Solesne Bourguin

The Malliavin–Stein Method on the Poisson Space

This chapter provides a detailed and unified discussion of a collection of recently introduced techniques, allowing one to establish limit theorems with explicit rates of convergence, by combining the Stein’s and Chen–Stein methods with Malliavin calculus. Some results concerning multiple integrals are discussed in detail.

On the law of the solution to a stochastic heat equation with fractional noise in time

Abstract We study the law of the solution to the stochastic heat equation with additive Gaussian noise which behaves as the fractional Brownian motion in time and is white in space. We prove a decomposition of the solution in terms of the bifractional Brownian motion. Our result is an extension of a result by Swanson.

Poisson convergence on the free Poisson algebra

Based on recent findings by Bourguin and Peccati, we give a fourth moment type condition for an element of a free Poisson chaos of arbitrary order to converge to a free (centered) Poisson distribution. We also show that free Poisson chaos of order strictly greater than one do not contain any non-zero free Poisson random variables. We are also able to give a sufficient and necessary condition for an element of the first free Poisson chaos to have a free Poisson distribution. Finally, depending on the parity of the considered free Poisson chaos, we provide a general counterexample to the naive universality of the semicircular Wigner chaos established by Deya and Nourdin as well as a transfer principle between the Wigner and the free Poisson chaos.

Vector-valued semicircular limits on the free Poisson chaos

In this note, we prove a multidimensional counterpart of the central limit theorem on the free Poisson chaos recently proved by Bourguin and Peccati (2014). A noteworthy property of convergence toward the semicircular distribution on the free Poisson chaos is obtained as part of the limit theorem: component-wise convergence of sequences of vectors of multiple integrals with respect to a free Poisson random measure toward the semicircular distribution implies joint convergence. This result complements similar findings for the Wiener chaos by Peccati and Tudor (2005), the classical Poisson chaos by Peccati and Zheng (2010) and the Wigner chaos by Nourdin, Peccati and Speicher (2013).

Malliavin Calculus and Self Normalized Sums

We study the self-normalized sums of independent random variables from the perspective of the Malliavin calculus. We give the chaotic expansion for them and we prove a Berry–Esseen bound with respect to several distances.

Berry-Esséen Bounds for Long Memory Moving Averages via Stein's Method and Malliavin Calculus

Using the Stein method on Wiener chaos introduced in Nourdin and Peccati [10], we prove Berry-Esséen bounds for long memory moving averages.

Asymptotic Cramér type decomposition for Wiener and Wigner integrals

We investigate generalizations of the Cramer theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.

U-Statistics on the Spherical Poisson Space

We review a recent stream of research on normal approximations for linear functionals and more general U-statistics of wavelets/needlets coefficients evaluated on a homogeneous spherical Poisson field. We show how, by exploiting results from Peccati and Zheng (Electron J Probab 15(48):1487–1527, 2010) based on Malliavin calculus and Stein’s method, it is possible to assess the rate of convergence to Gaussianity for a triangular array of statistics with growing dimensions. These results can be exploited in a number of statistical applications, such as spherical density estimations, searching for point sources, estimation of variance, and the spherical two-sample problem.