Workshop on Stein’s Method
(31 Mar – 4 Apr 2008)
~ Abstracts ~
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)
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.
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
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.
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.
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.
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.
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.
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.
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 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.
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.
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.