We propose a new version of Stein's method of exchangeable pairs, which, given a suitable exchangeable pair (W, W ′ ) of real-valued random variables, suggests the approximation of the law of W by a suitable absolutely continuous distribution. This distribution is characterized by a first order linear differential Stein operator, whose coefficients γ and η are motivated by two regression properties satisfied by the pair (W, W ′ ). Furthermore, the general theory of Stein's method for such an absolutely continuous distribution is developed and a general characterization result as well as general bounds on the solution to the Stein equation are given. This abstract approach is a certain extension of the theory developed in the papers [5] and [13], which only consider the framework of the density approach, i.e. η ≡ 1. As an illustration of our technique we prove a general plug-in result, which bounds a certain distance of the distribution of a given random variable W to a Beta distribution in terms of a given exchangeable pair (W, W ′ ) and provide new bounds on the solution to the Stein equation for the Beta distribution, which complement the existing bounds from [18]. The abstract plug-in result is then applied to derive bounds of order n −1 for the distance between the distribution of the relative number of drawn red balls after n drawings in a Pólya urn model and the limiting Beta distribution measured by a certain class of smooth test functions.

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We prove a fourth moment bound without remainder for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result-that has been elusive for several years-shows that the so-called 'fourth moment phenomenon', first discovered by Nualart and Peccati [Ann. Probab. 33 (2005) 177-193] in the context of Gaussian fields, also systematically emerges in a Poisson framework. Our main findings are based on Stein's method, Malliavin calculus and Mecketype formulae, as well as on a methodological breakthrough, consisting in the use of carré-du-champ operators on the Poisson space for controlling residual terms associated with add-one cost operators. Our approach can be regarded as a successful application of Markov generator techniques to probabilistic approximations in a nondiffusive framework: as such, it represents a significant extension of the seminal contributions by Ledoux [Ann. Probab. 40 (2012) 2439-2459 and Azmoodeh, Campese and Poly [J. Funct. Anal. 266 (2014) 2341-2359]. To demonstrate the flexibility of our results, we also provide some novel bounds for the Gamma approximation of nonlinear functionals of a Poisson measure.

We study the Stein equation associated with the one-dimensional Gamma distribution, and provide novel bounds, allowing one to effectively deal with test functions supported by the whole real line. We apply our estimates to derive new quantitative results involving random variables that are non-linear functionals of random fields, namely: (i) a non-central quantitative de Jong theorem for sequences of degenerate Ustatistics satisfying minimal uniform integrability conditions, significantly extending previous findings by de

We introduce a simple iterative technique for bounding derivatives of solutions of Stein equations Lf = h − Eh(Z), where L is a linear differential operator and Z is the limit random variable. Given bounds on just the solutions or certain lower order derivatives of the solution, the technique allows one to deduce bounds for derivatives of any order, in terms of supremum norms of derivatives of the test function h. This approach can be readily applied to many Stein equations from the literature. We consider a number of applications; in particular, we derive new bounds for derivatives of any order of the solution of the general variancegamma Stein equation. Finally, we present a connection between Stein equations and Poisson equations, from which we first recognised the importance of the iterative technique to Stein's method.1 for all functions f belonging to some measure determining class. For continuous random variables, L is a differential operator; for discrete random variables, L is a difference operator. Such characterisations have often been derived via Stein's density approach [53], [54] or the generator approach of Barbour and Götze [3], [32]. The scope of the density approach has recently been extended by [38], and other techniques for obtaining Stein characterisations are discussed in that work.The characterisation (1.1) leads to the so-called Stein equation:where the test function h is real-valued. The second step of Stein's method, which will be the focus of this paper, concerns the problem of obtaining a solution f to the Stein equation (1.2) and then establishing estimates for f and some of its lower order derivatives (for continuous distributions). Evaluating (1.2) at a random variable W and taking expectations gives3) and thus the problem of bounding the quantity Eh(W ) − Eh(Z) reduces to solving (1.2) and bounding the right-hand side of (1.3). The third step of Stein's method concerns the problem of bounding the expectation E[Lf (W )]. For continuous limit distributions, such bounds are usually obtained via Taylor expansions and coupling techniques. For many classical distributions, the problem of obtaining the first two necessary ingredients is relatively tractable. As a result, over the years, Stein's method has been adapted to many standard distributions, including the beta [11], [30], gamma [28], [39], exponential [5], [19], [45], Laplace [48] and, more generally, the class of variance-gamma distributions [21]. The method has also been adapted to distributions arising from specific problems, such as preferential attachment graphs [46], the Curie-Weiss model [6] and statistical mechanics [13], [14]. For a comprehensive overview of the literature, see [38]. Estimates for solutions of Stein equationsThe Stein equations of many classical distributions, such as the normal, beta and gamma, are linear first order ODEs with simple coefficients. As a result, the problem of solving the Stein equation and bounding the derivatives of the solution is reasonably tractable.In fact, for Stein characterisation...