Workshop on New Directions in Stein's Method - IMS

Workshop on New Directions in Stein's Method
(18 - 29 May 2015)

~ Abstracts ~

Matrix concentration inequalities via the method of exchangeable pairs
Joel A. Tropp, California Institute of Technology, USA

This talk discusses a new way to establish moment inequalities for the spectral norm of a random matrix. The analysis is based on a matrix extension of the scalar concentration theory developed by Sourav Chatterjee using Stein's method of exchangeable pairs. This approach offers a truly elementary proof of the noncommutative Khintchine inequality. It also delivers matrix versions of the classical inequalities due to Hoeffding, Bernstein, Rosenthal, McDiarmid, and more.

Multivariate approximation in total variation
Andrew Barbour, The University of Melbourne, Australia

When approximating the distribution of an integer valued random variable, it is natural to try to measure the error with respect to total variation distance. For Poisson approximation, this has been extraordinarily successful. In recent years, it has also been shown that approximation in total variation can be established in many circumstances in which a normal approximation is good, strengthening the usual results based on Kolmogorov distance.

In our talks, we present a recipe for establishing total variation approximation in 2 or more dimensions, using the discrete normal family, and illustrate it in the context of Stein's method of exchangeable pairs.
Joint work with Malwina Luczak

Bounds to the normal for group sequential statistics with covariates
Jay Bartroff, University of Southern California, USA

Group sequential methods, in which accumulating data is evaluated intermittently in stages, are the dominant statistical approach in biomedical clinical trials, among other applications. The limiting joint multivariate normal distribution of certain statistics, such as log-likelihood ratios or MLEs, at each analysis is widely used to compute critical values and p-values, although there are no existing explicit bounds for this approximation. We apply existing multivariate Berry-Esseen bounds based on Stein's method to the joint distribution of MLEs of a vector parameter at each analysis to obtain bounds to the normal. Our setting is a general parametric regression setup which allows the i-th observation to be the i-th subject's response regressed on their covariates.

Approximating the CLT using Stein's method
Ben Berckmoes, University of Antwerp, Belgium

In this talk, we establish an approximate CLT as a new tool to study the asymptotic behavior of the sample mean estimator based on observations which are subject to certain mechanisms of contamination. Stein's method plays a key role in the proof of the approximate CLT.
(joint work with Bob Lowen, Geert Molenberghs and Jan Van Casteren)

Stein's method for steady-state diffusion approximation of waiting times in a G/G/1 queue
Anton Braverman, Cornell University, USA

We consider the problem of a single-server queue with general customer arrival process and general service time distribution. We show that as traffic intensity to the queue approaches one, the steady-state customer waiting time can be approximated by an exponential distribution. Based on joint work with Jim Dai.

Monotone couplings: Stein's method and beyond
Fraser Daly, Heriot-Watt University, UK

We use stochastic orderings to describe dependence structures that are of interest in many settings, and examine how these descriptions can be used to derive inequalities using Stein's method and in other areas of probability.

Our first example is related to the Poisson distribution, where we formulate a negative dependence assumption related to the monotone couplings and negative relation that play a crucial role in Stein's method for Poisson approximation. We show how this negative dependence assumption also leads to bounds on the Poincaré (inverse spectral gap) constant and concentration inequalities. Several generalizations and extensions will also be discussed.

Finally, we use the same techniques to derive bounds in approximation by geometric sums, with applications to queueing models and birth-death processes. Here, our stochastic ordering condition corresponds to a lower bound on the hazard rate of the distribution of interest.

Parts of this talk are based on joint work with Oliver Johnson, Claude Lefèvre and Sergey Utev.

The Stein-Dirichlet-Malliavin method and applications to stochastic geometry
Laurent Decreusefond, Paris Telecom, France

We show how Malliavin calculus can enhance the general procedure of the Stein's method on arbitrary probability spaces. The versatility of this new approach is illustrated on two examples from stochastic geometry: superposition of determinantal point processes and Poisson polytopes.
This talk is based on arXiv:1207.3517, arXiv:1406.5484 and some new results still under development.

Stein's method for alpha-stable distributions
Partha Dey, University of Illinois at Urbana-Champaign, USA

Stein's method is a semi-classical tool for establishing distributional convergence, particularly effective in problems involving complex dependencies. We show how to use the method to find rate of convergence to alpha-stable distributions where the second moment is infinite. The main ingredient being analysis of certain non-local operators arising out of SDEs driven by Levy processes.

Error bounds for the random sums CLT by Stein's method of normal approximation
Christian Döbler, University of Luxembourg, Luxembourg

We use common probabilistic tools to derive competitive error bounds in the CLT for non-randomly centered random sums using the framework of Stein's method of normal approximation. When specializing to popular distributions of the summation index as the Poisson or Binomial, our bounds turn out to yield the best known if not optimal rates of convergence.

Malliavin-Stein method for Variance-Gamma approximation on Wiener space
Peter Eichelsbacher, Ruhr-University of Bochum, Germany

In a joint work with Christoph Th\"ale, we combine Malliavin calculus with Stein's method to derive bounds for the Variance-Gamma approximation of functionals of isonormal Gaussian processes, in particular of random variables living inside a fixed Wiener chaos induced by such a process. The bounds are presented in terms of Malliavin operators and norms of contractions. We show that a sequence of distributions of random variables in the second Wiener chaos converges to a Variance-Gamma distribution if and only if their moments of order two to six converge to that of a Variance-Gamma distributed random variable (six moment theorem). Moreover, simplified versions for Laplace or symmetrized Gamma distributions are presented. Also multivariate extensions and a universality result for homogeneous sums are considered.

Poisson approximation for two scan statistics with rates of convergence
Xiao Fang, National University of Singapore

As an application of Stein's method for Poisson approximation, we prove rates of convergence for the tail probabilities of two scan statistics that have been suggested for detecting local signals in sequences of independent random variables subject to possible change-points. Our formulation deals simultaneously with ordinary and with large deviations. This is joint work with David Siegmund.

Conditional distribution approximation with birth death processes
Han Liang Gan, Washington University in St. Louis, USA

In this talk, we will focus on approximating the distribution of the number of rare events given at least 1 rare event has occurred, and to this end, formulate Stein's method for conditional distribution approximation and discuss how this problem relates to birth death processes. We will also discuss conditional point process approximation, and the difficulties that generalising to a spatial point process generates. This is joint work with Aihua Xia.

Random graphs, the Chen-Stein method and networking applications
Ayalvadi Ganesh, University of Bristol, UK

Communication and computer networks are large, complex and geographically distributed. It is often necessary to perform tasks such as routing or resource allocation based only on limited, local knowledge of the network. This motivates an interest in decentralised algorithms and their analysis. For the purposes of this analysis, the network is often treated as a random graph. The objective is to develop simple algorithms whose performance is, in a quantifiable sense, close to optimal. We illustrate the use of random graphs models and their analysis with two examples.

First, we consider a small world network model in which nodes are uniformly distributed over the unit square, and connected to other nodes lying within a certain range; in addition, each node also has a number of randomly chosen long-range contacts. We study the connectivity properties of this model. Second, we again consider nodes uniformly distributed on the unit square. Now, the problem is to colour the nodes so as to maximise the minimum distance between nodes assigned the same colour. We analyse two simple algorithms for this problem, namely random colouring and greedy colouring. In both examples, the Chen-Stein method provides sharp estimates of some of the probabilities arising in the analysis.

Rates of convergence for multivariate normal approximations by Stein's method
Robert Gaunt, University of Oxford, UK

In this talk, we see how Stein?s method for normal approximation can be extended relatively easily to the approximation of statistics that are asymptotically distributed as functions of multivariate normal random variables. We obtain some general bounds and a surprising result regarding the rate of convergence. We end with an application to the rate of convergence of Pearson's chi-square statistic.

Part of this talk is based on joint work with Alastair Pickett and Gesine Reinert.

Progress using Stein's method for strong embeddings
Larry Goldstein, University of Southern California, USA

The celebrated KMT strong embedding of Koml\'os, Major and Tusn\'ady gives conditions under which the partial sum process of n mean zero, variance one, i.i.d. random variables and a Brownian motion evaluated at integer time points up to n can be closely coupled. By using arguments based on Stein kernels that are less technically difficult than those previously used in this area, Chatterjee arxiv:0711.0501 provides a proof of the KMT embedding in the case where the summand variables take the values +1,-1 with equal probability. We extend the method developed there to the case where the variables take any values in a fixed finite set not containing zero and have vanishing third moment.

This work is joint with Chinmoy Bhattacharjee (USC, Los Angeles)

Some applications of Stein's method to number theory
Adam Harper, Jesus College, UK

There is a very rich interplay between probability and number theory, both in model problems and heuristics, and in solving number-theoretic problems by reformulating them in terms of random variables on suitable spaces. One often ends up with random variables that are "almost" independent, and in modern problems these random variables can be high dimensional and very precise information about them is required.

I will survey work of myself and others where Stein's method has been applied to number theory. For example, I will try to mention work on the Erd\H{o}s--Kac central limit theorem, on random multiplicative functions, on prime number races, and on the behaviour of the Riemann zeta function on the critical line. In most of these areas there remain interesting open problems that might be attacked with Stein's method.

Stein's method and subsequence problems
Christian Houdré, Georgia Institute of Technology, USA

I will first describe some recent work of \"Umit I\c{s}lak and myself showing the normal convergence of the length of the longest common subsequences (LCS) of two random words. The methodology of proof relies on Stein's method and thus provides at once a (suboptimal) rate of convergence. However, the main achievement is in the convergence result itself which answers a long standing open question.

I will then present a few subsequence comparison problems (LCS of two independent uniform random permutations, longest increasing subsequences in random words, ...) where convergence towards (new) laws related to random matrices occurs. In these problems, rates are often unknown or suboptimal and this certainly motivates the need for the exploration of Stein's methodology in these contexts.

Gaussian and bootstrap approximations to suprema of empirical processes
Kengo Kato, University of Tokyo, Japan

We present a new approach to Gaussian and Bootstrap approximations of suprema of empirical processes indxed by infinite classes of functions. These approximation results depend essentially on Stein's method and Gaussian comparison inequalities, and the approach is different from the Hungarian type approximation. We illustrate applications of these approximations to nonparametric statistical inference (and some other statistical problems). This talk is based on joint work with Victor Chernozhukov and Denis Chetverikov.

A second order Poincar\'e inequality for functionals of general Poisson processes
Gunter Last, Karlsruhe Institute of Technology, Germany

In this talk we consider square-integrable functionals of a general Poisson process. We start with a covariance identity based on a Mehler-type formula for the (abstract) Ornstein-Uhlenbeck semigroup. We will then combine this tool with seminal results by Peccati, Sol\'e, Taqqu and Utzet (2010) (obtained by a combination of Stein's method and Malliavin calculus), to derive Berry-Esseen bounds for the normal approximation of Poisson functionals. These bounds do only involve the first two difference operators of the functional. Chatterjee (2009) was the first who established such a second order Poincar\'e inequality in a Gaussian setting. In the second part of the talk we will present several applications to stochastic geometry. In particular we plan to discuss the Poisson Voronoi tessellation, Gilbert and nearest neighbour graphs, and the random connection model. In all these cases the obtained rates are optimal. A significant part of this talk is based on joint work with Giovanni Peccati and Matthias Schulte.

On bounds in multivariate Poisson approximations
Tran Loc Hung, University of Finance and Marketing, Vietnam

The main purpose of this paper is to establish some bounds in multivariate Poisson approximation for random sums of independent random vectors via a probability distance.

Keywords: Random sum, d-dimensional random variables, Poisson approximation, Integer-valued random variable, Poisson-binomial random variable, Probability distance, Linear operator.

Mathematics Subject Classification : 60B12, 60F05, 60G50.

On the probability approximation of spatial point processes
Giovanni Luca Torrisi, Consiglio Nazionale delle Ricerche, Italy

We give quantitative Gaussian and Poisson approximations for the innovation of a spatial point process with conditional intensity. To this aim we use a Malliavin-Stein-Chen approach. This method is based on the following two steps. First, we provide an integration by parts (IBP) formula which relates the innovation with the discrete Malliavin derivative. Second, we derive the probability approximations by combining the IBP formula with the Stein and Chen methods. Our findings extend the corresponding results on the Poisson space due to Peccati, Sole, Taqqu and Utzet (2011) and Peccati (2011). The general formulas presented in this talk are applied to stationary, inhibitory and finite range Gibbs point processes with pair potential, providing explicit error bounds and quantitative limit theorems.

Stein's method and the many-worlds interpretation of quantum mechanics
Ian McKeague, Columbia University, USA

It has been conjectured that quantum effects arise from the interaction of finitely many classical "worlds". The wave function is then thought to be recoverable from observations of particles in these worlds, without knowing the world from which any particular observation originates. In this talk we discuss how Stein's method can be used to obtain such a result as the number of worlds goes to infinity. We examine the ground-state configuration of a parabolic potential well, and show that the sequential bootstrap of the mean particle location converges to an Ornstein-Uhlenbeck process, thus matching the ground-state solution of Schroedinger's equation in this case. This is joint work with Bruce Levin.

Steining the steiner formula
Ivan Nourdin, Université du Luxembourg, Luxembourg

Recent empirical research indicates that many convex optimization problems with random constraints exhibit a phase transition as the number of those constraints increases. In this talk, we will explicitly connect this phase transition with the asymptotic Gaussian fluctuations of the intrinsic volumes of the descent cone that is canonically attached to the convex optimization problem at hand. Our approach will be based on a variety of techniques, including (1) Steiner formulae for closed convex cones, (2) Stein's method and second order Poincaré inequality, (3) concentration estimates, and (4) Fourier analysis. This is a joint work with Larry Goldstein (Univ. of Southern California) and Giovanni Peccati (Univ. of Luxembourg).

Concentration inequalities by Stein couplings
Daniel Paulin, National University of Singapore

We show some new concentration inequalities for sums of dependent random variables using Stein couplings. Applications are given to the number of isolated vertices in Erdos-Renyi random graphs, the number of edges in geometric random graphs, and randomly chosen large subgraphs of huge fixed graphs.

Beating log-Sobolev, one Stein's kernel at a time
Giovanni Peccati, University of Luxembourg, Luxembourg

I will present a new set of functional inequalities involving the following four parameters associated with a given multidimensional distribution: the relative entropy, the relative Fisher information, the 2-Wasserstein distance, and the Stein discrepancy (which naturally appears within the framework Stein's method for normal approximations). Our results improve the classical log-Sobolev and Talagrand's transport inequalities, and provide new key tools in order to deal with high-dimensional quantitative central limit theorems on a Gaussian space. Some open problems will be discussed in the last part of the talk. Joint works with M. Ledoux (Toulouse), I. Nourdin (Luxembourg) and Y. Swan (Liège).

Coupling the binary tree with continuum random tree and more
Erol Peköz, Boston University, USA

Some progress on obtaining rates of convergence for the uniform binary plane tree to the continuum random tree using coupling will be discussed. Also, normal approximation using biasing will be discussed.

Isolated points in the random connection model
Mathew Penrose, University of Bath, UK

The soft random geometric graph has n vertices placed at random in a region of Euclidean space and each pair of points connected with a probability that
depends on how far apart they are. We discuss Poisson approximation for the number of isolated vertices in this model as n becomes large with the connection function function varying with n. One reason to be interested is because absence of isolated points is necessary for connectivity and is often asymptotically sufficient, in probability.

Mixing times for abelian sandpiles
John Pike, Cornell University, USA

I will discuss recent work with Dan Jerison and Lionel Levine on convergence rates for a Markov chain related to sandpile groups of simple connected graphs. One of the primary aims of this talk is to give an introduction to the abelian sandpile model, which is a source of many intriguing questions that may be amenable to Stein's method techniques.

On the use of Markov triplets for limit theorems and moment inequalities
Guillaume Poly, Université de Rennes, France

We will make an exposition of recent results of probabilistic approximation obtained by spectral theory of Markov operators. We will discuss the existing links with various correlation inequalities,asymptotic independence, U-conjecture and the polarization constant problem.

Stein approximation for Ito and Skorohod integrals by Edgeworth type expansions
Nicolas Privault, Nanyang Technological University

In this talk we will derive Edgeworth-type expansions for Skorohod and Ito integrals with respect to Brownian motion, based on cumulant operators defined by the Malliavin calculus. As a consequence we will obtain Stein approximation bounds for stochastic integrals, which apply to SDE solutions and to multiple stochastic integrals.

Multivariate normal approximation: permutation statistics, local dependence and beyond
Martin Raic, University of Ljubljana and University of Primorska, Slovenia

This talk will mainly discuss two classical topics which have been extensively studied: statistics based on random permutations (so called combinatorial limit theorem, double-indexed permutation statistics etc.) and dependence structures that can be described in terms of dependence graphs. We shall bound the error in the multivariate CLT expressed by probabilities of convex sets; an optimal rate of convergence will be derived in terms of third moments.

Both results appear as a special case of a general result, which is stated for a framework based on random vector measures and generalized Palm distributions. The former can be considered as generalizations of sums of random variables, while the latter can be thought of as conditional distributions. This allows a possibility to apply the result in more advanced set-ups, such as functionals on various random processes.

Bounds with data and an almost sure central limit theorem using Stein's method
Gesine Reinert, University of Oxford, UK

Stein's method lends itself to explicit bounds for distributional comparisons where the bounds, for Lipschitz-continuous test functions, may depend on the data in an explicit way. In this talk we shall discuss two examples. Firstly we can bound the distance between model and posterior in Bayesian analysis , and secondly we give a proof of the almost sure central limit theorem by Lacey and Philipp.

This is joint work with Christophe Ley (Brussels) and Yvik Swan (Liege) for the first example, and with George Deligiannidis (Oxford) and Larry Goldstein (Los Angeles) for the second example.

Limit theorems for Poisson U-statistics
Matthias Reitzner, University of Osnabruck, Germany

Assume that $\eta$ is a Poisson point process. A Poisson U-statistic with kernel $f$ is the sum of $f(x_1, \dots, x_k)$ over all $k$-tuples of $\eta$. Since many interesting functionals in Stochastic Geometry can be written as U-statistics they play an important role. We investigate some properties of Poisson U-statistics, and the limit behaviour of Poisson U-statistics, in particular concentration inequalities.

Stein couplings for normal approximation
Adrian Roellin, National University of Singapore

We present and discuss some couplings used for normal approximation with Stein's method that do not fit into the framework of well-known couplings, such as exchangeable pairs, local approach and size-biasing. We also show how these new couplings can be used to obtain elegant variance asymptotics. The aim of this work (which is joint with Louis Chen) is to broaden the applicability of Stein's method.

Distributional limits for (i) a new family of Polya urn models with immigration or (ii) preferential attachment networks where entering nodes having a random number of initial edges
Nathan Ross, The University of Melbourne, Australia

An urn starts with b black and w white balls. At discrete time steps, a ball is drawn and returned to the urn along with another of the same color. In addition, at random times a black ball is added (immigrated) to the urn; the number of draws between these immigrations are iid. As the number of draws goes to infinity, what is the behavior of the number of white balls in the urn? In this talk we discuss how to answer this question using Stein's method techniques, in particular identifying the limits as unique fixed points of (probabilistic) distributional fixed point equations. We also show how this process and these results are related to a preferential attachment random graph process where entering nodes have a random number of initial edges.

Joint work with Erol Pekoz and Adrian Roellin.

On the weak convergence of Poisson-Mixture sums via Stein's method
Uwe Schmock, Vienna University of Technology, Austria

We study the weak limit behavior of random sums of independent random variables, where the number of summands has a Poisson-mixture distribution. Since we consider Poisson-mixture sums, we are able to analyze the weak limit behavior of a whole class of random series at once. To this end we derive a conditional version of Stein's equation and utilize techniques established in theory of Stein's method. We prove the convergence of this class to appropriate normal variance mixture distributions, and we derive their Wasserstein as well as Kolmogorov distance. Furthermore, we give applications and discuss our results.
(Based on joint work with Peter Eichelsbacher and Piet Porkert.)

Stein's method for Gibbs point process approximation
Dominic Schuhmacher, Georg-August-Universität Göttingen, Germany

Finite Gibbs point processes are flexible models in spatial statistics. We develop Stein's method for total variation approximation by a Gibbs process satisfying a certain stability condition. For this Barbour's generator approach is used. The Stein factors are obtained by constructing an explicit coupling of two spatial birth-death processes with identical transition kernels but started at different configurations. The Stein factors can then be bounded by the expected coupling time, which in turn can be bounded by the hitting time at zero of a non-spatial birth-death process.

We obtain upper bounds of the form of an explicit constant times an L_1-type distance between the conditional intensities of the point processes. As applications we consider the total variation distance between two pairwise interaction processes and the hard core process approximation of an area-interaction process with small interaction parameter.

Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry
Matthias Schulte, Karlsruhe Institute of Technology, Germany

A Poisson or a binomial process on an abstract state space and a symmetric function $f$ acting on $k$-tuples of its points are considered. They induce a point process on the target space of $f$. The main result is a functional limit theorem which provides an upper bound for the Kantorovich-Rubinstein distance between the image process and a Poisson process on the target space. The technical background are a version of Stein's method for Poisson process approximation, a Glauber dynamic representation for the Poisson process and the Malliavin formalism. As applications of the main result, error bounds for approximations of U-statistics by Poisson, compound Poisson and stable random variables are derived and examples from stochastic geometry are investigated. This is joint work with Laurent Decreusefond and Christoph Th\"ale.

Cramer type moderate deviations by Stein's method
Qi-Man Shao, Chinese University of Hong Kong, Hong Kong

This talk will give a brief survey on the recent developments of Cramer type moderate deviations by Stein's method. The focus will be on non-normal approximation. Applications and future problems will also be discussed.

A canonical approach to Stein's density approach
Yvik Swan, Université de Liège, Belgium

We propose a canonical definition of the Stein operator and Stein class of a distribution. The resulting Stein identity highlights the unifying theme behind the literature on Stein's method. Multivariate extensions will be discussed.

On Stein operators for discrete approximations
Neelesh Upadhye, Indian Institute of Technology Madras, India

In this talk, we discuss a new method based on probability generating functions, to derive Stein operators for discrete approximations. This method is then used to obtain multiple Stein operators for various random variables closely related to Poisson, binomial and negative binomial distributions. A well-known perturbation approach for Stein's method is used to obtain total variation bounds for the distributions discussed above. The importance of such approximations is illustrated, for example, by the binomial convoluted with Poisson approximation to sums of independent and dependent indicator random variables.

Multivariate approximation in total variation
Aihua Xia, The University of Melbourne, Australia

When approximating the distribution of an integer valued random variable, it is natural to try to measure the error with respect to total variation distance. For Poisson approximation, this has been extraordinarily successful. In recent years, it has also been shown that approximation in total variation can be established in many circumstances in which a normal approximation is good, strengthening the usual results based on Kolmogorov distance.

In our talks, we present a recipe for establishing total variation approximation in 2 or more dimensions, using the discrete normal family, and illustrate it in the context of Stein's method of exchangeable pairs.
Joint work with Malwina Luczak

Stein's method and functionals of convex hulls
Joe Yukich, Lehigh University, USA

It is well-known that Stein's method yields rates of normal convergence for k-face functionals and volumes of convex hulls of i.i.d. uniform samples as well as convex hulls of Gaussian samples. In this talk we interpret functionals of convex hulls as functionals of a non-linear transform of the convex hull. The non-linear transform has some magical properties:
(i) it sends functionals of the convex hull into sums of stabilizing functionals which are shown to be asymptotically normal by Stein's method,
(ii) it yields a functional CLT for defect volumes of convex hulls,
(iii) it produces a scaling limit of the boundary of convex hulls, and
(iv) it provides variance asymptotics for the k-face and volume functional of the convex hull, resolving an open problem.
This is based on joint work with Pierre Calka.

Best viewed with IE 7 and above