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We take another look at using Stein's method to establish uniform Berry–Esseen bounds for Studentized nonlinear statistics, highlighting variable censoring and an exponential randomized concentration inequality for a sum of censored variables as the essential tools to carry out the arguments involved. As an important application, we prove a uniform Berry–Esseen bound for Studentized U-statistics in a form that exhibits the dependence on the degree of the kernel. [ABSTRACT FROM AUTHOR]

This research explored the number of returns to the origin within the framework of a symmetric simple random walk. Our primary objective was to approximate the distribution of return events to the origin by utilizing the half-normal distribution, which is chosen for its appropriateness as a limit distribution for nonnegative values. Employing the Stein's method in conjunction with concentration inequalities, we derived an exponential non-uniform bound for the approximation error. This bound signifies a significant advancement in contrast to existing bounds, encompassing both the uniform bounds proposed by Döbler [1] and polynomial non-uniform bounds presented by Sama-ae, Chaidee, and Neammanee [2], and Siripraparat and Neammanee [3].

We analyze the p-random Wishart tensor ( p ≥ 2 ) associated with an initial n × d random matrix X n , d whose entries are independent and belong to the Wiener chaos. The order of the Wiener chaos is constant in each row of the matrix, and it may change from one row to another. Therefore, the entries of our starting matrix are not identically distributed (we only assume that their second moments coincide). We prove that, in high-dimensional regime (i.e. when n and d are large enough), the multidimensional random vector corresponding to the p-Wishart tensor is close in distribution to a standard multidimensional Gaussian vector. We also evaluate the Wasserstein distance between the probability distributions of these two random vectors. By using the techniques of the Stein–Malliavin calculus, we show that when the dimensions n and d are large enough and p is fixed, this Wasserstein distance is of order less than n 2 p - 1 d . [ABSTRACT FROM AUTHOR]

We consider the normal approximation of Kabanov–Skorohod integrals on a general Poisson space. Our bounds are for the Wasserstein and the Kolmogorov distance and involve only difference operators of the integrand of the Kabanov–Skorohod integral. The proofs rely on the Malliavin–Stein method and, in particular, on multiple applications of integration by parts formulae. As examples, we study some linear statistics of point processes that can be constructed by Poisson embeddings and functionals related to Pareto optimal points of a Poisson process. [ABSTRACT FROM AUTHOR]

Abstract In a recent paper the authors proved a nonuniform local limit theorem concerning normal approximation of the point probabilities P ( S = k ) $P(S=k)$ when S = ∑ i = 1 n X i $S=\sum_{i=1}^{n}X_{i}$ and X 1 , X 2 , … , X n $X_{1},X_{2},\ldots ,X_{n}$ are independent Bernoulli random variables that may have different success probabilities. However, their main result contained an undetermined constant, somewhat limiting its applicability. In this paper we give a nonuniform bound in the same setting but with explicit constants. Our proof uses Stein’s method and, in particular, the K-function and concentration inequality approaches. We also prove a new uniform local limit theorem for Poisson binomial random variables that is used to help simplify the proof in the nonuniform case.

In a recent paper the authors proved a nonuniform local limit theorem concerning normal approximation of the point probabilities P (S = k) when S = ∑ i = 1 n X i and X 1 , X 2 , ... , X n are independent Bernoulli random variables that may have different success probabilities. However, their main result contained an undetermined constant, somewhat limiting its applicability. In this paper we give a nonuniform bound in the same setting but with explicit constants. Our proof uses Stein's method and, in particular, the K-function and concentration inequality approaches. We also prove a new uniform local limit theorem for Poisson binomial random variables that is used to help simplify the proof in the nonuniform case. [ABSTRACT FROM AUTHOR]

The autocovariance and cross‐covariance functions naturally appear in many time series procedures (e.g. autoregression or prediction). Under assumptions, empirical versions of the autocovariance and cross‐covariance are asymptotically normal with covariance structure depending on the second‐ and fourth‐order spectra. Under non‐restrictive assumptions, we derive a bound for the Wasserstein distance of the finite‐sample distribution of the estimator of the autocovariance and cross‐covariance to the Gaussian limit. An error of approximation to the second‐order moments of the estimator and an m‐dependent approximation are the key ingredients to obtain the bound. As a worked example, we discuss how to compute the bound for causal autoregressive processes of order 1 with different distributions for the innovations. To assess our result, we compare our bound to Wasserstein distances obtained via simulation. [ABSTRACT FROM AUTHOR]

The problem of testing composite hypotheses with respect to the equal parameters of a normal distribution using the constrained Bayesian method is discussed. Hypotheses are tested using the maximum likelihood and Stein's methods. The optimality of our decision rule is shown by the following criteria: the mixed directional false discovery rate, the false discovery rate, and the Type I and Type II errors, under the conditions of providing a desired level of constraint. The algorithms for implementing the proposed methods and the computational tools for their application are included. Simulation results show validity of the theoretical results along with their superiority over the classical Bayesian method. [ABSTRACT FROM AUTHOR]

The exponential random graph model (ERGM) is a central object in the study of clustering properties in social networks as well as canonical ensembles in statistical physics. Despite some breakthrough works in the mathematical understanding of ERGM, most notably by Bhamidi, Bresler, and Sly [Ann. Appl. Probab. 21 (2011), pp. 2146–2170] through the analysis of a natural Heat-bath Glauber dynamics, and by Chatterjee and Diaconis [Ann. Statist. 41 (2013), pp. 2428–2461; Eldan [Geom. Funct. Anal. 28 (2018), pp. 1548–1596; and Eldan and Gross [Ann. Appl. Probab. 28 (2018), pp. 3698–3735] via a large deviation theoretic perspective, several basic questions have remained unanswered owing to the lack of exact solvability unlike the much studied Curie-Weiss model (Ising model on the complete graph). In this paper, we establish a series of new concentration of measure results for the ERGM throughout the entire sub-critical phase , including a Poincaré inequality, Gaussian concentration for Lipschitz functions, and a central limit theorem. In addition, a new proof of a quantitative bound on the W_1-Wasserstein distance to Erdős-Rényi graphs, previously obtained by Reinert and Ross [Ann. Appl. Probab. 29 (2019), pp. 3201–3229], is also presented. The arguments rely on translating temporal mixing properties of Glauber dynamics to static spatial mixing properties of the equilibrium measure and have the potential of being useful in proving similar functional inequalities for other Gibbsian systems beyond the perturbative regime. [ABSTRACT FROM AUTHOR]

In this paper, we present a rate of convergence in the uniform norm for the densities of spatial averages of the solution to the d-dimensional parabolic Anderson model driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. The proof is based on the combination of Malliavin calculus techniques and the Stein's method for normal approximations. [ABSTRACT FROM AUTHOR]

Much like higher-order Taylor expansions allow one to approximate functions to a higher degree of accuracy, we demonstrate that, by accounting for higher-order terms in the Taylor expansion of a Markov process generator, one can derive novel diffusion approximations that achieve a higher degree of accuracy compared with the classical ones used in the literature over the last 50 years. We derive and analyze new diffusion approximations of stationary distributions of Markov chains that are based on second- and higher-order terms in the expansion of the Markov chain generator. Our approximations achieve a higher degree of accuracy compared with diffusion approximations widely used for the last 50 years while retaining a similar computational complexity. To support our approximations, we present a combination of theoretical and numerical results across three different models. Our approximations are derived recursively through Stein/Poisson equations, and the theoretical results are proved using Stein's method. Funding: X. Fang is partially supported by Hong Kong RGC [Grants 24301617, 14302418, and 14304917], a CUHK direct grant, and a CUHK start-up grant. J. G. Dai is partially supported by NSF [Grant CMMI-1537795]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2022.2362. [ABSTRACT FROM AUTHOR]

In a recent paper, Gaunt (Ann I H Poincare Probab Stat 56:1484–1513, 2020) extended Stein's method to limit distributions that can be represented as a function g : R d → R of a centred multivariate normal random vector Σ 1 / 2 Z with 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 solution of the Stein equation and use these inequalities to obtain general bounds on the distance, measured using smooth test functions, between the distributions of g (W n) and g (Z) , where W n is a standardised sum of random vectors with independent components and Z is a standard d-dimensional multivariate normal random vector. We apply these general bounds to obtain bounds for the Chi-square approximation of the family of power divergence statistics (special cases include the Pearson and likelihood ratio statistics), for the case of two cell classifications, that improve on existing results in the literature. [ABSTRACT FROM AUTHOR]

By a delicate analysis for the Stein's equation associated with the α -stable law approximation with α ∈ (0 , 2) , we prove a quantitative stable central limit theorem in Wasserstein-type distance, which generalizes the results in the series of work (Chen et al. in J Theor Probab 34(3):1382–1407, 2021; Chen et al. in J Theor Probab 35(2):1137–1186 2022; Xu in Ann Appl Probab 29(1):458–504, 2019) 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. [ABSTRACT FROM AUTHOR]

We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals with respect to Hawkes processes by a normally distributed random variable. In the case of deterministic and nonnegative integrands, our estimates involve only the third moment of the integrand in addition to a variance term using a squared norm of the integrand. As a consequence, we are able to observe a "third moment phenomenon" in which the vanishing of the first cumulant can lead to faster convergence rates. Our results are also applied to compound Hawkes processes, and improve on the current literature where estimates may not converge to zero in large time or have been obtained only for specific kernels such as the exponential or Erlang kernels. [ABSTRACT FROM AUTHOR]

We use Stein's method to establish the rates of normal approximation in terms of the total variation distance for a large class of sums of score functions of samples arising from random events driven by a marked Poisson point process on $\mathbb{R}^d$. As in the study under the weaker Kolmogorov distance, the score functions are assumed to satisfy stabilisation and moment conditions. At the cost of an additional non-singularity condition, we show that the rates are in line with those under the Kolmogorov distance. We demonstrate the use of the theorems in four applications: Voronoi tessellations, k -nearest-neighbours graphs, timber volume, and maximal layers. [ABSTRACT FROM AUTHOR]

Abstract We prove a nonuniform local limit theorem concerning approximation of the point probabilities P ( S = k ) $P(S=k)$ , where S = ∑ i = 1 n X i $S=\sum_{i=1}^{n}X_{i}$ , and X 1 , … , X n $X_{1},\ldots ,X_{n}$ are independent Bernoulli random variables with possibly different success probabilities. Our proof uses Stein’s method, in particular, the zero bias transformation and concentration inequality approaches.

We prove a nonuniform local limit theorem concerning approximation of the point probabilities P (S = k) , where S = ∑ i = 1 n X i , and X 1 , ... , X n are independent Bernoulli random variables with possibly different success probabilities. Our proof uses Stein's method, in particular, the zero bias transformation and concentration inequality approaches. [ABSTRACT FROM AUTHOR]

We investigate the optimal rate of convergence in the multidimensional normal approximation of vector-valued Wiener-Itô integrals whose components all belong to a fixed Wiener chaos with coinciding orders. By combining Malliavin calculus, Stein's method for normal approximation and the method of cumulants, we obtain the optimal rate of convergence with respect to a suitable smooth distance. As applications, we derive the optimal rates of convergence for complex Wiener-Itô integrals, vector-valued Wiener-Itô integrals with kernels of step functions and vector-valued Toeplitz quadratic functionals. [ABSTRACT FROM AUTHOR]

A Galton-Watson process in a varying environment is a discrete time branching process where the offspring distributions vary among generations. It is known that in the critical case, these processes have a Yaglom limit, that is, a suitable normalization of the process conditioned on nonextinction converges in distribution to a standard exponential random variable. In this manuscript, we provide the rate of convergence of the Yaglom limit with respect to the Wasserstein metric. [ABSTRACT FROM AUTHOR]

In this paper, we develop Stein’s method for binomial approximation using the stop-loss metric that allows one to obtain a bound on the error term between the expectation of call functions. We obtain the results for a locally dependent collateralized debt obligation (CDO), under certain conditions on moments. The results are also exemplified for an independent CDO. Finally, it is shown that our bounds are sharper than the existing bounds. [ABSTRACT FROM AUTHOR]

We study the coupon collector's problem with group drawings. 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 $k_n=({n/s})(\alpha\log(n)+x)$ , where $0 and $x\in\mathbb{R}$. The same coupling construction is used to retrieve a quantitative Poisson limit theorem in the boundary case $\alpha=1$ , again using Stein's method. [ABSTRACT FROM AUTHOR]

In this paper, we develop Stein's method for negative binomial distribution using call function defined by fz(k) = (k - z)+ = max{k - z, 0}, for k ≥ 0 and z ≥ 0. We obtain error bounds between E [ fz(Nr,p)] and E [ fz(V )], where Nr,p follows negative binomial distribution and V is the sum of locally dependent random variables, using certain conditions on moments. We demonstrate our results through an interesting application, namely, collateralized debt obligation (CDO), and compare the bounds with the existing bounds.

We propose a new class of goodness‐of‐fit tests for the logistic distribution based on a characterization related to the density approach in the context of Stein's method. This characterization‐based test is a first of its kind for the logistic distribution. The asymptotic null distribution of the test statistic is derived and it is shown that the test is consistent against fixed alternatives. The finite sample power performance of the newly proposed class of tests is compared to various existing tests by means of a Monte Carlo study. It is found that this new class of tests are especially powerful when the alternative distributions are heavy tailed, like Student's t and Cauchy, or for skew alternatives such as the log‐normal, gamma and chi‐square distributions. [ABSTRACT FROM AUTHOR]

In this paper, we estimate the difference |Eh(Zn) −Eh(Y)| between the expectations of real finite Lipschitz function h of the sum Zn = (X1 + ⋯ + Xn)/Bn, where B n 2 = E(X1 + ⋯ + Xn)2> 0, and a standard normal random variable Y, where real centered random variables X1,X2,... satisfy the φ-mixing condition, defined between the "past" and " future", or are m-dependent. In particular cases, under the condition ∑ r = 1 ∞ r φ (r) < ∞ or ∑ r = 1 ∞ r φ 1 / 2 (r) < ∞ , the obtained upper bounds for φ-mixing random variables are of order O(n−1/2). In addition, we refine the previously known upper bounds of order O((m + 1)1+δL2+δ,n), where L2+δ,n is the Lyapunov fraction of order 2 + δ, for m-dependent random variables, supplementing them with explicit constants. We also separately present the case of independent r.v.s. [ABSTRACT FROM AUTHOR]

We obtain estimates for the Kolmogorov distance to appropriately chosen gaussians, of linear functions ∑ i ∈ [ n ] d θ i X i of random tensors X = ⟨ X i : i ∈ [ n ] d ⟩ which are symmetric and exchangeable, and whose entries have bounded third moment and vanish on diagonal indices. These estimates are expressed in terms of intrinsic (and easily computable) parameters associated with the random tensor X and the given coefficients ⟨ θ i : i ∈ [ n ] d ⟩ , and they are optimal in various regimes. The key ingredient—which is of independent interest—is a combinatorial CLT for high-dimensional tensors which provides quantitative non-asymptotic normality under suitable conditions, of statistics of the form ∑ (i 1 , ⋯ , i d) ∈ [ n ] d ζ (i 1 , ⋯ , i d , π (i 1) , ⋯ , π (i d)) where ζ : [ n ] d × [ n ] d → R is a deterministic real tensor, and π is a random permutation uniformly distributed on the symmetric group S n . Our results extend, in any dimension d, classical work of Bolthausen who covered the one-dimensional case, and more recent work of Barbour/Chen who treated the two-dimensional case. [ABSTRACT FROM AUTHOR]

We prove concentration inequalities of the form P (Y ≥ t) ≤ exp (− B (t)) for a random variable Y with mean zero and variance σ 2 using a coupling technique from Stein's method that is so-called approximate zero bias couplings. Applications to the Hoeffding's statistic where the random permutation has the Ewens distribution with parameter θ > 0 are also presented. A few simulation experiments are then provided to visualize the tail probability of the Hoeffding's statistic and our bounds. Based on the simulation results, our bounds work well especially when θ ≤ 1. [ABSTRACT FROM AUTHOR]

We introduce a new version of Stein’s method of comparison of operators specifically tailored to the problem of bounding the L 1 (a.k.a. Wasserstein-1) distance between continuous and discrete distributions on the real line. Our approach rests on a new family of weighted discrete derivative operators, which we call bespoke derivatives. We also propose new bounds on the derivatives of the solutions of Stein equations for integrated Pearson random variables; this is a crucial step in Stein’s method. We apply our result to several examples, including the central limit theorem, Pólya–Eggenberger urn models, the empirical distribution of the ground state of a many-interacting-worlds harmonic oscillator, the stationary distribution for the number of genes in the Moran model, and the stationary distribution of the Erlang-C system. Whenever our bounds can be compared with bounds from the literature, our constants are sharper. [ABSTRACT FROM AUTHOR]

Consider a class of stochastic pseudo-partial differential equation on R of the form @ @t u(t; x) = Lu(t; x) + (u(t; x)) _W (t; x); where L is a nonlocal pseudo-differential operator and _W denotes a Gaussian noise which is white in time and has a covariance function of fractional Brownian motion with Hurst index H 2 [1=2; 1) with respect to the spatial variable. If, in addition, u(0; x) = 1; 8x 2 R, we prove that the normalized spatial average of the solution from R to R (R > 0) converges in total variance distance to a standard normal distribution as R tends to infinity. We also establish a functional version of this central limit theorem. The Malliavin-Stein's method plays an important role. [ABSTRACT FROM AUTHOR]

We propose a kernel-based nonparametric test of relative goodness of fit, where the goal is to compare two models, both of which may have unobserved latent variables, such that the marginal distribution of the observed variables is intractable. The proposed test generalizes the recently proposed kernel Stein discrepancy (KSD) tests (Liu et al. Proceedings of the 33rd international conference on machine learning (pp. 276–284); Chwialkowski et al. (2016), In Proceedings of the 33rd international conference on machine learning (pp. 2606–2615); Yang et al. (2018), In Proceedings of the 35th international conference on machine learning (pp. 5561–5570)) to the case of latent variable models, a much more general class than the fully observed models treated previously. The new test, with a properly calibrated threshold, has a well-controlled type-I error. In the case of certain models with low-dimensional latent structures and high-dimensional observations, our test significantly outperforms the relative maximum mean discrepancy test, which is based on samples from the models and does not exploit the latent structure. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we obtain a nonuniform Berry–Esseen bound for a normal approximation via the Stein method and the exchangeable-pair coupling technique where the boundedness condition of the difference between the exchangeable pair is not required. As applications of the result, we obtain nonuniform bounds for the normal approximations in two well-known applications that are the independence test and the quadratic form. Our results suggest that the obtained bounds for the two applications are sharper than other existing bounds.

In this paper, we study asymptotic properties of the maximum likelihood estimator (MLE) for the speed of a stochastic wave equation. We follow a well-known spectral approach to write the solution as a Fourier series, then we project the solution to a N-finite dimensional space and find the estimator as a function of the time and N. We then show consistency of the MLE using classical stochastic analysis. Afterward, we prove the asymptotic normality using the Malliavin–Stein method. We also study asymptotic properties of a discretized version of the MLE for the parameter. We provide this asymptotic analysis of the proposed estimator as the number of Fourier modes, N, used in the estimation and the observation time go to infinity. Finally, we illustrate the theoretical results with some numerical experiments. [ABSTRACT FROM AUTHOR]

In Holm and Alouini (2004), an explicit formula for the probability density function of λ 1 − 1 Z r + λ 2 − 1 Z ˜ r was obtained, where λ 1 ≠ λ 2 ≠ 0 , Z r , Z ˜ r are independent identical random variables with Γ (r , 1) distribution. In this paper, we use a recent technique from Gaunt (2018) involving Stein's method literature to obtain some simple and new proofs, which could also be used in other problems as a general method. [ABSTRACT FROM AUTHOR]

The Dirichlet-Ferguson measure is a random probability measure that has seen widespread use in Bayesian nonparametrics. Our main results can be seen as a first step towards the development of a stochastic analysis of the Dirichlet-Ferguson measure. We define a gradient that acts on functionals of the measure and derive its adjoint. The corresponding integration by parts formula is used to prove a covariance representation formula for square integrable functionals of the Dirichlet-Ferguson measure and to provide a quantitative central limit theorem for the first chaos. Our findings are illustrated by a variety of examples. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a nonuniform Berry–Esseen bound for a normal approximation via the Stein method and the exchangeable-pair coupling technique where the boundedness condition of the difference between the exchangeable pair is not required. As applications of the result, we obtain nonuniform bounds for the normal approximations in two well-known applications that are the independence test and the quadratic form. Our results suggest that the obtained bounds for the two applications are sharper than other existing bounds. [ABSTRACT FROM AUTHOR]

Bayesian inference problems require sampling or approximating high-dimensional probability distributions. The focus of this paper is on the recently introduced Stein variational gradient descent methodology, a class of algorithms that rely on iterated steepest descent steps with respect to a reproducing kernel Hilbert space norm. This construction leads to interacting particle systems, the mean-field limit of which is a gradient flow on the space of probability distributions equipped with a certain geometrical structure. We leverage this viewpoint to shed some light on the convergence properties of the algorithm, in particular addressing the problem of choosing a suitable positive definite kernel function. Our analysis leads us to considering certain nondifferentiable kernels with adjusted tails. We demonstrate significant performance gains of these in various numerical experiments. [ABSTRACT FROM AUTHOR]

We study stability of the eigenvalues of the generator of a one dimensional reversible diffusion process satisfying some natural conditions. The proof is based on Stein's method. In particular, these results are applied to the Normal distribution (via the Ornstein–Uhlenbeck process), to Gamma distributions (via the Laguerre process) and to Beta distributions (via the Jacobi process). [ABSTRACT FROM AUTHOR]

The Dirichlet-Ferguson measure is a cornerstone in nonparametric Bayesian statistics and the study of the distributional properties of expectations with respect to such measure is an important line of research initiated in Cifarelli and Regazzini (1979a,b) and still very active, see Letac and Piccioni (2018) and Lijoi and Prünster (2009). In this paper we provide explicit upper bounds for the d3, the d2 and the convex distances between random vectors whose components are means of the Dirichlet-Ferguson measure and a random vector distributed according to the multivariate Gaussian law. These results are applied to the Gaussian approximation of linear transformations of random vectors with the Dirichlet distribution, yielding presumably optimal rates on the d3 and the d2 distances and presumably suboptimal rates on the convex and the Kolmogorov distances. [ABSTRACT FROM AUTHOR]

We propose a new functional analytic approach to Stein's method of exchangeable pairs that does not require the pair at hand to satisfy any approximate linear regression property. We make use of this theory in order to derive abstract bounds on the normal and Gamma approximation of certain functionals in the Wasserstein distance. Moreover, we illustrate the relevance of this approach by means of three instances of situations to which it can be applied: Functionals of independent random variables, finite population statistics and functionals on finite groups. In the independent case, and in particular for symmetric U-statistics, we demonstrate in which respect this approach yields systematically better bounds than those in the existing literature. Finally, we apply our results to provide Wasserstein bounds in a CLT for subgraph counts in geometric random graphs based on n i.i.d. points in Euclidean space as well as to the normal approximation of Pearson's statistic. [ABSTRACT FROM AUTHOR]

In extending Stein's method to new target distributions, the first step is to find a Stein operator that suitably characterises the target distribution. In this paper, we introduce a widely applicable technique for proving sufficiency of these Stein characterisations, which can be applied when the Stein operators are linear differential operators with polynomial coefficients. The approach involves performing an asymptotic analysis to prove that only one characteristic function satisfies a certain differential equation associated to the Stein characterisation. We use this approach to prove that all Stein operators with linear coefficients characterise their target distribution, and verify on a case-by-case basis that all polynomial Stein operators in the literature with coefficients of degree at most two are characterising. For X denoting a standard Gaussian random variable and Hp the p-th Hermite polynomial, we also prove, amongst other examples, that the Stein operators for Hp(X), p = 3; 4;: ::; 8, with coefficients of minimal possible degree characterise their target distribution, and that the Stein operators for the products of p = 3; 4;: ::; 8 independent standard Gaussian random variables are characterising (in both settings the Stein operators for the cases p = 1; 2 are already known to be characterising). We leverage our Stein characterisations of H3(X) and H4(X) to derive characterisations of these target distributions in terms of iterated Gamma operators from Malliavin calculus, that are natural in the context of the Malliavin-Stein method. [ABSTRACT FROM AUTHOR]