Below are talk titles, slides, abstracts, and links to papers covering background material.
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Abstract:
With the goal of finding new and improved concentration inequalities for random variables that are threshold- and equality-type counts of multivariate occupancy models, we describe a general method for size biasing such models which produces bounded couplings whenever the model’s margin distributions are lattice log concave and satisfy some other weak conditions. Examples on which the method succeeds include degree counts in an Erdos-Renyi random graph, the number of neighbors and the volume covered by multi-way intersections in germ grain models from stochastic geometry, bin occupancy counts in the multinomial model, and population sizes under multivariate hypergeometric sampling. Via the work of Ghosh and Goldstein (2011) and Arratia and Baxendale (2014) relating bounded size-biased couplings to concentration inequalities, our couplings provide new concentration of measure results for the above models having the Poisson tail rate, many of which improve on those achieved by competing methods. We will compare our results with those achieved by McDiarmid’s inequality, negative association, and self bounding and certifiable functions. This work is joint with Larry Goldstein and Umit Islak.
Background paper:
- Bounded size biased couplings, log concave distributions and concentration of measure for occupancy models [ArXiv]
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Abstract:
The entropy method has proved to be a handy device to derive concentration inequalities for functions of independent random variables. So far, it has delivered the tightest general inequalities concerning suprema of bounded empirical processes, conditional Rademacher averages or self-bounded functionals like VC-entropies. The entropy method is made of two ingredients: the product property of entropy, variance, and intermediate Phi-entropies; Herbst’s argument which consists of taking advantage of a differential inequality satisfied by a logarithmic moment generating function.
It has also been pointed out that off-the-shelf exponential Efron-Stein inequalities delivered by the entropy method may not be as tight as they should, they may fail to capture the so-called super-concentration phenomenon.
We show that in the simplest setting of this phenomenon, suprema of Gaussian vectors, complementing the entropy method with appropriate representations provides a pedestrian derivation of tight “super”-concentration inequalities. The argument works for order statistics from log-concave distributions. It consists of combining Rényi’s representation for order statistics, log-concavity arguments and the entropy method.
Combining Renyi’s representation, Karamata’s representation for regularly varying functions and sharp concentration inequalities for smooth functions of independent exponentially distributed random variables (due to Talagrand, Maurey, Bobkov and Ledoux) allows us to derive concentration inequalities for tail index estimators that have been popular in Extreme Value Theory for decades. This provides a canvas for constructing adaptive tail index estimators (a notoriously difficult task).
The analysis of occupancy scores in finite and infinite urn schemes provides a setting where the product property of entropy/variance is challenged by negative association (or Poissonization) arguments, or by Stein’s method. We show how variants of the entropy method can be used to derive simple yet sharp concentration inequalities for occupancy scores and for the missing mass. Tail and moment bounds are benchmarked in the regular variation setting introduced by S. Karlin.
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Abstract:
The one-parameter family of Jack_α measures on partitions of n is an important discrete analog of Dyson’s β ensembles of random matrix theory. Except for α = ½, 1, 2, which have group theoretic interpretations, the Jack_α measure is difficult to analyze. In the case α = 1, the Jack measure agrees with the Plancherel measure on the irreducible representations of the symmetric group S_n, parametrized by the partitions of n. The normal approximation for the character ratio evaluated at the transposition (12) under the Plancherel measure has been well studied, notably by Fulman (2005, 2006) and Shao and Su (2006). A generalization of the character ratio under the Jack_ α measure has also been studied by Fulman (2004, 2006) and Fulman and Goldstein (2011). In this talk, we present results on both uniform and non-uniform error bounds on the normal approximation for the Jack_ α measure for α > 0. Our results improve those in the literature and come close to solving a conjecture of Fulman (2004). Our proofs use Stein’s method and zero-bias coupling. This talk is based on joint work with Le Van Thanh.
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Abstract:
For a nonnegative random variable Y with finite nonzero mean a, we say that Y^s has the Y-size bias distribution if
E[Yf(Y)] = a E[f(Y^s)]
for all smooth functions f. If Y can be coupled to Y^s having the Y-size bias distribution such that for some constant c we have Y^s < Y + c almost surely, then the deviations of Y from its mean satisfy a concentration of measure inequality of Poisson type. Applications of the method yield Poisson tail concentration results for the number of local maxima of a random function on a graph, urn occupancy statistics in multinomial allocation models, and the volume contained in k-way intersections of n balls placed uniformly over a volume n subset of d dimensional space. The two final examples are members of a class of occupancy models with log concave marginals for which bounded size bias couplings may be constructed more generally.
Similarly, concentration bounds can be shown using the zero bias coupling, that is, when one can construct a bounded coupling of a mean zero random variable Y with finite nonzero variance v^2 to a Y* satisfying
E[Yf(Y)] = v^2 E[f'(Y*)]
for all smooth f. Such couplings can be used to demonstrate concentration in Hoeffding’s combinatorial central limit theorem under diverse assumptions on the permutation distribution.
The bounds produced by these couplings may offer various improvements to those obtained using other methods available.
This work is joint with Jay Bartroff, Subhankar Ghosh and Umit Islak.Background papers:
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Abstract:
We develop connections between Stein’s approximation method, logarithmic Sobolev and transport inequalities by introducing a new class of functional inequalities involving the relative entropy, the Stein (factor or) matrix, the relative Fisher information and the Wasserstein distance with respect to a given reference distribution on ${\bf R}^d$. For the Gaussian model, these results improve upon the classical logarithmic Sobolev inequality and the Talagrand quadratic transportation cost inequality. As a by-product, they also produce bounds for normal entropic convergence expressed in terms of the Stein matrix. Further examples of illustrations include multidimensional gamma distributions, the uniform distribution on a compact interval, as well as families of log-concave densities. The tools rely on semigroup interpolation and bounds, in particular by means of the iterated gradients of the Markov generator with invariant measure the distribution under consideration. In a second part, motivated by the recent investigation by Nourdin, Peccati and Swan on Wiener chaoses, we address the issue of entropic bounds on multidimensional functionals $F$ with the Stein matrix via a set of data on $F$ and its gradients rather than on the Fisher information of the density. A natural framework for this investigation is given by the Markov Triple structure $(E, \mu, \Gamma)$ in which abstract Malliavin-type arguments may be developed and extend the Wiener chaos setting.
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Abstract:
The limit distributions of degree counts for proportional attachment models are unique fixed point of certain distributional transformations that are described in terms of size biasing and multiplying by mixtures of beta variables. From this fact, couplings can be used to obtain rates of convergence of degree counts to their distributional limits and properties of the limits can be read.
Background papers:
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Abstract:
We propose a canonical definition of the Stein operator of a univariate distribution through a skew-adjoint relationship for a differential operator or a difference operator. The resulting Stein identity not only comprises the known examples but also highlights the unifying theme as well as the flexibility in specifying the Stein identity. The identity naturally links in with a coupling approach, which we detail for sums of independent random variables. We apply our approach to the comparison of several pairs of distributions : normal vs normal, normal vs Student, maximum of random variables vs Gumbel and exponential, and beta-binomial vs Beta.
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Abstract:
The limiting distributions of degree counts in preferential attachment graphs can be understood through urn models. These limits also appear in random walks and trees and in this talk we make the connection to urns explicit as well as derive new results for all of these models.
Background paper:
- Generalized gamma approximation with rates for urns, walks and trees [ArXiv]
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Abstract:
We derive a simple representation for the limit distribution of the joint degree counts in proportional attachment random graphs and provide optimal rates of convergence to these limits. The results hold for models with any general initial seed graph and any fixed number of outgoing edges.
Background paper:
- Joint degree distributions of preferential attachment random graphs [ArXiv]
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Background papers:
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Abstract:
Given a random variable F regular enough in the sense of the Malliavin calculus, we measure the distance between its law and almost any continuous probability law on the real line. The bounds are given in terms of the Malliavin derivative of the random variable F and are based by the properties of the Ito diffussion which has as invariant measure the law considered. We also characterize the convergence in distribution of a sequence of random variables in a Wiener chaos of a fixed order to a probability distribution which is the invariant measure! of a diffusion process. This class of target distributions includes the most known continuous probability distributions. Our results are given in terms of the Malliavin calculus and of the coefficients of the associated diffusion process and extend the standard Fourth Moment Theorem by Nualart and Peccati.
Background papers: