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Guangqu Zheng

Publications and source records attributed to Guangqu Zheng.

35 records · Page 2Linked to original sources
arXiv2021

A simplified second-order Gaussian Poincaré inequality in discrete setting with applications

In this paper, a simplified second-order Gaussian Poincaré inequality for normal approximation of functionals over infinitely many Rademacher random variables is derived. It is based on a new bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, which is established by means of the discrete Malliavin-Stein method and is of independent interest. As an application, the number of vertices with prescribed degree and the subgraph counting statistic in the Erdös-Rényi random graph are discussed. The number of vertices of fixed degree is also studied for percolation on the Hamming hypercube. Moreover, the number of isolated faces in the Linial-Meshulam-Wallach random $κ$-complex and infinite weighted 2-runs are treated.

math.PR↗
arXiv2020

Averaging 2d stochastic wave equation

We consider a 2D stochastic wave equation driven by a Gaussian noise, which is temporally white and spatially colored described by the Riesz kernel. Our first main result is the functional central limit theorem for the spatial average of the solution. And we also establish a quantitative central limit theorem for the marginal and the rate of convergence is described by the total-variation distance. A fundamental ingredient in our proofs is the pointwise $L^p$-estimate of Malliavin derivative, which is of independent interest.

math.PR↗
arXiv2020

Spatial averages for the Parabolic Anderson model driven by rough noise

In this paper, we study spatial averages for the parabolic Anderson model in the Skorohod sense driven by rough Gaussian noise, which is colored in space and time. We include the case of a fractional noise with Hurst parameters $H_0$ in time and $H_1$ in space, satisfying $H_0 \in (1/2,1)$, $H_1\in (0,1/2)$ and $H_0 + H_1 > 3/4$. Our main result is a functional central limit theorem for the spatial averages. As an important ingredient of our analysis, we present a Feynman-Kac formula that is new for these values of the Hurst parameters.

math.PR↗
arXiv2020

The Probability of Intransitivity in Dice and Close Elections

We study the phenomenon of intransitivity in models of dice and voting. First, we follow a recent thread of research for $n$-sided dice with pairwise ordering induced by the probability, relative to $1/2$, that a throw from one die is higher than the other. We build on a recent result of Polymath showing that three dice with i.i.d. faces drawn from the uniform distribution on $\{1,\ldots,n\}$ and conditioned on the average of faces equal to $(n+1)/2$ are intransitive with asymptotic probability $1/4$. We show that if dice faces are drawn from a non-uniform continuous mean zero distribution conditioned on the average of faces equal to $0$, then three dice are transitive with high probability. We also extend our results to stationary Gaussian dice, whose faces, for example, can be the fractional Brownian increments with Hurst index $H\in(0,1)$. Second, we pose an analogous model in the context of Condorcet voting. We consider $n$ voters who rank $k$ alternatives independently and uniformly at random. The winner between each two alternatives is decided by a majority vote based on the preferences. We show that in this model, if all pairwise elections are close to tied, then the asymptotic probability of obtaining any tournament on the $k$ alternatives is equal to $2^{-k(k-1)/2}$, which markedly differs from known results in the model without conditioning. We also explore the Condorcet voting model where methods other than simple majority are used for pairwise elections. We investigate some natural definitions of "close to tied" for general functions and exhibit an example where the distribution over tournaments is not uniform under those definitions.

math.PR↗
arXiv2020

Central limit theorems for stochastic wave equations in dimensions one and two

Fix $d\in\{1,2\}$, we consider a $d$-dimensional stochastic wave equation driven by a Gaussian noise, which is temporally white and colored in space such that the spatial correlation function is integrable and satisfies Dalang's condition. In this setting, we provide quantitative central limit theorems for the spatial average of the solution over a Euclidean ball, as the radius of the ball diverges to infinity. We also establish functional central limit theorems. A fundamental ingredient in our analysis is the pointwise $L^p$-estimate for the Malliavin derivative of the solution, which is of independent interest. This paper is another addendum to the recent research line of averaging stochastic partial differential equations.

math.PR↗
arXiv2020

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 $γ_0$ 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 $γ_0$, 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.

math.PR↗
arXiv2019

A Central Limit Theorem for the stochastic wave equation with fractional noise

We study the one-dimensional stochastic wave equation driven by a Gaussian multiplicative noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in [1/2,1)$ in the spatial variable. We show that the normalized spacial average of the solution over $[-R,R]$ converges in total variation distance to a normal distribution, as $R$ tends to infinity. We also provide a functional central limit theorem.

math.PR↗
arXiv2019

Oscillatory Breuer-Major theorem with application to the random corrector problem

In this paper, we present an oscillatory version of the celebrated Breuer-Major theorem that is motivated by the random corrector problem. As an application, we are able to prove new results concerning the Gaussian fluctuation of the random corrector. We also provide a variant of this theorem involving homogeneous measures.

math.PR↗
arXiv2019

Gaussian fluctuations for the stochastic heat equation with colored noise

In this paper, we present a quantitative central limit theorem for the d-dimensional stochastic heat equation driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. We show that the spatial average of the solution over an Euclidean ball is close to a Gaussian distribution, when the radius of the ball tends to infinity. Our central limit theorem is described in the total variation distance, using Malliavin calculus and Stein's method. We also provide a functional central limit theorem.

math.PR↗
arXiv2018

Almost sure convergence on chaoses

We present several new phenomena about almost sure convergence on homogeneous chaoses that include Gaussian Wiener chaos and homogeneous sums in independent random variables. Concretely, we establish the fact that almost sure convergence on a fixed finite sum of chaoses forces the almost sure convergence of each chaotic component. Our strategy uses "{\it extra randomness}" and a simple conditioning argument. These ideas are close to the spirit of \emph{Stein's method of exchangeable pairs}. Some natural questions are left open in this note.

math.PR↗
arXiv2018

Asymptotic behavior of large Gaussian correlated Wishart matrices

We consider high-dimensional Wishart matrices $d^{-1}\mathcal{X}_{n,d}\mathcal{X}_{n,d}^T$, associated with a rectangular random matrix $\mathcal{X}_{n,d}$ of size $n\times d$ whose entries are jointly Gaussian and correlated. Even if we will consider the case of overall correlation among the entries of $\mathcal{X}_{n,d}$, our main focus is on the case where the rows of $\mathcal{X}_{n,d}$ are independent copies of a $n$-dimensional stationary centered Gaussian vector of correlation function $s$. When $s$ belongs to $\ell^{4/3}(\mathbb{Z})$, we show that a proper normalization of $d^{-1}\mathcal{X}_{n,d}\mathcal{X}_{n,d}^T$ is close in Wasserstein distance to the corresponding Gaussian ensemble as long as $d$ is much larger than $n^3$, thus recovering the main finding of [3,9] and extending it to a larger class of matrices. We also investigate the case where $s$ is the correlation function associated with the fractional Brownian noise of parameter $H$. This example is very rich, as it gives rise to a great variety of phenomena with very different natures, depending on how $H$ is located with respect to $1/2$, $5/8$ and $3/4$. Notably, when $H>3/4$, our study highlights a new probabilistic object, which we have decided to call the Rosenblatt-Wishart matrix. Our approach crucially relies on the fact that the entries of the Wishart matrices we are dealing with are double Wiener-Itô integrals, allowing us to make use of multivariate bounds arising from the Malliavin-Stein method and related ideas. To conclude the paper, we analyze the situation where the row-independence assumption is relaxed and we also look at the setting of random $p$-tensors ($p\geq 3$), a natural extension of Wishart matrices.

math.PR↗
arXiv2018

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. Döbler 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.

math.PR↗
arXiv2017

A Peccati-Tudor type theorem for Rademacher chaoses

In this article, we prove that in the Rademacher setting, a random vector with chaotic components is close in distribution to a centred Gaussian vector, if both the maximal influence of the associated kernel and the fourth cumulant of each component is small. In particular, we recover the univariate case recently established in Döbler and Krokowski (2017). Our main strategy consists in a novel adaption of the exchangeable pairs couplings initiated in Nourdin and Zheng (2017), as well as its combination with estimates via chaos decomposition.

math.PR↗
arXiv2017

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.

math.PR↗
arXiv2017

Convergence of random oscillatory integrals in the presence of long-range dependence and application to homogenization

This paper deals with the asymptotic behavior of random oscillatory integrals in the presence of long-range dependence. As a byproduct, we solve the corrector problem in random homogenization of one-dimensional elliptic equations with highly oscillatory random coefficients displaying long-range dependence, by proving convergence to stochastic integrals with respect to Hermite processes.

math.PR↗
arXiv2016

Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals

In this work, we study the normal approximation and almost sure central limit theorems for some functionals of an independent sequence of Rademacher random variables. In particular, we provide a new chain rule that improves the one derived by Nourdin, Peccati and Reinert(2010) and then we deduce the bound on Wasserstein distance for normal approximation using the (discrete) Malliavin-Stein approach. Besides, we are able to give the almost sure central limit theorem for a sequence of random variables inside a fixed Rademacher chaos using the Ibragimov-Lifshits criterion.

math.PR↗