
Progress in Stein's Method
(5 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 ~
Rubinstein distances on configurations spaces Laurent Decreusefond, École Nationale Supérieure des Télécommunications, France
We
provide upper bounds on several Rubinsteintype distances on
configuration spaces. Our inequalities involve the two wellknown
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.
« Back... 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.
« Back... 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).
« Back... Convergence to fractional Brownian motion and to the Telecom process Murad Taqqu, Boston University, USA
It
has become common practice to use heavytailed distributions in order
to describe the variations in time and space of the workloads in network
traffic. The asymptotic behavior of these workloads is complex;
different limit processes emerge depending on the specifics of the work
arrival structure and the nature of the asymptotic scaling. We will
describe these limits. The Stein approach to this problem is still open.
« Back... 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 wellknown 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 resampling, which give
rise to many new potential applications.
« Back... 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 birthdeath 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).
« Back... 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 TracyWidom law
of the largest singular values of the Jacobi emsemble. Moreover, the
circular law, the MarchenkoPastur law, the central limit theorem, and
the laws of large numbers for the spectral norms are also obtained.
« Back... 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.
« Back... 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.
« Back... 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.
« Back... Normal approximation for stochastic geometry and allocations Mathew Penrose, University of Bath, UK
We
discuss an approach based on Stein's method via sizebiased 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.
« Back... 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 BerryEsseen type bounds for the distribution
of the number vertices of given degree in a random graph. A sizebiased
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 squareroot rate. This is joint work with
Larry Goldstein at USC.
« Back... Centered Poisson and Binomial approximations for the PoissonBinomial 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.
« Back... 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), zerobiassed
couplings and Palm distributions. Roughly speaking, bounds of optimal
order can be derived under the assumption of boundedness of certain "key
ingredients".
« Back... 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.
« Back... 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; ShenOrr 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 overrepresented motifs is based on heavy simulations:
random graphs are first generated, then the pvalue is derived either
from the empirical distribution of the count or via a Gaussian
approximation of the zscore 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 pvalue and to
decide if an observed count is significantly high or not.
Beyond the pvalue 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 ErdösRényi 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 goodnessoffit
of this model on real biological networks is very satisfactory.
« Back...

