and Giovanni Peccati have recently established results in
stochastic analysis by which, for a scalr random variable X which is
differentiable in the sense of Malliavin calculus (|DX| is
square-integrable), an important quantity to study is the random
variableG=, where M is the pseudo-inverse of the so-called
Ornstein-Uhlenbeck semigroup generator on Wiener space. For instance G
is a random way of measuring the dispersion of X since Var[X]=E[G] ;
plus, G is constant if and only if X is Gaussian ; and comparing G to a
constant can yield comparisons of X to the Gaussian law, as in limit
theorems or density formulas. This presentation will review such
results and the required background from Malliavin calculus, explain
their relation to
Stein’s lemma and equation, and outline their applications to suprema
of Gaussian fields, and other extensions. The results represent work
in various papers by Airault, Malliavin, Nourdin, Peccati, and Viens,
listed here :

Airault, H. ; Malliavin, P. ; Viens, F. Stokes formula on the Wiener space
and n-dimensional Nourdin-Peccati analysis. Journal of Functional
Analysis, 258 no. 5 (2009), 1763-1783.

Nourdin, I. ; Peccati, G. Stein’s method on Wiener chaos. Probability
Theory and Related Fields, 145 (2008), no. 1, 75-118.

Nourdin, I. ; Viens, F. Density estimates and concentration inequalities
with Malliavin calculus. Electronic Journal of Probability, 14 (2009),

Viens, F. Stein’s lemma, Malliavin calculus, and tail bounds, with
application to polymer fluctuation exponent. Stochastic Processes and
their Applications 119 (2009), 3671-3698.

Cet exposé se tiendra à 11h00 en salle C20-13, 20ème étage, Université
Paris 1, Centre Pierre Mendes-France, 90 rue de Tolbiac, 75013 Paris
(métro : Olympiades).