Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any nonnegative random vector. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given degrees in certain random graphs is asymptotically multivariate normal and obtain a bound on the rate of convergence. Both examples demonstrate that this approach may be suitable for situations involving non-local dependence. We also present Theorem 1.4 for sums of vectors having a local type of dependence. We apply this theorem to obtain a multivariate normal approximation for the distribution of the random $p$-vector which counts the number of edges in a fixed graph both of whose vertices have the same given color when each vertex is colored by one of $p$ colors independently. All normal approximation results presented here do not require an ordering of the summands related to the dependence structure. This is in contrast to hypotheses of classical central limit theorems and examples, which involve e.g., martingale, Markov chain, or various mixing assumptions.
In this paper we present a general framework for Stein's method for multivariate continuous distributions. The approach gives a collection of Stein characterisations, among which we highlight score-Stein operators and kernel Stein operators. Applications include copulas and distance between posterior distributions. We give a general construction for Stein kernels for elliptical distributions and discuss Stein kernels in generality, highlighting connections with Fisher information and mass transport. Finally, a goodness-of-fit test based on Stein discrepancies is given.
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…
We establish uniform bounds on the low-order derivatives of Stein equation solutions for a broad class of multivariate, strongly log-concave target distributions. These "Stein factor" bounds deliver control over Wasserstein and related smooth function distances and are well-suited to analyzing the computable Stein discrepancy measures of Gorham and Mackey. Our arguments of proof are probabilistic and feature the synchronous coupling of multiple overdamped Langevin diffusions.
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.
By a delicate analysis for the Stein's equation associated to the $α$-stable law approximation with $α\in (0,2)$, we prove a quantitative stable central limit theorem in Wasserstein type distance, which generalizes the results in the series of work \cite{Xu19, CNX19+,CNXYZ19+} from the univariate case to the multiple variate case. From an explicit computation for Pareto's distribution, we see that the rate of our approximation is sharp. The analysis of the Stein's equation is new and has independent interest.
In this paper we establish a multivariate exchangeable pairs approach within the framework of Stein's method to assess distributional distances to potentially singular multivariate normal distributions. By extending the statistics into a higher-dimensional space, we also propose an embedding method which allows for a normal approximation even when the corresponding statistics of interest do not lend themselves easily to Stein's exchangeable pairs approach. To illustrate the method, we provide the examples of runs on the line as well as double-indexed permutation statistics.
The concentration inequality approach for normal approximation by Stein's method is generalized to the multivariate setting. We use this approach to prove a non-smooth function distance for multivariate normal approximation for standardized sums of $k$-dimensional independent random vectors $W=\sum_{i=1}^n X_i$ with an error bound of order $k^{1/2}γ$ where $γ=\sum_{i=1}^n E|X_i|^3$. For sums of locally dependent (unbounded) random vectors, we obtain a fourth moment bound which is typically of order $O_k(1/\sqrt{n})$, as well as a third moment bound which is typically of order $O_k(\log n/\sqrt…
We develop a multidimensional Stein methodology for non-degenerate self-decomposable random vectors in $\mathbb{R}^d$ having finite first moment. Building on previous univariate findings, we solve an integro-partial differential Stein equation by a mixture of semigroup and Fourier analytic methods. Then, under a second moment assumption, we introduce a notion of Stein kernel and an associated Stein discrepancy specifically designed for infinitely divisible distributions. Combining these new tools, we obtain quantitative bounds on smooth-Wasserstein distances between a probability measure in $\…
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…
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…
We use Stein's method to bound the Wasserstein distance of order $2$ between a measure $ν$ and the Gaussian measure using a stochastic process $(X_t)_{t \geq 0}$ such that $X_t$ is drawn from $ν$ for any $t > 0$. If the stochastic process $(X_t)_{t \geq 0}$ satisfies an additional exchangeability assumption, we show it can also be used to obtain bounds on Wasserstein distances of any order $p \geq 1$. Using our results, we provide optimal convergence rates for the multi-dimensional Central Limit Theorem in terms of Wasserstein distances of any order $p \geq 2$ under simple moment assumptions.
We use Stein's method to obtain a bound on the distance between scaled $p$-dimensional random walks and a $p$-dimensional (correlated) Brownian Motion. We consider dependence schemes including those in which the summands in scaled sums are weakly dependent and their $p$ components are strongly correlated. As an example application, we prove a functional limit theorem for exceedances in an $m$-scans process, together with a bound on the rate of convergence. We also find a bound on the rate of convergence of scaled U-statistics to Brownian Motion, representing an example of a sum of strongly dep…
In a recent paper, Gaunt 2020 extended Stein's method to limit distributions that can be represented as a function $g:\mathbb{R}^d\rightarrow\mathbb{R}$ of a centered multivariate normal random vector $Σ^{1/2}\mathbf{Z}$ with $\mathbf{Z}$ a standard $d$-dimensional multivariate normal random vector and $Σ$ a non-negative definite covariance matrix. In this paper, we obtain improved bounds, in the sense of weaker moment conditions, smaller constants and simpler forms, for the case that $g$ has derivatives with polynomial growth. We obtain new non-uniform bounds for the derivatives of the soluti…
In this paper we develop a framework for multivariate functional approximation by a suitable Gaussian process via an exchangeable pairs coupling that satisfies a suitable approximate linear regression property, thereby building on work by Barbour (1990) and Kasprzak (2020). We demonstrate the applicability of our results by applying it to joint subgraph counts in an Erdős-Renyi random graph model on the one hand and to vectors of weighted, degenerate $U$-processes on the other hand. As a concrete instance of the latter class of examples, we provide a bound for the functional approximation of a…
The purpose of this paper is to synthesize the approaches taken by Chatterjee-Meckes and Reinert-Röllin in adapting Stein's method of exchangeable pairs for multivariate normal approximation. The more general linear regression condition of Reinert-Röllin allows for wider applicability of the method, while the method of bounding the solution of the Stein equation due to Chatterjee-Meckes allows for improved convergence rates. Two abstract normal approximation theorems are proved, one for use when the underlying symmetries of the random variables are discrete, and one for use in contexts in whic…
We generalize the well-known zero bias distribution and the $λ$-Stein pair to an approximate zero bias distribution and an approximate $λ,R$-Stein pair, respectively. Berry Esseen type bounds to the normal, based on approximate zero bias couplings and approximate $λ,R$-Stein pairs, are obtained using Stein's method. The bounds are then applied to combinatorial central limit theorems where the random permutation has the Ewens $\mathcal{E}_θ$ distribution with $θ>0$ which can be specialized to the uniform distribution by letting $θ=1$. The family of the Ewens distributions appears in the context…
In this paper we extend Stein's method to the distribution of the product of $n$ independent mean zero normal random variables. A Stein equation is obtained for this class of distributions, which reduces to the classical normal Stein equation in the case $n=1$. This Stein equation motivates a generalisation of the zero bias transformation. We establish properties of this new transformation, and illustrate how they may be used together with the Stein equation to assess distributional distances for statistics that are asymptotically distributed as the product of independent central normal random…
Since the introduction of Stein's method in the early 1970s, much research has been done in extending and strengthening it; however, there does not exist a version of Stein's original method of exchangeable pairs for multivariate normal approximation. The aim of this article is to fill this void. We present three abstract normal approximation theorems using exchangeable pairs in multivariate contexts, one for situations in which the underlying symmetries are discrete, and real and complex versions of a theorem for situations involving continuous symmetry groups. Our main applications are proof…
Let $M_n$ be a random element of the unitary, special orthogonal, or unitary symplectic groups, distributed according to Haar measure. By a classical result of Diaconis and Shahshahani, for large matrix size $n$, the vector $ (\on{Tr}(M_n), \on{Tr}(M_n^2),..., \on{Tr}(M_n^d))$ tends to a vector of independent (real or complex) Gaussian random variables. Recently, Jason Fulman has demonstrated that for a single power $j$ (which may grow with $n$), a speed of convergence result may be obtained via Stein's method of exchangeable pairs. In this note, we extend Fulman's result to the multivariate c…
We use a multivariate version of Stein's method to establish a quantitative Lindeberg CLT for the Fourier transforms of random $N$-vectors. We achieve this by deducing a specific integral representation for the Hessian matrix of a solution to the Stein equation with test function $e_t(x) = \exp(- i \sum_{k=1}^N t_k x_k)$, where $t,x \in \mathbb{R}^N$.
This work explores and develops elements of Stein's method of approximation, in the infinitely divisible setting, and its connections to functional analysis. It is mainly concerned with multivariate self-decomposable laws without finite first moment and, in particular, with $α$-stable ones, $α\in (0,1]$. At first, several characterizations of these laws via covariance identities are presented. In turn, these characterizations lead to integro-differential equations which are solved with the help of both semigroup and Fourier methodologies. Then, Poincaré-type inequalities for self-decomposable …
By the continuous mapping theorem, if a sequence of $d$-dimensional random vectors $(\mathbf{W}_n)_{n\geq1}$ converges in distribution to a multivariate normal random variable $Σ^{1/2}\mathbf{Z}$, then the sequence of random variables $(g(\mathbf{W}_n))_{n\geq1}$ converges in distribution to $g(Σ^{1/2}\mathbf{Z})$ if $g:\mathbb{R}^d\rightarrow\mathbb{R}$ is continuous. In this paper, we develop Stein's method for the problem of deriving explicit bounds on the distance between $g(\mathbf{W}_n)$ and $g(Σ^{1/2}\mathbf{Z})$ with respect to smooth probability metrics. We obtain several bounds for t…
Consider the multivariate Stein equation $Δf - x\cdot \nabla f = h(x) - E h(Z)$, where $Z$ is a standard $d$-dimensional Gaussian random vector, and let $f\_h$ be the solution given by Barbour's generator approach. We prove that, when $h$ is $α$-Hölder ($0<α\leq1$), all derivatives of order $2$ of $f\_h$ are $α$-Hölder {\it up to a $\log$ factor}; in particular they are $β$-Hölder for all $β\in (0, α)$, hereby improving existing regularity results on the solution of the multivariate Gaussian Stein equation. For $α=1$, the regularity we obtain is optimal, as shown by an example given by Raič \c…
We investigate the multivariate central limit theorem for nonlinear statistics by means of Stein's method and Slepian's smart path interpolation method. Based on certain difference operators in theory of concentration inequalities, we obtain two explicit bounds for the rate of convergence. Applications to Rademacher functionals, the runs and quadratic forms are provided as well.
For integer valued random variables, the translated Poisson distributions form a flexible family for approximation in total variation, in much the same way that the normal family is used for approximation in Kolmogorov distance. Using the Stein--Chen method, approximation can often be achieved with error bounds of the same order as those for the CLT. In this paper, an analogous theory, again based on Stein's method, is developed in the multivariate context. The approximating family consists of the equilibrium distributions of a collection of Markov jump processes, whose analogues in one dimens…
This paper concerns portfolio selection with multiple assets under rough covariance matrix. We investigate the continuous-time Markowitz mean-variance problem for a multivariate class of affine and quadratic Volterra models. In this incomplete non-Markovian and non-semimartingale market framework with unbounded random coefficients, the optimal portfolio strategy is expressed by means of a Riccati backward stochastic differential equation (BSDE). In the case of affine Volterra models, we derive explicit solutions to this BSDE in terms of multi-dimensional Riccati-Volterra equations. This framew…
We provide a Lyapunov type bound in the multivariate central limit theorem for sums of independent, but not necessarily identically distributed random vectors. The error in the normal approximation is estimated for certain classes of sets, which include the class of measurable convex sets. The error bound is stated with explicit constants. The result is proved by means of Stein's method. In addition, we improve the constant in the bound of the Gaussian perimeter of convex sets.
We provide a new general theorem for multivariate normal approximation on convex sets. The theorem is formulated in terms of a multivariate extension of Stein couplings. We apply the results to a homogeneity test in dense random graphs and to prove multivariate asymptotic normality for certain doubly indexed permutation statistics.
We provide an explanation of the main ideas underlying Götze's main result in using Stein's method. We also provide detailed derivations of various intermediate estimates. Curiously, we are led to a different dimensional dependence of the constant than that given Götze's paper. We would like to dedicate this to Charles Stein on the occasion of his 90th birthday.
Berry-Esseen bounds for non-linear functionals of infinite Rademacher sequences are derived by means of the Malliavin-Stein method. Moreover, multivariate extensions for vectors of Rademacher functionals are shown. The results establish a connection to small-ball probabilities and shed new light onto the relation between central limit theorems on the Rademacher chaos and norms of contraction operators. Applications concern infinite weighted 2-runs, a combinatorial central limit theorem and traces of Bernoulli random matrices.
In a recent paper by the authors, a new approach--called the "embedding method"--was introduced, which allows to make use of exchangeable pairs for normal and multivariate normal approximation with Stein's method in cases where the corresponding couplings do not satisfy a certain linearity condition. The key idea is to embed the problem into a higher dimensional space in such a way that the linearity condition is then satisfied. Here we apply the embedding to U-statistics as well as to subgraph counts in random graphs.
We provide an abstract multivariate central limit theorem with the Lindeberg-type error bounded in terms of Lipschitz functions (Wasserstein 1-distance) or functions with bounded second or third derivatives. The result is proved by means of Stein's method. For sums of i.i.d. random vectors with finite third absolute moment, the optimal rate of convergence is established (that is, we eliminate the logarithmic factor in the case of Lipschitz test functions). We indicate how the result could be applied to certain other dependence structures, but do not derive bounds explicitly.
Let $W$ be a random variable with mean zero and variance $σ^2$. The distribution of a variate $W^*$, satisfying $EWf(W)=σ^2 Ef'(W^*)$ for smooth functions $f$, exists uniquely and defines the zero bias transformation on the distribution of $W$. The zero bias transformation shares many interesting properties with the well known size bias transformation for non-negative variables, but is applied to variables taking on both positive and negative values. The transformation can also be defined on more general random objects. The relation between the transformation and the expression $wf'(w)-σ^2 f''…
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…
Let $\mathbf{W}=(W_1,W_2,...,W_k)$ be a random vector with nonnegative coordinates having nonzero and finite variances. We prove concentration inequalities for $\mathbf{W}$ using size biased couplings that generalize the previous univariate results. Two applications on local dependence and counting patterns are provided.
We present, in a unified way, a Stein methodology for infinitely divisible laws (without Gaussian component) having finite first moment. Based on a correlation representation, we obtain a characterizing non-local Stein operator which boils down to classical Stein operators in specific examples. Thanks to this characterizing operator, we introduce various extensions of size bias and zero bias distributions and prove that these notions are closely linked to infinite divisibility. Combined with standard Fourier techniques, these extensions also allow obtaining explicit rates of convergence for co…
New bounds for the $k$-th order derivatives of the solutions of the normal and multivariate normal Stein equations are obtained. Our general order bounds involve fewer derivatives of the test function than those in the existing literature. We apply these bounds and local approach couplings to obtain an order $n^{-(p-1)/2}$ bound, for smooth test functions, for the distance between the distribution of a standardised sum of independent and identically distributed random variables and the standard normal distribution when the first $p$ moments of these distributions agree. We also obtain a bound …
In the seventies, Charles Stein revolutionized the way of proving the Central Limit Theorem by introducing a method that utilizes a characterization equation for Gaussian distribution. In the last 50 years, much research has been done to adapt and strengthen this method to a variety of different settings and other limiting distributions. However, it has not been yet extended to study conditional convergences. In this article, we develop a novel approach using Stein's method for exchangeable pairs to find a rate of convergence in Conditional Central Limit Theorem of the form $(X_n\mid Y_n=k)$, …
This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Itô integrals with respect to the compensated Poisson process. Second, a multivariate central limit theorem is shown for a vector whose components admit a finite chaos expansion of the type of a Poisson U-statistic. The approach is based on recent results of Peccati et al.\ combining Malliavin calculus and Stein's method, and also yields Berry-Esseen type bounds. As applications, moment formulae and central limit theorems…
Using Stein's method techniques, we develop a framework which allows one to bound the error terms arising from approximation by the Laplace distribution and apply it to the study of random sums of mean zero random variables. As a corollary, we deduce a Berry-Esseen type theorem for the convergence of certain geometric sums. Our results make use of a second order characterizing equation and a distributional transformation which is related to zero-biasing.
We extend Stein's celebrated Wasserstein bound for normal approximation via exchangeable pairs to the multi-dimensional setting. As an intermediate step, we exploit the symmetry of exchangeable pairs to obtain an error bound for smooth test functions. We also obtain a continuous version of the multi-dimensional Wasserstein bound in terms of fourth moments. We apply the main results to multivariate normal approximations to Wishart matrices of size $n$ and degree $d$, where we obtain the optimal convergence rate $\sqrt{n^3/d}$ under only moment assumptions, and to quadratic forms and Poisson fun…
This article derives quantitative limit theorems for multivariate Poisson and Poisson process approximations. Employing the solution of Stein's equation for Poisson random variables, we obtain an explicit bound for the multivariate Poisson approximation of random vectors in the Wasserstein distance. The bound is then utilized in the context of point processes to provide a Poisson process approximation result in terms of a new metric called $d_π$, stronger than the total variation distance, defined as the supremum over all Wasserstein distances between random vectors obtained by evaluating the …
Stein's method is applied to obtain a general Cramer-type moderate deviation result for dependent random variables whose dependence is defined in terms of a Stein identity. A corollary for zero-bias coupling is deduced. The result is also applied to a combinatorial central limit theorem, a general system of binary codes, the anti-voter model on a complete graph, and the Curie-Weiss model. A general moderate deviation result for independent random variables is also proved.
In this paper we extend the refined second-order Poincaré inequality for Poisson functionals from a one-dimensional to a multi-dimensional setting. Its proof is based on a multivariate version of the Malliavin-Stein method for normal approximation on Poisson spaces. We also present an application to partial sums of vector-valued functionals of heavy-tailed moving averages. The extension allows a functional with multivariate arguments, i.e. multiple moving averages and also multivariate values of the functional. Such a set-up has previously not been explored in the framework of stable moving av…
We derive explicit central moment inequalities for random variables that admit a Stein coupling, such as exchangeable pairs, size--bias couplings or local dependence, among others. The bounds are in terms of moments (not necessarily central) of variables in the Stein coupling, which are typically local in some sense, and therefore easier to bound. In cases where the Stein couplings have the kind of behaviour leading to good normal approximation, the central moments are closely bounded by those of a normal. We show how the bounds can be used to produce concentration inequalities, and compare th…
In the setting of intermittent Pomeau-Manneville maps with time dependent parameters, we show a functional correlation bound widely useful for the analysis of the statistical properties of the model. We give two applications of this result, by showing that in a suitable range of parameters the bound implies the conditions of the normal approximation methods of Stein and Rio. For a single Pomeau-Manneville map belonging to this parameter range, both methods then yield a multivariate central limit theorem with a rate of convergence.
Given a vector $F=(F_1,\dots,F_m)$ of Poisson functionals $F_1,\dots,F_m$, we investigate the proximity between $F$ and an $m$-dimensional centered Gaussian random vector $N_Σ$ with covariance matrix $Σ\in\mathbb{R}^{m\times m}$. Apart from finding proximity bounds for the $d_2$- and $d_3$-distances, based on classes of smooth test functions, we obtain proximity bounds for the $d_{convex}$-distance, based on the less tractable test functions comprised of indicators of convex sets. The bounds for all three distances are shown to be of the same order, which is presumably optimal. The bounds are …
We present an adaptation of Stein's method of normal approximation to the study of both discrete- and continuous-time dynamical systems. We obtain new correlation-decay conditions on dynamical systems for a multivariate central limit theorem augmented by a rate of convergence. We then present a scheme for checking these conditions in actual examples. The principal contribution of our paper is the method, which yields a convergence rate essentially with the same amount of work as the central limit theorem, together with a multiplicative constant that can be computed directly from the assumption…
Let $(\mathbf{W,W'})$ be an exchangeable pair of vectors in $\mathbb{R}^k$. Suppose this pair satisfies \beas E(\mathbf{W}'|\mathbf{W})=(I_k-Λ)\mathbf{W}+\mathbf{R(W)}. \enas If $||\mathbf{W-W'}||_2\le K$ and $\mathbf{R(W)}=0$, then concentration of measure results of following form is proved for all $\mathbf{w}\succeq 0$ when the moment generating function of $\mathbf{W}$ is finite. \beas P(\mathbf{W}\succeq\mathbf{w}),P(\mathbf{W}\preceq -\mathbf{w})\le \exp(-\frac{||\mathbf{w}||_2^2}{2K^2ν_1}), \enas for an explicit constant $ν_1$, where $\succeq$ stands for coordinate wise $\ge$ ordering. …
Using Stein's method, we prove an abstract result that yields multivariate central limit theorems with a rate of convergence for time-dependent dynamical systems. As examples we study a model of expanding circle maps and a quasistatic model. In both models we prove multivariate central limit theorems with a rate of convergence.
To improve the efficiency of Monte Carlo estimation, practitioners are turning to biased Markov chain Monte Carlo procedures that trade off asymptotic exactness for computational speed. The reasoning is sound: a reduction in variance due to more rapid sampling can outweigh the bias introduced. However, the inexactness creates new challenges for sampler and parameter selection, since standard measures of sample quality like effective sample size do not account for asymptotic bias. To address these challenges, we introduce a new computable quality measure based on Stein's method that quantifies …
In this paper the coupon collector's problem with group drawings is studied. Assume there are $ n $ different coupons. At each time precisely $ s $ of the $ n $ coupons are drawn, where all choices are supposed to have equal probability. The focus lies on the fluctuations, as $n\to\infty$, of the number $Z_{n,s}(k_n)$ of coupons that have not been drawn in the first $k_n$ drawings. Using a size-biased coupling construction together with Stein's method for normal approximation, a quantitative central limit theorem for $Z_{n,s}(k_n)$ is shown for the case that $k_n={n\over s}(α\log(n)+x)$, where…
