26 research outputs found
A quantitative central limit theorem for linear statistics of random matrix eigenvalues
It is known that the fluctuations of suitable linear statistics of Haar
distributed elements of the compact classical groups satisfy a central limit
theorem. We show that if the corresponding test functions are sufficiently
smooth, a rate of convergence of order almost can be obtained using a
quantitative multivariate CLT for traces of powers that was recently proven
using Stein's method of exchangeable pairs.Comment: Title modified; main result stated under slightly weaker conditions;
accepted for publication in the Journal of Theoretical Probabilit
The spectrum of the random environment and localization of noise
We consider random walk on a mildly random environment on finite transitive
d- regular graphs of increasing girth. After scaling and centering, the
analytic spectrum of the transition matrix converges in distribution to a
Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph
changes from a regular tree to the integers, the noise becomes localized.Comment: 18 pages, 1 figur
Lectures on Gaussian approximations with Malliavin calculus
In a seminal paper of 2005, Nualart and Peccati discovered a surprising
central limit theorem (called the "Fourth Moment Theorem" in the sequel) for
sequences of multiple stochastic integrals of a fixed order: in this context,
convergence in distribution to the standard normal law is equivalent to
convergence of just the fourth moment. Shortly afterwards, Peccati and Tudor
gave a multidimensional version of this characterization. Since the publication
of these two beautiful papers, many improvements and developments on this theme
have been considered. Among them is the work by Nualart and Ortiz-Latorre,
giving a new proof only based on Malliavin calculus and the use of integration
by parts on Wiener space. A second step is my joint paper "Stein's method on
Wiener chaos" (written in collaboration with Peccati) in which, by bringing
together Stein's method with Malliavin calculus, we have been able (among other
things) to associate quantitative bounds to the Fourth Moment Theorem. It turns
out that Stein's method and Malliavin calculus fit together admirably well.
Their interaction has led to some remarkable new results involving central and
non-central limit theorems for functionals of infinite-dimensional Gaussian
fields. The current survey aims to introduce the main features of this recent
theory. It originates from a series of lectures I delivered at the Coll\`ege de
France between January and March 2012, within the framework of the annual prize
of the Fondation des Sciences Math\'ematiques de Paris. It may be seen as a
teaser for the book "Normal Approximations Using Malliavin Calculus: from
Stein's Method to Universality" (jointly written with Peccati), in which the
interested reader will find much more than in this short survey.Comment: 72 pages. To be published in the S\'eminaire de Probabilit\'es. Mild
update: typos, referee comment
On the binary expansion of a random integer
It is shown that the distribution of the number of ones in the binary expansion of an integer chosen uniformly at random from the set 0, 1,…, n − 1 can be approximated in total variation by a mixture of two neighbouring binomial distributions, with error of order (log n)−1. The proof uses Stein's method
Approximating dependent rare events
10.3150/12-BEJSP18Bernoulli1941243-126
A non-uniform Berry-Esseen bound via Stein's method
Probability Theory and Related Fields1202236-25
From stein identities to moderate deviations
10.1214/12-AOP746Annals of Probability411262-29