27 research outputs found
Exchangeable pairs on Wiener chaos
In [14], Nourdin and Peccati combined the Malliavin calculus and Stein's
method of normal approximation to associate a rate of convergence to the
celebrated fourth moment theorem [19] of Nualart and Peccati. Their analysis,
known as the Malliavin-Stein method nowadays, has found many applications
towards stochastic geometry, statistical physics and zeros of random
polynomials, to name a few. In this article, we further explore the relation
between these two fields of mathematics. In particular, we construct
exchangeable pairs of Brownian motions and we discover a natural link between
Malliavin operators and these exchangeable pairs. By combining our findings
with E. Meckes' infinitesimal version of exchangeable pairs, we can give
another proof of the quantitative fourth moment theorem. Finally, we extend our
result to the multidimensional case.Comment: 19 pages, submitte
Fourth moment theorems on the Poisson space in any dimension
We extend to any dimension the quantitative fourth moment theorem on the
Poisson setting, recently proved by C. D\"obler and G. Peccati (2017). In
particular, by adapting the exchangeable pairs couplings construction
introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove
our results under the weakest possible assumption of finite fourth moments.
This yields a Peccati-Tudor type theorem, as well as an optimal improvement in
the univariate case. Finally, a transfer principle "from-Poisson-to-Gaussian"
is derived, which is closely related to the universality phenomenon for
homogeneous multilinear sums.Comment: Minor revision. to appear in Electron. J. Proba
Averaging Gaussian functionals
This paper consists of two parts. In the first part, we focus on the average of a functional over shifted Gaussian homogeneous noise and as the averaging domain covers the whole space, we establish a Breuer-Major type Gaussian fluctuation based on various assumptions on the covariance kernel and/or the spectral measure. Our methodology for the first part begins with the application of Malliavin calculus around Nualart-Peccati’s Fourth Moment Theorem, and in addition we apply the Fourier techniques as well as a soft approximation argument based on Bessel functions of first kind.
The same methodology leads us to investigate a closely related problem in the second part. We study the spatial average of a linear stochastic heat equation driven by space-time Gaussian colored noise. The temporal covariance kernel γ0 is assumed to be locally integrable in this paper. If the spatial covariance kernel is nonnegative and integrable on the whole space, then the spatial average admits the Gaussian fluctuation; with some extra mild integrability condition on γ0, we are able to provide a functional central limit theorem. These results complement recent studies on the spatial average for SPDEs. Our analysis also allows us to consider the case where the spatial covariance kernel is not integrable: For example, in the case of the Riesz kernel, the first chaotic component of the spatial average is dominant so that the Gaussian fluctuation also holds true
Averaging Gaussian functionals
This paper consists of two parts. In the first part, we focus on the average
of a functional over shifted Gaussian homogeneous noise and as the averaging
domain covers the whole space, we establish a Breuer-Major type Gaussian
fluctuation based on various assumptions on the covariance kernel and/or the
spectral measure. Our methodology for the first part begins with the
application of Malliavin calculus around Nualart-Peccati's Fourth Moment
Theorem, and in addition we apply the Fourier techniques as well as a soft
approximation argument based on Bessel functions of first kind.
The same methodology leads us to investigate a closely related problem in the
second part. We study the spatial average of a linear stochastic heat equation
driven by space-time Gaussian colored noise. The temporal covariance kernel
is assumed to be locally integrable in this paper. If the spatial
covariance kernel is nonnegative and integrable on the whole space, then the
spatial average admits Gaussian fluctuation; with some extra mild integrability
condition on , we are able to provide a functional central limit
theorem. These results complement recent studies on the spatial average for
SPDEs. Our analysis also allows us to consider the case where the spatial
covariance kernel is not integrable: For example, in the case of the Riesz
kernel, the first chaotic component of the spatial average is dominant so that
the Gaussian fluctuation also holds true.Comment: Version 1:60pages; Version 2: 64 pages, Theorem 1.9 (functional
version of Theorem 1.7) is ne
Hyperbolic Anderson model with L\'evy white noise: spatial ergodicity and fluctuation
In this paper, we study one-dimensional hyperbolic Anderson models (HAM)
driven by space-time L\'evy white noise in a finite-variance setting. Motivated
by recent active research on limit theorems for stochastic partial differential
equations driven by Gaussian noises, we present the first study in this L\'evy
setting. In particular, we first establish the spatial ergodicity of the
solution and then a quantitative central limit theorem (CLT) for the spatial
averages of the solution to HAM in both Wasserstein distance and Kolmogorov
distance, with the same rate of convergence. To achieve the first goal (i.e.
spatial ergodicity), we exploit some basic properties of the solution and apply
a Poincar\'e inequality in the Poisson setting, which requires delicate moment
estimates on the Malliavin derivatives of the solution. Such moment estimates
are obtained in a soft manner by observing a natural connection between the
Malliavin derivatives of HAM and a HAM with Dirac delta velocity. To achieve
the second goal (i.e. CLT), we need two key ingredients: (i) a univariate
second-order Poincar\'e inequality in the Poisson setting that goes back to
Last, Peccati, and Schulte (Probab. Theory Related Fields, 2016) and has been
recently improved by Trauthwein (arXiv:2212.03782); (ii) aforementioned moment
estimates of Malliavin derivatives up to second order. We also establish a
corresponding functional central limit theorem by (a) showing the convergence
in finite-dimensional distributions and (b) verifying Kolmogorov's tightness
criterion. Part (a) is made possible by a linearization trick and the
univariate second-order Poincar\'e inequality, while part (b) follows from a
standard moment estimate with an application of Rosenthal's inequality.Comment: 43 page