Nguyen Tuan Minh
In this paper, we give an upper bound for a probabilistic distance between a Gaussian vector and a vector of U-statistics of Poisson point processes by applying Malliavin-Stein inequality on the Poisson space.
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Initiated around the year 2007, the Malliavin-Stein approach to probabilistic approximations combines Stein's method with infinite-dimensional integration by parts formulae based on the use of Malliavin-type operators. In the last decade, Malliavin-Stein techniques have allowed researchers to establish new quantitative limit theorems in a variety of domains of theoretical and applied stochastic analysis. The aim of this survey is to illustrate some of the latest developments of the Malliavin-Stein method, with specific emphasis on extensions and generalisations in the framework of Markov semig…
Using the Stein method on Wiener chaos introduced by Nourdin and Peccati we prove Berry-Esséen bounds for long memory moving averages.
Non-uniform Berry-Esseen bounds via Malliavin-Stein method
本文利用Malliavin-Stein方法建立非均匀Berry-Esseen界，并应用于多重Wiener-Itô积分与布朗运动指数泛函。
The fourth moment theorem provides error bounds of the order $\sqrt{{\mathbb E}(F^4) - 3}$ in the central limit theorem for elements $F$ of Wiener chaos of any order such that ${\mathbb E}(F^2) = 1$. It was proved by Nourdin and Peccati (2009) using Stein's method and the Malliavin calculus. It was also proved by Azmoodeh, Campese and Poly (2014) using Stein's method and Dirichlet forms. This paper is an exposition on the connections between Stein's method and the Malliavin calculus and between Stein's method and Dirichlet forms, and on how these connections are exploited in proving the fourth…
The Stein's method is a popular method used to derive upper-bounds of distances between probability distributions. It can be viewed, in certain of its formulations, as an avatar of the semi-group or of the smart-path method used commonly in Gaussian analysis. We show how this procedure can be enriched by Malliavin calculus leading to a functional approach valid in infinite dimensional spaces.
In this paper, following Nourdin-Peccati's methodology, we combine the Malliavin calculus and Stein's method to provide general bounds on the Wasserstein distance between functionals of a compound Hawkes process and a given Gaussian density. To achieve this, we rely on the Poisson embedding representation of an Hawkes process to provide a Malliavin calculus for the Hawkes processes, and more generally for compound Hawkes processes. As an application, we close a gap in the literature by providing the first Berry-Esséen bounds associated to Central Limit Theorems for the compound Hawkes process.
Given a random variable $F$ regular enough in the sense of the Malliavin calculus, we are able to measure the distance between its law and almost any continuous probability law on the real line. The bounds are given in terms of the Malliavin derivative of $F$. Our approach is based on the theory of Itô diffusions and the stochastic calculus of variations. Several examples are considered in order to illustrate our general results.
In this paper, we provide upper bounds on the d2 distance between a large class of functionals of a multivariate compound Hawkes process and a given Gaussian vector. This is proven using Malliavin's calculus defined on an underlying Poisson embedding. The upper bound is then used to infer the speed of convergence of Central Limit Theorems for the multivariate compound Hawkes process with exponential kernels as the observation time T goes to infinity.
We combine Stein's method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main findings by Peccati and Tudor (2005), Nualart and Ortiz-Latorre (2007), Peccati (2007) and Nourdin and Peccati (2007b, 2008); in particular, they apply to approximations by means of Gaussian vectors with an arbitrary, positive definite covariance matrix. Among several examples, we provide an application to a functional version of the Breuer-Major CLT for field…
New bounds on the total variation distance between the law of integer valued functionals of possibly non-symmetric and non-homogeneous infinite Rademacher sequences and the Poisson distribution are established. They are based on a combination of the Chen-Stein method and a discrete version of Malliavin calculus. We give some applications to shifted discrete multiple stochastic integrals.
We combine Malliavin calculus with Stein's method to derive bounds for the Variance-Gamma approximation of functionals of isonormal Gaussian processes, in particular of random variables living inside a fixed Wiener chaos induced by such a process. The bounds are presented in terms of Malliavin operators and norms of contractions. We show that a sequence of distributions of random variables in the second Wiener chaos converges to a Variance-Gamma distribution if and only if their moments of order two to six converge to that of a Variance-Gamma distributed random variable (six moment theorem). M…
We provide an overview of some recent techniques involving the Malliavin calculus of variations and the so-called ``Stein's method'' for the Gaussian approximations of probability distributions. Special attention is devoted to establishing explicit connections with the classic method of moments: in particular, we use interpolation techniques in order to deduce some new estimates for the moments of random variables belonging to a fixed Wiener chaos. As an illustration, a class of central limit theorems associated with the quadratic variation of a fractional Brownian motion is studied in detail.
Stein's method is a method of probability approximation which hinges on the solution of a functional equation. For normal approximation the functional equation is a first order differential equation. Malliavin calculus is an infinite-dimensional differential calculus whose operators act on functionals of general Gaussian processes. Nourdin and Peccati (2009) established a fundamental connection between Stein's method for normal approximation and Malliavin calculus through integration by parts. This connection is exploited to obtain error bounds in total variation in central limit theorems for …
Berry-Esseen bounds for large-time asymptotics of one-dimensional diffusion processes via Malliavin-Stein method
本文研究一维随机微分方程解在时间趋于无穷时的渐近行为，利用Malliavin-Stein方法证明了中心化缩放解分布与标准正态分布间全变差距离的Berry-Esseen界，并获得了时间参数的最优收敛速率。
Suppose that $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}^d}$ is the solution to a $d$-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance that satisfies Dalang's condition. The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form $N^{-d} \int_{[0,N]^d} g(u(t\,,x))\, \mathrm{d} x$, as $N\rightarrow\infty$, where $g$ is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Stein's method for …
Mathematical research with GPT-5: a Malliavin-Stein experiment
受GPT-5被报道解决凸优化公开问题的启发，研究者在Malliavin--Stein框架下开展对照实验，评估其能否将四阶矩定理拓展为带显式收敛速率的定量表述，填补了相关文献空白。
We investigate Stein-Malliavin approximations for nonlinear functionals of geometric interest of Gaussian random eigenfunctions on the unit $d$ -dimensional sphere ${\mathbb{S}}^{d},$ $d\geq 2.$ All our results are established in the high energy limit, i.e. for eigenfunctions corresponding to growing eigenvalues. More precisely, we provide an asymptotic analysis for the variance of random eigenfunctions, and also establish rates of convergence for various probability metrics for Hermite subordinated processes, arbitrary polynomials of finite order and square integral nonlinear transforms; the …
A new Berry-Esseen bound for non-linear functionals of non-symmetric and non-homogeneous infinite Rademacher sequences is established. It is based on a discrete version of the Malliavin-Stein method and an analysis of the discrete Ornstein-Uhlenbeck semigroup. The result is applied to sub-graph counts and to the number of vertices having a prescribed degree in the Erdős-Renyi random graph. A further application deals with a percolation problem on trees.
Following a strategy recently developed by Ivan Nourdin and Giovanni Peccati, we provide a general technique to compare the tail of a given random variable to that of a reference distribution. This enables us to give concrete conditions to ensure upper and/or lower bounds on the random variable's tail of various power or exponential types. The Nourdin-Peccati strategy analyzes the relation between Stein's method and the Malliavin calculus, and is adapted to dealing with comparisons to the Gaussian law. By studying the behavior of the solution to general Stein equations in detail, we show that …
On any denumerable product of probability spaces, we construct a Malliavin gradient and then a divergence and a number operator. This yields a Dirichlet structure which can be shown to approach the usual structures for Poisson and Brownian processes. We obtain versions of almost all the classical functional inequalities in discrete settings which show that the Efron-Stein inequality can be interpreted as a Poincar{é} inequality or that Hoeffding decomposition of U-statistics can be interpreted as a chaos decomposition. We obtain a version of the Lyapounov central limit theorem for independent …
In this paper we introduce a Hilbert space-valued Malliavin calculus for Poisson random measures. It is solely based on elementary principles from the theory of point processes and basic moment estimates, and thus allows for a simple treatment of the Malliavin operators. The main part of the theory is developed for general Poisson random measures, defined on a $σ$-finite measure space, with minimal conditions. The theory is shown to apply to a space-time setting, suitable for studying stochastic partial differential equations. As an application, we analyze the weak order of convergence of spac…
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 Orti…
A Poisson-Alekseev-Gröbner formula through Malliavin calculus for Poisson random integrals
本文为Poisson随机测度驱动的SDE建立了Alekseev-Gröbner公式，将全局误差表示为系数差，并通过新的Malliavin导数定义处理预期项，适用于随机连续流过程。
Malliavin calculus for marked binomial processes: portfolio optimisation in the trinomial model and compound Poisson approximation
本文构建带标记二项过程的Markov-Malliavin结构与混沌展开体系，将其应用于复合泊松近似的Chen-Stein方法重构及三叉树模型投资组合优化研究。
We investigate solutions of backward stochastic differential equations (BSDE) with time delayed generators driven by Brownian motions and Poisson random measures, that constitute the two components of a Levy process. In this new type of equations, the generator can depend on the past values of a solution, by feeding them back into the dynamics with a time lag. For such time delayed BSDE, we prove existence and uniqueness of solutions provided we restrict on a sufficiently small time horizon or the generator possesses a sufficiently small Lipschitz constant. We study differentiability in the va…
We combine the notion of free Stein kernel and the free Malliavin calculus to provide quantitative bounds under the free (quadratic) Wasserstein distance in the multivariate semicircular approximations for self-adjoint vector-valued multiple Wigner integrals. On the way, we deduce an HSI inequality for a modified non-microstates free entropy with respect to the potential associated with these semicircular families in the case of non-degeneracy of the covariance matrix. The strategy of the proofs is based on functional inequalities involving the free Stein discrepancy. We obtain a bound which d…
We consider a random variable X satisfying almost-sure conditions involving G:=<DX,-DL^{-1}X> where DX is X's Malliavin derivative and L^{-1} is the inverse Ornstein-Uhlenbeck operator. A lower- (resp. upper-) bound condition on G is proved to imply a Gaussian-type lower (resp. upper) bound on the tail P[X>z]. Bounds of other natures are also given. A key ingredient is the use of Stein's lemma, including the explicit form of the solution of Stein's equation relative to the function 1_{x>z}, and its relation to G. Another set of comparable results is established, without the use of Stein's lemm…
We combine Stein's method with a version of Malliavin calculus on the Poisson space. As a result, we obtain explicit Berry-Esséen bounds in Central Limit Theorems (CLTs) involving multiple Wiener-Itô integrals with respect to a general Poisson measure. We provide several applications to CLTs related to Ornstein-Uhlenbeck Lévy processes.
We develop the asymptotic expansion theory for vector-valued sequences (F N) N $\ge$1 of random variables in terms of the convergence of the Stein-Malliavin matrix associated to the sequence F N. Our approach combines the classical Fourier approach and the recent theory on Stein method and Malliavin calculus. We find the second order term of the asymptotic expansion of the density of F N and we illustrate our results by several examples. 2010 AMS Classification Numbers: 62M09, 60F05, 62H12
We prove the chain rule in the more general framework of the Wiener-Poisson space, allowing us to obtain the so-called Nourdin-Peccati bound. From this bound we obtain a second-order Poincare-type inequality that is useful in terms of computations. For completeness we survey these results on the Wiener space, the Poisson space, and the Wiener-Poisson space. We also give several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated fields (where the subordinated field can be processes like fractional Brownian motion or the solution of the Orn…
We obtain Stein approximation bounds for stochastic integrals with respect to a Poisson random measure over ${\Bbb R}^d$, $d\geq 2$. This approach relies on third cumulant Edgeworth-type expansions based on derivation operators defined by the Malliavin calculus for Poisson random measures. The use of third cumulants can exhibit faster convergence rates than the standard Berry-Esseen rate for some sequences of Poisson stochastic integrals.
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…
In this paper, we construct a Malliavin derivative for functionals of square-integrable Lévy processes and derive a Clark-Ocone formula. The Malliavin derivative is defined via chaos expansions involving stochastic integrals with respect to Brownian motion and Poisson random measure. As an illustration, we compute the explicit martingale representation for the maximum of a Lévy process.
We consider functionals which are weighted averages of the avoidance function of a Poisson process. Using the approach to Stein's method based on Malliavin calculus for Poisson functionals we provide explicit bounds for the Wasserstein distance between these standardized functionals and the standard normal distribution. Our approach relies on closed-form expressions for the action of some Malliavin type operators on avoidance functionals of Poisson processes. As a result we provide Berry-Esseen bounds in the CLT for the volume of the union of balls of a fixed radius around random Poisson cente…
Relative Error of Scaled Poisson Approximation via Stein's Method
该论文研究独立泊松变量加权和的缩放泊松近似，特别是尾部分布的相对误差，并用改进的Stein-Chen方法建立了误差界。
In this paper a Malliavin calculus for Lévy processes based on a family of true derivative operators is developed. The starting point is an extension to Lévy processes of the pioneering paper by Carlen and Pardoux [8] for the Poisson process, and our approach includes also the classical Malliavin derivative for Gaussian processes. We obtain a sufficient condition for the absolute continuity of functionals of the Lévy process. As an application, we analyze the absolute continuity of the law of the solution of some stochastic differential equations.
We apply the Dirichlet forms version of Malliavin calculus to stochastic differential equations with jumps. As in the continuous case this weakens significantly the assumptions on the coefficients of the SDE. In spite of the use of the Dirichlet forms theory, this approach brings also an important simplification which was not available nor visible previously : an explicit formula giving the carré du champ matrix, i.e. the Malliavin matrix. Following this formula a new procedure appears, called the lent particle method which shortens the computations both theoretically and in concrete examples.
We establish various bounds on the solutions to a Stein equation for Poisson approximation in Wasserstein distance with non-linear transportation costs. The proofs are a refinement of those in [Barbour and Xia (2006)] using the results in [Liu and Ma (2009)]. As a corollary, we obtain an estimate of Poisson approximation error measured in L^2-Wasserstein distance.
Stein's method has been widely used for probability approximations. However, in the multi-dimensional setting, most of the results are for multivariate normal approximation or for test functions with bounded second- or higher-order derivatives. For a class of multivariate limiting distributions, we use Bismut's formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Stein's exchangeable pair approach, we obtain a general theorem for multivariate approximations with near optimal error bounds on the Wass…
Stein's (1972) method is a very general tool for assessing the quality of approximation of the distribution of a random element by another, often simpler, distribution. In applications of Stein's method, one needs to establish a Stein identity for the approximating distribution, solve the Stein equation and estimate the behaviour of the solutions in terms of the metrics under study. For some Stein equations, solutions with good properties are known; for others, this is not the case. Barbour and Xia (1999) introduced a perturbation method for Poisson approximation, in which Stein identities for…
One major obstacle in applications of Stein's method for compound Poisson approximation is the availability of so-called magic factors (bounds on the solution of the Stein equation) with favourable dependence on the parameters of the approximating compound Poisson random variable. In general, the best such bounds have an exponential dependence on these parameters, though in certain situations better bounds are available. In this paper, we extend the region for which well-behaved magic factors are available for compound Poisson approximation in the Kolmogorov metric, allowing useful compound Po…
In this paper, we apply the Stein's method in the context of point processes, namely when the target measure is the distribution of a finite Poisson point process. We show that the so-called Kantorovich-Rubinstein distance between such a measure and another finite point process is bounded by the $L^1$-distance between their respective Papangelou intensities. Then, we deduce some convergence rates for sequences of point processes approaching a Poisson or a Cox point process.
In the first part of the paper we use a new Fourier technique to obtain a Stein characterizations for random variables in the second Wiener chaos. We provide the connection between this result and similar conclusions that can be derived using Malliavin calculus. We also introduce a new form of discrepancy which we use, in the second part of the paper, to provide bounds on the 2-Wasserstein distance between linear combinations of independent centered random variables. Our method of proof is entirely original. In particular it does not rely on estimation of bounds on solutions of the so-called S…
The main purpose of the paper is to investigate the possibility of applying Chen-Stein approach to estimate the $χ^2$ distance between Poisson distribution and a sum of independent indicators. Earlier results concerning $χ^2$ distance between above mentioned distributions either used analytical approach heavily based on the analysis of the generating functions or on rather lengthy and complicated elementary calculations. Applying Chen-Stein approach we succeed in providing a very quick proof of upper bounds for $χ^2$ distance that are of comparable strength to the earlier estimates obtained by…
Stable limit theorems on the Poisson space
该研究利用泊松空间上的Malliavin演算证明泊松点过程泛函的稳定极限定理，给出条件高斯极限的定量界，推广了多项已有经典与最新的极限定理成果，并给出随机过程方向的应用。
This exposition explains the basic ideas of Stein's method for Poisson random variable approximation and Poisson process approximation from the point of view of the immigration-death process and Palm theory. The latter approach also enables us to define local dependence of point processes [Chen and Xia (2004)] and use it to study Poisson process approximation for locally dependent point processes and for dependent superposition of point processes.
We establish a general inequality on the Poisson space, yielding an upper bound for the distance in total variation between the law of a regular random variable with values in the integers and a Poisson distribution. Several applications are provided, in particular: (i) to deduce a set of sufficient conditions implying that a sequence of (suitably shifted) multiple Wiener-Itô integrals converges in distribution to a Poisson random variable, and (ii) to compute explicit rates of convergence for the Poisson approximation of statistics associated with geometric random graphs with sparse connectio…
This survey is a preliminary version of a chapter of the forthcoming book "Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Itô Chaos Expansions and Stochastic Geometry" edited by Giovanni Peccati and Matthias Reitzner. The paper develops some basic theory for the stochastic analysis of Poisson process on a general $σ$-finite measure space. After giving some fundamental definitions and properties (as the multivariate Mecke equation) the paper presents the Fock space representation of square-integrable functions of a Poisson process in terms of iterated difference ope…
This paper uses the generator comparison approach of Stein's method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. The "standard" generator comparison approach starts with the Poisson equation for the diffusion, and the main technical difficulty is to obtain bounds on the derivatives of the solution to the Poisson equation, also known as Stein factor bounds. In this paper we propose starting with the Poisson equation of the Markov chain; we term this the prelimit approach. Although one still needs Stein factor bounds, they now correspond to fini…
Density convergence of spatial average of solution to a one dimensional stochastic wave equation
该研究针对由特定高斯乘性噪声驱动的一维随机波动方程解的空间平均，结合Malliavin微积分与Stein方法，建立了其密度与标准正态密度间均匀距离的收敛速率，解决了相关技术挑战。
In this paper we prove an estimate for the total variation distance, in the framework of the Breuer-Major theorem, using the Malliavin-Stein method, assuming the underlying function $g$ to be once weakly differentiable with $g$ and $g'$ having finite moments of order four with respect to the standard Gaussian density. This result is proved by a combination of Gebelein's inequality and some novel estimates involving Malliavin operators.
In this article, we provide an extension of the Chen-Stein inequality for Poisson approximation in the total variation distance for sums of independent Bernoulli random variables in two ways. We prove that we can improve the rate of convergence (hence the quality of the approximation) by using explicitly constructed signed or positive probability measures, and that we can extend the setting to possibly dependent random variables. The framework which allows this is that of mod-Poisson convergence, and more precisely those mod-Poisson convergent sequences whose residue functions can be expressed…
We apply the Stein-Chen method to problems from extreme value theory. On the one hand, the Stein-Chen method for Poisson approximation allows us to obtain bounds on the Kolmogorov distance between the law of the maximum of i.i.d. random variables, following certain well known distributions, and an extreme value distribution. On the other hand, we introduce marked point processes of exceedances (MPPE's) whose i.i.d. marks can be either univariate or multivariate. We use the Stein-Chen method for Poisson process approximation to determine bounds on the error of the approximation, in some appropr…
Multidimensional Stein method and quantitative asymptotic independence
本文发展多维Stein-Malliavin演算，度量随机向量与高斯独立向量的Wasserstein距离，并在Wiener混沌中证明渐近联合收敛，给出定量独立性结果。
The framework of Stein's method for Poisson process approximation is presented from the point of view of Palm theory, which is used to construct Stein identities and define local dependence. A general result (Theorem \refimportantproposition) in Poisson process approximation is proved by taking the local approach. It is obtained without reference to any particular metric, thereby allowing wider applicability. A Wasserstein pseudometric is introduced for measuring the accuracy of point process approximation. The pseudometric provides a generalization of many metrics used so far, including the t…
We combine Malliavin calculus with Stein's method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. We also prove results concerning random variables admitting a possibly infinite Wiener chaotic decomposition. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener-Itô integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry-Esséen bounds in…
Based on Stein's method, we derive upper bounds for Poisson process approximation in the $L_1$-Wasserstein metric $d_2^{(p)}$, which is based on a slightly adapted $L_p$-Wasserstein metric between point measures. For the case $p=1$, this construction yields the metric $d_2$ introduced in [Barbour and Brown Stochastic Process. Appl. 43 (1992) 9--31], for which Poisson process approximation is well studied in the literature. We demonstrate the usefulness of the extension to general $p$ by showing that $d_2^{(p)}$-bounds control differences between expectations of certain $p$th order average stat…
In this paper we use a Malliavin-Stein type method to investigate Poisson and normal approximations for the measurable functions of infinitely many independent random variables. We combine Stein's method with the difference operators in theory of concentration inequalities to obtain explicit bounds on Wasserstein, Kolmogorov and total variation distances. When restricted to the functions of a finite number of independent random variables, our method provides new bounds in the normal approximation. Meanwhile, our bounds in Poisson approximation are first to obtain explicitly.
We study multi-dimensional normal approximations on the Poisson space by means of Malliavin calculus, Stein's method and probabilistic interpolations. Our results yield new multi-dimensional central limit theorems for multiple integrals with respect to Poisson measures -- thus significantly extending previous works by Peccati, Solé, Taqqu and Utzet. Several explicit examples (including in particular vectors of linear and non-linear functionals of Ornstein-Uhlenbeck Lévy processes) are discussed in detail.
We compute explicit bounds in the Gaussian approximation of functionals of infinite Rademacher sequences. Our tools involve Stein's method, as well as the use of appropriate discrete Malliavin operators. Although our approach does not require the classical use of exchangeable pairs, we employ a chaos expansion in order to construct an explicit exchangeable pair vector for any random variable which depends on a finite set of Rademacher variables. Among several examples, which include random variables which depend on infinitely many Rademacher variables, we provide three main applications: (i) t…
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