The celebrated Nualart-Peccati criterion [Ann. Probab. 33 (2005) 177-193] ensures the convergence in distribution toward a standard Gaussian random variable N of a given sequence {Xn} n≥1 of multiple Wiener-Itô integrals of fixed order, if E[X

We build upon recent advances on the distributional aspect of Stein's method to propose a novel and flexible technique for computing Stein operators for random variables that can be written as products of independent random variables. We show that our results are valid for a wide class of distributions including normal, beta, variance-gamma, generalized gamma and many more. Our operators are kth degree differential operators with polynomial coefficients; they are straightforward to obtain even when the target density bears no explicit handle. As an application, we derive a new formula for the density of the product of k independent symmetric variance-gamma distributed random variables. -Tilman, Allée de la découverte 12, B-8000 Liège, Belgium yswan@uliege.be. variable of interest W and taking the supremum over all h in some class of functions H leads to the estimatewhere the final supremum is taken over all f h that solve (1). The third and final step of the method involves developing appropriate strategies for bounding the expectation on the right hand side of (2). This is of interest because many important probability metrics (such as the Kolmogorov and Wasserstein metrics) are of the form d H (L(W ), L(X)). Moreover, in many settings bounding the expectation IE[Af h (W )] is relatively tractable, and as a result Stein's method has found application in disciplines as diverse as random graph theory [5], number theory [22], statistical mechanics [13] and quantum mechanics [29]. We refer to the survey paper [37] as well as to the monographs [30,9] for a deeper look into some of the fruits of Charles Stein's seminal insights, particularly in the case where the target is the normal distribution. The linchpin of the method is the operator A whose properties are crucial to the success of the whole enterprise. In the sequel, we concentrate exclusively on differential Stein operators (some operators in the literature are integral or even fractional, see e.g. [43,3]) and adopt the following lax definition:Definition 1.1. A linear differential operator A acting on a class F of functions is a Stein operator for X if (i) Af ∈ L 1 (X) and (ii) IE [Af (X)] = 0 for all f ∈ F.There are infinitely many Stein operators for any given target distribution. For instance, if the distribution is known (even if only up to a normalizing constant) then the "canonical" theory from [26] applies, leading to entire families of operators. This approach provides natural first order polynomial operators e.g. for target distributions which belong to the Pearson family [38] or which satisfy a diffusive assumption [11,25]. In some cases, one may rather apply a duality argument. For instance the p.d.f. γ(x) = (2π) −1/2 e −x 2 /2 of the standard normal distribution satisfies the first order ODE γ ′ (x) + xγ(x) = 0 leading, by integration by parts, to the already mentioned operator Af (x) = f ′ (x) − xf (x). This is particularly useful for densities defined implicitely via ODEs. Such are by no means the only methods for deriving differential Stein ope...

Taking a decision on the feasibility and estimating the duration of patients' recruitment in a clinical trial are very important but very hard questions to answer, mainly because of the huge variability of the system. The more elaborated works on this topic are those of Anisimov and co-authors, where they investigate modelling of the enrolment period by using Gamma-Poisson processes, which allows to develop statistical tools that can help the manager of the clinical trial to answer these questions and thus help him to plan the trial. The main idea is to consider an ongoing study at an intermediate time, denoted t(1). Data collected on [0,t(1)] allow to calibrate the parameters of the model, which are then used to make predictions on what will happen after t(1). This method allows us to estimate the probability of ending the trial on time and give possible corrective actions to the trial manager especially regarding how many centres have to be open to finish on time. In this paper, we investigate a Pareto-Poisson model, which we compare with the Gamma-Poisson one. We will discuss the accuracy of the estimation of the parameters and compare the models on a set of real case data. We make the comparison on various criteria : the expected recruitment duration, the quality of fitting to the data and its sensitivity to parameter errors. We discuss the influence of the centres opening dates on the estimation of the duration. This is a very important question to deal with in the setting of our data set. In fact, these dates are not known. For this discussion, we consider a uniformly distributed approach. Finally, we study the sensitivity of the expected duration of the trial with respect to the parameters of the model : we calculate to what extent an error on the estimation of the parameters generates an error in the prediction of the duration.

In this paper, we propose a general means of estimating the rate at which convergences in law occur. Our approach, which is an extension of the classical Stein-Tikhomirov method, rests on a new pair of linear operators acting on characteristic functions. In principle, this method is admissible for any approximating sequence and any target, although obviously the conjunction of several favorable factors is necessary in order for the resulting bounds to be of interest. As we briefly discuss, our approach is particularly promising whenever some version of Stein's method applies. We apply our approach to two examples. The first application concerns convergence in law towards targets F ∞ which belong to the second Wiener chaos (i.e. F ∞ is a linear combination of independent centered chi-squared rvs). We detail an application to U -statistics. The second application concerns convergence towards targets belonging to the generalized Dickman family of distributions. We detail an application to a theorem from number theory. In both cases our method produces bounds of the correct order (up to a logarithmic loss) in terms of quantities which occur naturally in Stein's method.

A consequence of de Finetti's representation theorem is that for every infinite sequence of exchangeable 0-1 random variables (X k ) k≥1 , there exists a probability measure µ on the Borel sets of [0, 1] such thatX n = n −1 n i=1 X i converges weakly to µ. For a wide class of probability measures µ having smooth density on (0, 1), we give bounds of order 1/n with explicit constants for the Wasserstein distance between the law ofX n and µ. This extends a recent result by Goldstein and Reinert [10] regarding the distance between the scaled number of white balls drawn in a Pólya-Eggenberger urn and its limiting distribution. We prove also that, in the most general cases, the distance between the law ofX n and µ is bounded below by 1/n and above by 1/ √ n (up to some multiplicative constants). For every δ ∈ [1/2, 1], we give an example of an exchangeable sequence such that this distance is of order 1/n δ .

We provide a general result for finding Stein operators for the product of two independent random variables whose Stein operators satisfy a certain assumption, extending a recent result of Gaunt, Mijoule and Swan [13]. This framework applies to non-centered normal and non-centered gamma random variables, as well as a general sub-family of the variance-gamma distributions. Curiously, there is an increase in complexity in the Stein operators for products of independent normals as one moves, for example, from centered to non-centered normals. As applications, we give a simple derivation of the characteristic function of the product of independent normals, and provide insight into why the probability density function of this distribution is much more complicated in the non-centered case than the centered case.

This paper provides a general framework for Stein's density method for multivariate continuous distributions. The approach associates to any probability density function a canonical operator and Stein class, as well as an infinite collection of operators and classes which we call standardizations. These in turn spawn an entire family of Stein identities and characterizations for any continuous distribution on R d , among which we highlight those based on the score function and the Stein kernel. A feature of these operators is that they do not depend on normalizing constants. A new definition of Stein kernel is introduced and examined; integral formulas are obtained through a connection with mass transport, as well as ready-to-use explicit formulas for elliptical distributions. The flexibility of the kernels is used to compare in Stein discrepancy (and therefore 2-Wasserstein distance) between two normal distributions, Student and normal distributions, as well as two normal-gamma distributions. Upper and lower bounds on the 1-Wasserstein distance between continuous distributions are provided, and computed for a variety of examples: comparison between different normal distributions (improving on existing bounds in some regimes), posterior distributions with different priors in a Bayesian setting (including logistic regression), centred Azzalini-Dalla Valle distributions. Finally the notion of weak Stein equation and weak Stein factors is introduced, and new bounds are obtained for Lipschitz test functions if the distribution admits a Poincaré constant, which we use to compare in 1-Wasserstein distance between different copulas on the unit square.MSC 2010 classification: 60B10, 60B12

We investigate the sparse spikes deconvolution problem onto spaces of algebraic polynomials. Our framework encompasses the measure reconstruction problem from a combination of noiseless and noisy moment measurements. We study a TVnorm regularization procedure to localize the support and estimate the weights of a target discrete measure in this frame. Furthermore, we derive quantitative bounds on the support recovery and the amplitude errors under a Chebyshev-type minimal separation condition on its support. Incidentally, we study the localization of the knots of non-uniform splines when a Gaussian perturbation of their innerproducts with a known polynomial basis is observed (i.e. a small degree polynomial approximation is known) and the boundary conditions are known. We prove that the knots can be recovered in a grid-free manner using semidefinite programming.