## Progress in Stein's Method (05 Jan - 6 Feb 2009)

Jointly organized with Department of Mathematics and Department of Statistics and Applied Probability in celebration of 80th Anniversary of Faculty of Science

## ~ Abstracts ~

Bounds on the normal approximation for the number of vertices of given degree
Jay Bartroff, University of Southern California, USA

Stein's method is used to derive Berry-Esseen type bounds for the distribution of the number vertices of given degree in a random graph. A size-biased coupling is used whose difference between the biased and original random variables is unbounded as the number of vertices grows. The resulting bounds achieve the inverse square-root rate. This is joint work with Larry Goldstein at USC.

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Estimates for pseudomoments
Vydas Cekanavicius, Vilnius University, Lithuania

It is not easy to solve the Stein equation for unbounded functions. We show that, when estimating the difference of moments, one can use indirect step by step procedure beginning from the estimates for the total variation. Approach will be discussed for the Poisson perturbation model.

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Stein's method and convex orderings
Fraser Daly, University of Zurich, Switzerland

We apply Stein's method in conjunction with some stochastic ordering assumptions, considering approximation by the equilibrium distribution of a birth-death process. These stochastic ordering assumptions give a natural framework for deriving simple bounds using Stein's method. With these conditions, bounds may typically be expressed as a difference of moments. Our main example will be Poisson approximation for a sum of indicators. This is joint work with C. Lefèvre (Brussels) and S. Utev (Nottingham).

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Rubinstein distances on configurations spaces
Laurent Decreusefond, École Nationale Supérieure des Télécommunications, France

We provide upper bounds on several Rubinstein-type distances on configuration spaces. Our inequalities involve the two well-known gradients, in the sense of Malliavin calculus, which can be defined on such spaces. Actually, we show that depending on the distance between configurations which is considered, it is one gradient or the other which is the most effective. Some applications to distance estimates between Poisson and other more sophisticated processes are provided, and an investigation of such distances to functional inequalities completes this work.

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Stein's method and symmetric spaces
Jason Fulman, University of Southern California, USA

We show how Stein's method can be used to study spherical functions of symmetric spaces. In the discrete setting, we describe a normal approximation example related to random walks on matchings, and an exponential approximation example (joint with Chatterjee and Rollin). In the continuous setting, we illustrate the method on random matrices from Dyson's circular ensembles.

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Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles
Tiefeng Jiang, University of Minnesota, USA

I will first present tools to approximate the entries of a large dimensional real and complex Jacobi ensembles with independent complex Gaussian random variables. Based on this, we obtain the Tracy-Widom law of the largest singular values of the Jacobi emsemble. Moreover, the circular law, the Marchenko-Pastur law, the central limit theorem, and the laws of large numbers for the spectral norms are also obtained.

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Density estimates with Stein's method and Malliavin calculus
Ivan Nourdin, Université Pierre et Marie Curie (Paris VI), France

Let X be an (isonormal) Gaussian process, consider a random variable Z which is measurable with respect to X (for instance, Z=sup X) and assume that the law of Z is absolutely continuous with respect to the Lebesgue measure. In this talk, I will explain how, following the same strategy as in the paper "Stein's method on Wiener chaos" (by Giovanni Peccati and myself, 2008), we can combine Malliavin calculus with Stein's method in order to derive, this time, a new formula for the density of Z. Then, I will present several applications.
My talk will be based on the paper "Density estimates and concentration inequalities with Malliavin calculus", jointly written with Frederi Viens.

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Limit theorems on the Poisson space: decoupling, Stein's method and low influences
Giovanni Peccati, University of Paris Ouest, France

we shall discuss the problem of proving (stable) limit theorems involving functionals of a Poisson random measure. Our approach combines three points of view: Stein's method, Decoupling and Malliavin calculus. In particular, we shall show that the use of Stein's method, as applied to random variables living in a fixed chaos, leads to explicit bounds, admitting a direct interpretation in terms of ''low influences'' - a combinatorial concept implicitly appearing in the statements of invariance principles by Rotar' and Mossel, O'Donnel and Oleszkiewicz. This talk is based on joint works with J.L.
Sol? (Barcelona), F. Utzet (Barcelona) and M.S. Taqqu (Boston), and can be seen as a "Poisson counterpart" to the theory developed in the work "Stein's method on Wiener chaos" (by I. Nourdin and myself, 2008).

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Centered Poisson and Binomial approximations for the Poisson-Binomial
Erol Pekoz, Boston University School of Management, USA

Two approaches for fitting centered Poisson and binomial approximations in statistical applications will be discussed, and a variation of Stein?s method for the centered binomial will be developed and shown to yield sharper error bounds.

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Normal approximation for stochastic geometry and allocations
Mathew Penrose, University of Bath, UK

We discuss an approach based on Stein's method via size-biased couplings to three normal approximation problems: the covered area for a union of randomly centred disks in in a spatial region, the number of such disks which are isolated, and the number of isolated objects in a classical occupancy model. Unlike many approaches to such problems, we do not use any kind of Poissonization. Some of the work discussed is joint with
Larry Goldstein; see Arxiv:0812.3084.

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Stein's method and Cramér type large deviations
Martin Raic, University Of Ljubljana, Slovenia

We shall focus on the relative error in the normal approximation of large deviation probabilities. Our main abstract result can be applied to many constructions which are in general used in Stein's method for the normal approximation. These include the decompositions of Barbour, Karonski and Rucinski (which include local dependence), zero-biassed couplings and Palm distributions. Roughly speaking, bounds of optimal order can be derived under the assumption of boundedness of certain "key ingredients".

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Stein's method and stochastic analysis of Rademacher functionals
Gesine Reinert, University of Oxford, UK

Recently Nourdin and Peccati have related Stein?s method to Malliavin calculus. Here we consider the discrete case: 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.
The method allows us to treat random variables which depend on infinitely many Rademacher variables, such as weighted infinite runs of length 2. It also sheds new light on how to construct multivariate exchangeable pairs satisfying a linearity condition in conditional expectation.

This is joint work with Ivan Nourdin and Giovanni Peccati.

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Normal approximation with Stein's method: a unifying approach
Adrian Roellin, National University of Singapore

In the framework of Stein's method, a variety of couplings have been proposed and used for normal approximation. In the past, for each specific type of coupling a separate abstract theorems had to be proved under specific assumptions on the involved random variables.
Especially for results with respect to the Kolmogorov metric, several techniques are available to control the smoothness of the random variable under consideration, which inevitably has led to a variety of in fact similar theorems. This seems unsatisfying from both the practical and theoretical point of view. We present a unifying approach which essentially makes no assumptions on the involved random variables.
We show how well-known approaches such as the local approach, exchangeable pairs and size biasing, can not only be expressed, but also simplified in our framework. We also present a few new couplings, such as interpolation to independence and conditional re-sampling, which give rise to many new potential applications.

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On some examples where Stein's method could be improved
Bero Roos, University of Leicester, UK

Though Stein's method is very useful in many different situations, some of the results in the literature are not the best possible. We discuss some approximations by (compound) Poisson distributions and approximations of random sums.

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Statistics of biological network motifs: A compound Poisson approximation for their count in random graphs?
Sophie Schbath, Institut National de la Recherche Agronomique, France

Getting and analyzing biological interaction networks is at the core of systems biology. To help understanding these complex networks, many recent works have suggested focusing on motifs which occur more frequently than expected in random (Milo et al., 2002; Shen-Orr et al., 2002; Prill et al., 2005). Such motifs seem indeed to reflect functional or computational units which combine to regulate the cellular behavior as a whole. The common method that has been used for now to detect significantly over-represented motifs is based on heavy simulations:
random graphs are first generated, then the p-value is derived either from the empirical distribution of the count or via a Gaussian approximation of the z-score calculated thanks to the empirical mean and variance of the count.

To identify exceptional motifs in a given network, we propose a statistical and analytical method which does not require any simulation (Picard et al., 2008). For this, we first provide an analytical expression of the mean and variance of the count under any stationary random graph model. Then we approximate the motif count distribution by a compound Poisson distribution whose parameters are derived from the mean and variance of the count. Thanks to simulations, we show that the quality of such compound Poisson approximation is very good and highly better than a Gaussian or a Poisson one. The compound Poisson distribution can then be used to get an approximate p-value and to decide if an observed count is significantly high or not.

Beyond the p-value calculation, the assessment of the motif exceptionality in a given network relies on the choice of a suitable random graph model. This model should indeed fit some relevant characteristics of the observed network. The sequence degree is usually an important feature to take into account. Unfortunately the well known and well studied Erdös-Rényi model does not fit correctly biological networks, in particular it does not consider heterogeneities. We then emphasize the recent and promising mixture model for random graphs proposed by Daudin et al. (2008). This model assumes that nodes are spread into several classes of connectivity and that the probability for two nodes to be connected depends on their classes. The goodness-of-fit of this model on real biological networks is very satisfactory.

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Convergence to fractional Brownian motion and to the Telecom process