arXiv subjects

Guangqu Zheng

Publications and source records attributed to Guangqu Zheng.

At least 19 recordsLinked to original sources

arXiv2026

Coalescence and fluctuations of O'Connell$\unicode{x2013}$Yor polymers via the free-energy correlation profile

We establish an identity relating the spatial derivative of a two-point free-energy correlation to two fundamental geometric quantities: the annealed probability that two independently sampled polymers, in the same environment, meet before reaching the prescribed terminal level, and the exit-point location of a single polymer. Our result is inspired in part by recent integration-by-parts work of Gu and Quastel for the KPZ equation. A crucial step in their continuous setting relies on an ingenious application of Itô's formula, which has no direct analogue in our semi-discrete setting. Instead, we exploit intrinsic symmetries of the polymer model together with the memoryless property. We also provide two applications: (i) we recover the horizontal Burke property by proving that the anchored stationary horizontal free-energy profile is a two-sided Brownian motion; (ii) in the zero-temperature limit, we obtain the corresponding identity for Brownian last-passage percolation.

math.PR↗

arXiv2026

Gamma approximation and Poisson--Gaussian invariance principle on Poisson chaos

We study centered Gamma approximation and a same-kernel Poisson--Gaussian invariance principle on fixed Poisson chaoses. For Gamma approximation, a martingale-core argument extends the carré-du-champ and $d_2$ estimates of Döbler and Peccati (Ann. Probab., 2018) from regular kernels to every fourth-integrable chaos element. In the diffuse regime, characterized by vanishing fourth add-one energy, this yields an exact four-moment criterion under uniform integrability of fourth powers. In the rare-jump regime, ordinary moments do not determine the approximation mechanism: convergence of the full moment sequence may coexist with a nonvanishing fourth add-one energy, and we construct such centered Gamma limits in every fixed chaos order. The invariance principle is independent of the Gamma target. For Poisson and Gaussian multiple integrals with the same kernel, we bound both smooth-test discrepancies and the Wasserstein distance in terms of the variance and the fourth add-one energy. Thus, vanishing fourth add-one energy is an intrinsic Lindeberg condition under which the two chaoses are asymptotically indistinguishable in distribution. Combined with a moment-transfer estimate and the Gaussian fourth-moment theorem, this gives an alternative proof of the qualitative normal fourth-moment theorem on a fixed Poisson chaos. A rainbow example shows that the Lindeberg condition is essential: the Gaussian analogue may be asymptotically normal while the Poisson integral converges to a centered compound-Poisson law. The same comparison also explains the different behavior of even and odd chaos orders for diffuse centered Gamma limits.

math.PR↗

arXiv2026

Four-moment criteria for Poisson convergence on Poisson and Rademacher chaoses

In this paper, we establish Poisson limit theorems on Poisson and Rademacher chaoses. Our principal result is a total-variation bound, valid in both settings, for an integer-valued functional whose highest-order chaos is dominant. The approximation error is the sum of the pure-chaos four-moment terms and an explicit remainder controlled by the $L^4$-size of the lower-order chaoses. On Poisson space the pure-chaos term is controlled solely by the moment defect, whereas on Rademacher space an additional maximal-influence correction is required. When the lower-order remainder vanishes, we recover exactly the corresponding total-variation bound for a shifted pure chaos. As consequences, for nonnegative integer-valued shifts of random variables in a fixed Poisson chaos, convergence of the first four moments is equivalent to convergence in total variation to a Poisson law, together with uniform integrability of the fourth powers; on a fixed Rademacher chaos, the analogous conclusion holds under a vanishing maximal-influence condition. The proof reveals a unit-jump rigidity phenomenon: the four-moment defect simultaneously suppresses unwanted spectral components and rules out non-unit jumps. We show, through an explicit quadratic counterexample, that the maximal-influence condition in the Rademacher setting is necessary for a general Poisson limit theorem. Finally, for every order $q\geq2$, we construct pure Poisson-chaos sequences with vanishing moment defect that converge weakly to a centered Poisson law. These examples show that the exact higher-order rigidity is not uniform once the lattice condition is removed. In both the Poisson and Rademacher settings, our proofs follow a unified strategy combining the Chen--Stein method, exchangeable pairs, and Ledoux's spectral viewpoint.

math.PR↗

arXiv2026

A Kolmogorov fourth-moment bound on Poisson chaos via a martingale core

For any finite family of Poisson multiple integrals and any finite $p\geq2$, we construct a common increasing filtration generated by finitely many exact Poisson counts such that the associated conditional expectations converge in $L^p$, remain in their original chaoses, and have bounded step kernels with finite-measure support. This finite-count martingale core allows regular fixed-chaos identities and estimates to be extended under the sole assumption of a finite fourth moment. In particular, if $F$ lives in a Poisson chaos with unit variance and finite fourth moment, we prove that the Kolmogorov distance between $F$ and a standard normal is bounded by $15.6(\mathbb{E}[F^4]-3)^{1/2}$. This removes Assumptions $\mathbf A$ and $\mathbf A^{\textbf{loc}}$ from the Kolmogorov bound of Döbler and Peccati (Ann. Probab., 2018). We also obtain quantitative $L^4$ estimates for all iterated Malliavin derivatives and, for $F$ in a Poisson chaos, the fourth moment assumption of $F$ forces the $L^4$-integrability of its kernel.

math.PR↗

arXiv2026

Breuer-Major-Donsker invariance principle

We prove a Breuer-Major-type Donsker's invariance principle for stationary Gaussian sequences under the natural \emph{finite-variance assumption} on the test function. This result, which we call the \emph{Breuer-Major-Donsker} principle, or simply the \emph{BMD principle}, removes the additional moment assumption imposed in the functional Breuer-Major theorem of Nourdin and Nualart (\emph{Probab. Theory Related Fields}, 2020). Our method does not rely on the Malliavin-calculus estimates used by Nourdin and Nualart, in particular Meyer's inequality. Instead, it is based on a predictable-martingale decomposition of the partial-sum process, which is of independent interest. We also make systematic use of \emph{non-determinism}, a central notion in Gaussian prediction theory. In the non-deterministic case, the martingale part is handled by the martingale functional central limit theorem, while the predictable remainder gains integrability above order two through Ornstein-Uhlenbeck smoothing. In the deterministic case, the martingale part vanishes, and the smoothing mechanism is no longer available along the full sequence. Nevertheless, under an additional mild assumption on the covariance function, a suitable decimation recovers non-determinism and reduces the proof to the non-deterministic case.

math.PR↗

arXiv2026

Almost sure central limit theorems via chaos expansions and related results

In this work, we investigate the asymptotic behavior of integral functionals of stationary Gaussian random fields as the integration domain tends to be the whole space. More precisely, using the Wiener chaos expansion and Malliavin-Stein method, we establish an {\it almost sure central limit theorem} (ASCLT) only under mild conditions on the covariance function of the underlying stationary Gaussian fields. In this setting, we additionally derive a {\it quantitative central limit theorem} with rate of convergence in quadratic Wasserstein distance, and show certain regularity property for the said integral functionals. In particular, we solve an open question on the {\it Malliavin differentiability of the excursion volume of Berry's random wave model}. As a key consequence of our analysis, we obtain the exact asymptotic rate (as a function of the exponent $q$) for the $q$-th moment of Bessel functions, thus confirming a conjecture based on existing numerical simulations. In the end, we provide two applications of our result: (i) ASCLT in the context of Breuer-Major central limit theorems, (ii) ASCLT for Berry's random wave model. Our approach does not require any knowledge on the regularity properties of random variables (e.g., Malliavin differentiability) and hence not only complements the existing literature, but also leads to novel results that are of independent interest.

math.PR↗

arXiv2025

Hyperbolic $P(Φ)_2$-model on the plane

We study the hyperbolic $Φ^{k+1}_2$-model on the plane. By establishing coming down from infinity for the associated stochastic nonlinear heat equation (SNLH) on the plane, we first construct a $Φ^{k+1}_2$-measure on the plane as a limit of the $Φ^{k+1}_2$-measures on large tori. We then study the canonical stochastic quantization of the $Φ^{k+1}_2$-measure on the plane thus constructed, namely, we study the defocusing stochastic damped nonlinear wave equation forced by an additive space-time white noise (= the hyperbolic $Φ^{k+1}_2$-model) on the plane. In particular, by taking a limit of the invariant Gibbs dynamics on large tori constructed by the first two authors with Gubinelli and Koch (2021), we construct invariant Gibbs dynamics for the hyperbolic $Φ^{k+1}_2$-model on the plane. Our main strategy is to develop further the ideas from a recent work on the hyperbolic $Φ^3_3$-model on the three-dimensional torus by the first two authors and Okamoto (2021), and to study convergence of the so-called enhanced Gibbs measures, for which coming down from infinity for the associated SNLH with positive regularity plays a crucial role. By combining wave and heat analysis together with ideas from optimal transport theory, we then conclude global well-posedness of the hyperbolic $Φ^{k+1}_2$-model on the plane and invariance of the associated Gibbs measure. As a byproduct of our argument, we also obtain invariance of the limiting $Φ^{k+1}_2$-measure on the plane under the dynamics of the parabolic $Φ^{k+1}_2$-model.

math.AP↗

arXiv2025

Functional second-order Gaussian Poincaré inequalities

In this paper, we work in the framework of Hilbert-valued Wiener structures and derive a functional version of the second-order Gaussian Poincaré inequality that leads to abstract bounds for Gaussian process approximation in $d_2$ distance. Our abstract bounds are flexible and can be applied in various examples including functional Breuer-Major central limit theorems, shallow neural networks, and spatial statistics of SPDEs solutions.

math.PR↗

arXiv2025

On the deep-water and shallow-water limits of the intermediate long wave equation from a statistical viewpoint

(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study convergence problems for the intermediate long wave equation (ILW), with the depth parameter $δ> 0$, in the deep-water limit ($δ\to \infty$) and the shallow-water limit ($δ\to 0$) from a statistical point of view. In particular, we establish convergence of invariant Gibbs dynamics for ILW in both the deep-water and shallow-water limits. For this purpose, we first construct the Gibbs measures for ILW, $0 < δ< \infty$. As they are supported on distributions, a renormalization is required. With the Wick renormalization, we carry out the construction of the Gibbs measures for ILW. We then prove that the Gibbs measures for ILW converge in total variation to that for the Benjamin-Ono equation (BO) in the deep-water limit. In the shallow-water regime, after applying a scaling transformation, we prove that, as $δ\to 0$, the Gibbs measures for the scaled ILW converge weakly to that for the Korteweg-de Vries equation (KdV). We point out that this second result is of particular interest since the Gibbs measures for the scaled ILW and KdV are mutually singular (whereas the Gibbs measures for ILW and BO are equivalent). We also discuss convergence of the associated dynamical problem. Lastly, we point out that our results also apply to the generalized ILW equation in the defocusing case, converging to the generalized BO in the deep-water limit and to the generalized KdV in the shallow-water limit. In the non-defocusing case, however, our results can not be extended to a nonlinearity with a higher power due to the non-normalizability of the corresponding Gibbs measures.

math.AP↗

arXiv2025

Central limit theorem for stochastic nonlinear wave equation with pure-jump Lévy white noise

In this paper, we study the random field solution to the stochastic nonlinear wave equation (SNLW) with constant initial conditions and multiplicative noise $σ(u)\dot{L}$, where the nonlinearity is encoded in a Lipschitz function $σ: \mathbb{R}\to\mathbb{R}$ and $\dot{L}$ denotes a pure-jump Lévy white noise on $\mathbb{R}_+\times\mathbb{R}$ with finite variance. Combining tools from Itô calculus and Malliavin calculus, we are able to establish the Malliavin differentiability of the solution with sharp moment bounds for the Malliavin derivatives. As an easy consequence, we obtain the spatial ergodicity of the solution to SNLW that leads to a law of large number result for the spatial integrals of the solution over $[-R, R]$ as $R\to\infty$. One of the main results of this paper is the obtention of the corresponding Gaussian fluctuation with rate of convergence in Wasserstein distance. To achieve this goal, we adapt the discrete Malliavin-Stein bound from Peccati, Solé, Taqqu, and Utzet ({\it Ann. Probab.}, 2010), and further combine it with the aforementioned moment bounds of Malliavin derivatives and Itô tools. Our work substantially improves our previous results (\textit{Trans.~Amer.~Math.~Soc.}, 2024) on the linear equation that heavily relied on the explicit chaos expansion of the solution. In current work, we also establish a functional version, an almost sure version of the central limit theorems, and the (quantitative) asymptotic independence of spatial integrals from the solution. The asymptotic independence result is established based on an observation of L. Pimentel (\textit{Ann.~Probab.}, 2022) and a further adaptation of Tudor's generalization (\textit{Trans.~Amer.~Math.~Soc.}, 2025) to the Poisson setting.

math.PR↗

arXiv2025

Almost sure central limit theorems for parabolic/hyperbolic Anderson models with Gaussian colored noises

This short note is devoted to establishing the almost sure central limit theorem for the parabolic/hyperbolic Anderson models driven by colored-in-time Gaussian noises, completing recent results on quantitative central limit theorems for stochastic partial differential equations. We combine the second-order Gaussian Poincaré inequality with Ibragimov and Lifshits' method of characteristic functions, effectively overcoming the challenge from the lack of Itô tools in this colored-in-time setting, and achieving results that are inaccessible with previous methods.

math.PR↗

arXiv2024

Almost sure central limit theorem for the hyperbolic Anderson model with Lévy white noise

In this paper, we present an almost sure central limit theorem (ASCLT) for the hyperbolic Anderson model (HAM) with a Lévy white noise in a finite-variance setting, complementing a recent work by Balan and Zheng (\emph{Trans.~Amer.~Math.~Soc.}, 2024) on the (quantitative) central limit theorems for the solution to the HAM. We provide two different proofs: one uses the Clark-Ocone formula and takes advantage of the martingale structure of the white-in-time noise, while the other is obtained by combining the second-order Gaussian Poincaré inequality with Ibragimov and Lifshits' method of characteristic functions. Both approaches are different from the one developed in the PhD thesis of C. Zheng (2011), allowing us to establish the ASCLT without lengthy computations of star contractions. Moreover, the second approach is expected to be useful for similar studies on SPDEs with colored-in-time noises, whereas the former, based on Itô calculus, is not applicable.

math.PR↗

arXiv2024

Hyperbolic Anderson model with Lévy white noise: spatial ergodicity and fluctuation

In this paper, we study one-dimensional hyperbolic Anderson models (HAM) driven by space-time pure-jump Lévy 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 Lévy 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 Poincaré 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 Poincaré 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 Poincaré inequality, while part (b) follows from a standard moment estimate with an application of Rosenthal's inequality.

math.PR↗

arXiv2023

Stein's method, smoothing and functional approximation

Stein's method for Gaussian process approximation can be used to bound the differences between the expectations of smooth functionals $h$ of a càdlàg random process $X$ of interest and the expectations of the same functionals of a well understood target random process $Z$ with continuous paths. Unfortunately, the class of smooth functionals for which this is easily possible is very restricted. Here, we prove an infinite dimensional Gaussian smoothing inequality, which enables the class of functionals to be greatly expanded -- examples are Lipschitz functionals with respect to the uniform metric, and indicators of arbitrary events -- in exchange for a loss of precision in the bounds. Our inequalities are expressed in terms of the smooth test function bound, an expectation of a functional of $X$ that is closely related to classical tightness criteria, a similar expectation for $Z$, and, for the indicator of a set $K$, the probability $\mathbb{P}(Z \in K^θ\setminus K^{-θ})$ that the target process is close to the boundary of $K$.

math.PR↗

arXiv2022

The hyperbolic Anderson model: Moment estimates of the Malliavin derivatives and applications

In this article, we study the hyperbolic Anderson model driven by a space-time \emph{colored} Gaussian homogeneous noise with spatial dimension $d=1,2$. Under mild assumptions, we provide $L^p$-estimates of the iterated Malliavin derivative of the solution in terms of the fundamental solution of the wave solution. To achieve this goal, we rely heavily on the \emph{Wiener chaos expansion} of the solution. Our first application are \emph{quantitative central limit theorems} for spatial averages of the solution to the hyperbolic Anderson model, where the rates of convergence are described by the total variation distance. These quantitative results have been elusive so far due to the temporal correlation of the noise blocking us from using the Itô calculus. A \emph{novel} ingredient to overcome this difficulty is the \emph{second-order Gaussian Poincaré inequality} coupled with the application of the aforementioned $L^p$-estimates of the first two Malliavin derivatives. Besides, we provide the corresponding functional central limit theorems. As a second application, we establish the absolute continuity of the law for the hyperbolic Anderson model. The $L^p$-estimates of Malliavin derivatives are crucial ingredients to verify a local version of Bouleau-Hirsch criterion for absolute continuity. Our approach substantially simplifies the arguments for the one-dimensional case, which has been studied in the recent work by Balan, Quer-Sardanyons and Song (2019).

math.PR↗

arXiv2021

Stein's method, Gaussian processes and Palm measures, with applications to queueing

We develop a general approach to Stein's method for approximating a random process in the path space $D([0,T]\to R^d)$ by a real continuous Gaussian process. We then use the approach in the context of processes that have a representation as integrals with respect to anunderlying point process, deriving a general quantitative Gaussian approximation. The error bound is expressed in terms of couplings of the original process to processes generated from the reduced Palm measures associated with the point process. As applications, we study certain $\text{GI}/\text{GI}/\infty$ queues in the "heavy traffic" regime.

math.PR↗

arXiv2021

Quantitative central limit theorems for the parabolic Anderson model driven by colored noises

In this paper, we study the spatial averages of the solution to the parabolic Anderson model driven by a space-time Gaussian homogeneous noise that is colored in time and space. We establish quantitative central limit theorems (CLT) of this spatial statistics under some mild assumptions, by using the Malliavin-Stein approach. The highlight of this paper is the obtention of rate of convergence in the colored-in-time setting, where one can not use Itô's calculus due to the lack of martingale structure. In particular, modulo highly technical computations, we apply a modified version of second-order Gaussian Poincaré inequality to overcome this lack of martingale structure and our work improves the results by Nualart-Zheng (2020 \emph{Electron. J. Probab.}) and Nualart-Song-Zheng (2021 \emph{ALEA, Lat. Am. J. Probab. Math. Stat.}).

math.PR↗

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↗