Workshop on Stein’s Method

(31 Mar – 4 Apr 2008)

~ Abstracts ~

Multivariate exchangeable pairs in Stein's method for multivariate normal approximation
Gesine Reinert, University of Oxford, UK

It is only very recently that multivariable exchangeable pairs have been explored in the framework of Stein's method. Here we establish a multivariate exchangeable pairs 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, the joint count of edges, two-stars and triangles in Bernoulli random graphs, and complete $U$-statistics.

(joint work with A. Röllin)


 

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How to remove the exchangeability condition in Stein’s method
Adrian Röllin, University of Oxford, UK

Although exchangeable pairs are thought to be a coupling at the heart of Stein's method, we show that the exchangeability condition can be easily omitted in many standard settings, requiring only that the two random variables have the same distribution. We discuss the case of the normal and the Poisson distribution. The results in the Poisson case suggest that an additional exchangeability assumption can be interpreted as a smoothness condition. We also give connections to other approaches such as Barbour's generator interpretation and the zero biasing coupling.

 

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Stein’s Method and the spectrum of Markov chains
Jason Fulman, University of Southern California, USA

It is shown that Stein's method of exchangeable pairs is well suited for studying the spectrum of a fixed Markov chain. We focus on examples in which there is a lot of symmetry (such as the random transposition walk on
the symmetric group), as other researchers have investigated these by different techniques. The definition of the exchangeable pair used in our approach appears complicated, but fits perfectly with Fourier analysis to allow the exact computation of error terms in Stein's method. While mostly developed for normal approximation, this approach can be used for other limiting distributions too.

 

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Central limit theorems for convex hulls
Joe Yukich, Lehigh University, USA

Using Stein's method we show that the random point measures induced by the vertices in the convex hull of a Poisson sample on the unit ball, when properly scaled and centered, converge to those of a mean zero Gaussian field. We establish limiting variance and covariance asymptotics in terms of the density of the Poisson sample. Similar results hold for the point measures induced by the maximal points in a Poisson sample. The approach involves introducing a generalized spatial birth growth process allowing for cell overlap.

This is joint work with T. Schreiber.

 

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Metrics for point processes approximation
Dominic Schuhmacher, The University of Western Australia, Australia

In this talk, I compare four different metrics between point process distributions: the total variation metric and three somewhat similar Wasserstein metrics. I briefly recapitulate the main ideas of Stein's method for Poisson process approximation and then present the Stein factors for the various metrics as well as concrete upper bounds for Poisson process approximation of the hard core process.

 

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Stein’s method and infinitesimal symmetries
Elisabeth Meckes, Cornell University, USA

I will describe univariate and multivariate versions of a modification of the method of exchangeable pairs, for situations which involve continuous groups of symmetries. I will discuss applications of the method to the distribution of eigenfunctions of the Laplacian on Riemannnian manifolds, and to the approximation of Haar-distributed random matrices by Gaussian random matrices.

 

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Stein factors from generating functions and compound geometric approximation
Fraser Daly, University of Nottingham, UK

Stein factors, giving bounds of a good order on our Stein operator, are an essential ingredient of Stein's method. In this talk, a new method of finding Stein factors for discrete distributions will be presented, based on generating functions and an application of Cauchy's formula. This generating function approach will be applied to derive sharp bounds on a Stein operator for the compound geometric distribution.

The Stein factors obtained are combined with a suitable coupling argument to give bounds in compound geometric approximation. Applications include approximation of Markov chain hitting times.



 

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Closeness of convolutions of probability measures
Bero Roos, University of Leicester, UK

We derive new explicit bounds for the total variation distance between two convolution products of n probability distributions, one of which having identical convolution factors. Approximations by finite signed measures of arbitrary order are considered as well. We are interested in bounds with magic factors, i.e. roughly speaking n also appears in the denominator. Special emphasis is given to the approximation by the n-fold convolution of the arithmetic mean of the distributions under consideration. As an application, we consider the multinomial approximation of the generalized multinomial distribution. It turns out that here the order of some bounds given in Loh (1992) and Roos (2001) can significantly be improved. In particular, it follows that a dimension factor can be dropped. In the course of proof, we use a basic Banach algebra technique for measures on a measurable Abelian group. Though this method was already used by Le Cam (1960), our central arguments seem to be new. We also derive new smoothness bounds for convolutions of probability distributions, which might be of independent interest. It should be mentioned that Loh used Stein's method in a more general situation of dependent random variables. However, it seems to be unclear, whether Stein's method can be used to reproduce the results of the present paper.

References:

Loh, W.-L. (1992). Stein's method and multinomial approximation. Ann. Appl. Probab., 2, 536-554.

Roos, B. (2001). Multinomial and Krawtchouk approximations to the generalized multinomial distribution. Theory Probab. Appl., 46, 103-117.



 

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Stein's method on a Gaussian space, I: general theory
Giovanni Peccati, Université Pierre et Marie Curie (Paris VI), France

We will show that one can combine Stein's method with Malliavin calculus, in order to prove bounds in the normal and gamma approximation of functionals of infinite-dimensional Gaussian fields. We also present some techniques allowing to establish the optimality of these bounds. Our results provide a substantial refinement of the central and non-central limit theorems on Wiener chaos, recently proved by Nourdin, Nualart, Peccati and Tudor (2005-2007).
We discuss two applications: (i) to infinite-dimensional Poincaré-type inequalities, and (ii) to high-frequency limit theorems for functionals of spherical random fields. This is based on joint works with I. Nourdin.



 

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Stein's method on a Gaussian space, II: applications
Ivan Nourdin, Université Pierre et Marie Curie (Paris VI), France

We will present several refinements and applications of the general results, recently proved by I. Nourdin and G. Peccati (2007, 2008), concerning the normal approximation of functionals of Gaussian fields. Our starting point is a result concerning the multidimensional normal approximation of smooth random vectors defined on a Gaussian space. We then show how to obtain (possibly optimal) upper bounds in a functional version of a CLT, due to Breuer and Major, involving processes subordinated to a fractional Brownian motion. Further applications involve Toeplitz quadratic functionals of stationary (continuous-time) Gaussian fields, and a new almost sure CLT associated with the aforementioned Breuer-Major convergence result. This talk is based on joint works with G. Peccati and A. Reveillac.




 

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Discretized normal approximation
Louis Chen, National University of Singapore, Singapore

Let the mean and variance of W be 0 and 1 respectively. The zero-biased distribution of W is defined to be the distribution of W* which satisfies the equation EWf(W) = Ef'(W*) for all bounded functions f with bounded derivatives f'. Using this equation and Stein's method, one can easily obtain a simple error bound on the total variation distance between the distribution of W* and N(0,1). We use this result to study discretized normal approximation for sums of integer-valued random variables.



 

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Approximating probabilities from point processes
Tim Brown, La Trobe University, Australia

Stein’s method has been successfully used to approximate a difficult probability for a point process by that for a simpler point process and to bound the approximation error.  Unfortunately, most of the compelling applications have been using simple functions of the point process – such as numbers of points in a region – or relatively inapplicable more complex functions.  The reason centres on the domain of applicability of Stein “magic factors” for point processes.  After reviewing these issues, a simple example is considered where exact calculation is difficult.  The example shows the potential importance of combining Stein’s method with simulation for calculating approximations, thereby avoiding difficult issues of continuity of real valued functions on the carrier space of point processes.



 

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