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2016 | Buch

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High Dimensional Probability VII

The Cargèse Volume

herausgegeben von: Christian Houdré, David M. Mason, Patricia Reynaud-Bouret, Jan Rosiński

Verlag: Springer International Publishing

Buchreihe : Progress in Probability

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This volume collects selected papers from the 7th High Dimensional Probability meeting held at the Institut d'Études Scientifiques de Cargèse (IESC) in Corsica, France.

High Dimensional Probability (HDP) is an area of mathematics that includes the study of probability distributions and limit theorems in infinite-dimensional spaces such as Hilbert spaces and Banach spaces. The most remarkable feature of this area is that it has resulted in the creation of powerful new tools and perspectives, whose range of application has led to interactions with other subfields of mathematics, statistics, and computer science. These include random matrices, nonparametric statistics, empirical processes, statistical learning theory, concentration of measure phenomena, strong and weak approximations, functional estimation, combinatorial optimization, and random graphs.

The contributions in this volume show that HDP theory continues to thrive and develop new tools, methods, techniques and perspectives to analyze random phenomena.

Inhaltsverzeichnis

Frontmatter

Inequalities and Convexity

Frontmatter

Stability of Cramer’s Characterization of Normal Laws in Information Distances

Abstract

Optimal stability estimates in the class of regularized distributions are derived for the characterization of normal laws in Cramer’s theorem with respect to relative entropy and Fisher information distance.

Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

V.N. Sudakov’s Work on Expected Suprema of Gaussian Processes

Abstract

It is noted that the late Volodya N. Sudakov (1934–2016) first published a statement in 1973 and proof in 1976 that the expected supremum of a centered Gaussian process is bounded above by a constant times a metric entropy integral. In particular, the present author (R.M. Dudley) defined such an integral but did not state nor prove such a bound.

Richard M. Dudley

Optimal Concentration of Information Content for Log-Concave Densities

Abstract

An elementary proof is provided of sharp bounds for the varentropy of random vectors with log-concave densities, as well as for deviations of the information content from its mean. These bounds significantly improve on the bounds obtained by Bobkov and Madiman (Ann Probab 39(4):1528–1543, 2011).

Matthieu Fradelizi, Mokshay Madiman, Liyao Wang

Maximal Inequalities for Dependent Random Variables

Abstract

Maximal inequalities play a crucial role in many probabilistic limit theorem; for instance, the law of large numbers, the law of the iterated logarithm, the martingale limit theorem and the central limit theorem. Let X ₁, X ₂, … be random variables with partial sums S _k = X ₁ + ⋯ + X _k. Then a maximal inequality gives conditions ensuring that the maximal partial sum M _n = max_{1 ≤ i ≤ n} S _i is of the same order as the last sum S _n. In the literature there exist large number of maximal inequalities if X ₁, X ₂, … are independent but much fewer for dependent random variables. In this paper, I shall focus on random variables X ₁, X ₂, … having some weak dependence properties; such as positive and negative In-correlation, mixing conditions and weak martingale conditions.

Jørgen Hoffmann-Jørgensen

On the Order of the Central Moments of the Length of the Longest Common Subsequences in Random Words

Abstract

We investigate the order of the r-th, 1 ≤ r < +∞, central moment of the length of the longest common subsequences of two independent random words of size n whose letters are identically distributed and independently drawn from a finite alphabet. When all but one of the letters are drawn with small probabilities, which depend on the size of the alphabet, a lower bound is shown to be of order n ^r∕2. This result complements a generic upper bound also of order n ^r∕2.

Christian Houdré, Jinyong Ma

A Weighted Approximation Approach to the Study of the Empirical Wasserstein Distance

Abstract

We shall demonstrate that weighted approximation technology provides an effective set of tools to study the rate of convergence of the Wasserstein distance between the cumulative distribution function [c.d.f] and the empirical c.d.f.

David M. Mason

On the Product of Random Variables and Moments of Sums Under Dependence

Abstract

In this paper we compare the moments of products of dependent random vectors with the corresponding ones of independent vectors with the same marginal distributions. Various applications of this result are pointed out, including inequalities for the maximum of dependent random variables and moments of partial sums. The inequalities involve the generalized phi-mixing coefficient.

Magda Peligrad

The Expected Norm of a Sum of Independent Random Matrices: An Elementary Approach

Abstract

In contemporary applied and computational mathematics, a frequent challenge is to bound the expectation of the spectral norm of a sum of independent random matrices. This quantity is controlled by the norm of the expected square of the random matrix and the expectation of the maximum squared norm achieved by one of the summands; there is also a weak dependence on the dimension of the random matrix. The purpose of this paper is to give a complete, elementary proof of this important inequality.

Joel A. Tropp

Fechner’s Distribution and Connections to Skew Brownian Motion

Abstract

This note investigates two aspects of Fechner’s two-piece normal distribution: (1) connections with the mean-median-mode inequality and (strong) log-concavity; (2) connections with skew and oscillating Brownian motion processes. The developments here have been inspired by Wallis (Stat Sci 29:106–112, 2014) and rely on Chen and Zili (Sci China Math 58:97–108, 2015).

Jon A. Wellner

Limit Theorems

Frontmatter

Erdős-Rényi-Type Functional Limit Laws for Renewal Processes

Abstract

We prove functional limit laws for Erdős-Rényi-type increments of renewal processes.

Paul Deheuvels, Joseph G. Steinebach

Limit Theorems for Quantile and Depth Regions for Stochastic Processes

Abstract

Since contours of multi-dimensional depth functions often characterize the distribution, it has become of interest to consider structural properties and limit theorems for the sample contours [see Zuo and Serfling (Ann. Stat. 28(2):483–499, 2000) and Kong and Mizera (Stat. Sin. 22(4):1589–1610, 2012)]. In particular, Kong and Mizera have shown that for finite dimensional data, directional quantile envelopes coincide with the level sets of half-space (Tukey) depth. We continue this line of study in the context of functional data, when considering analogues of Tukey’s half-space depth (Tukey, Mathematics and the picturing of data, in Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974), vol. 2 (Canadian Mathematical Congress, Montreal, QC, 1975), pp. 523–531). This includes both a functional version of the equality of (directional) quantile envelopes and quantile regions as well as limit theorems for the sample quantile regions up to $\sqrt{n}$ asymptotics.

James Kuelbs, Joel Zinn

In Memory of Wenbo V. Li’s Contributions

Abstract

Wenbo V. Li was Professor of Mathematical Sciences at the University of Delaware. He died of a heart attack in January 2013. Wenbo made significant contributions to many of the areas in which he worked, especially to small value probability estimates. This note is a brief survey of Wenbo V. Li’s contributions, as well as discussion of the open problems he posed or found of interest.

Qi-Man Shao

Stochastic Processes

Frontmatter

Orlicz Integrability of Additive Functionals of Harris Ergodic Markov Chains

Abstract

For a Harris ergodic Markov chain (X _n)_n ≥ 0, on a general state space, started from the small measure or from the stationary distribution, we provide optimal estimates for Orlicz norms of sums ∑ _i = 0 ^τ f(X _i), where τ is the first regeneration time of the chain. The estimates are expressed in terms of other Orlicz norms of the function f (with respect to the stationary distribution) and the regeneration time τ (with respect to the small measure). We provide applications to tail estimates for additive functionals of the chain (X _n) generated by unbounded functions as well as to classical limit theorems (CLT, LIL, Berry-Esseen).

Radosław Adamczak, Witold Bednorz

Bounds for Stochastic Processes on Product Index Spaces

Abstract

In this paper we discuss the question of how to bound the supremum of a stochastic process with an index set of a product type. It is tempting to approach the question by analyzing the process on each of the marginal index sets separately. However it turns out that it is necessary to also study suitable partitions of the entire index set. We show what can be done in this direction and how to use the method to reprove some known results. In particular we point out that all known applications of the Bernoulli Theorem can be obtained in this way. Moreover we use the shattering dimension to slightly extend the application to VC classes. We also show some application to the regularity of paths of processes which take values in vector spaces. Finally we give a short proof of the Mendelson–Paouris result on sums of squares for empirical processes.

Witold Bednorz

Permanental Vectors and Selfdecomposability

Abstract

Exponential variables, gamma variables or squared centered Gaussian variables, are always selfdecomposable. Does this property extend to multivariate gamma distributions? We show here that for any d-dimensional centered Gaussian vector (η ₁, …, η _d) with a nonsingular covariance, the vector (η ₁ ², …, η _d ²) is not selfdecomposable unless its components are independent. More generally, permanental vectors with nonsingular kernels are not selfdecomposable unless their components are independent.

Nathalie Eisenbaum

Permanental Random Variables, M-Matrices and α-Permanents

Abstract

We explore some properties of a recent representation of permanental vectors which expresses them as sums of independent vectors with components that are independent gamma random variables.

Michael B. Marcus, Jay Rosen

Convergence in Law Implies Convergence in Total Variation for Polynomials in Independent Gaussian, Gamma or Beta Random Variables

Abstract

Consider a sequence of polynomials of bounded degree evaluated in independent Gaussian, Gamma or Beta random variables. We show that, if this sequence converges in law to a nonconstant distribution, then (1) the limit distribution is necessarily absolutely continuous with respect to the Lebesgue measure and (2) the convergence automatically takes place in the total variation topology. Our proof, which relies on the Carbery–Wright inequality and makes use of a diffusive Markov operator approach, extends the results of Nourdin and Poly (Stoch Proc Appl 123:651–674, 2013) to the Gamma and Beta cases.

Ivan Nourdin, Guillaume Poly

High Dimensional Statistics

Frontmatter

Perturbation of Linear Forms of Singular Vectors Under Gaussian Noise

Abstract

Let $A \in \mathbb{R}^{m\times n}$ be a matrix of rank r with singular value decomposition (SVD) A = ∑ _k = 1 ^r σ _k(u _k ⊗ v _k), where {σ _k, k = 1, …, r} are singular values of A (arranged in a non-increasing order) and $u_{k} \in \mathbb{R}^{m},v_{k} \in \mathbb{R}^{n},k = 1,\ldots,r$ are the corresponding left and right orthonormal singular vectors. Let $\tilde{A} = A + X$ be a noisy observation of A, where $X \in \mathbb{R}^{m\times n}$ is a random matrix with i.i.d. Gaussian entries, $X_{ij} \sim \mathcal{N}(0,\tau ^{2}),$ and consider its SVD $\tilde{A} =\sum _{ k=1}^{m\wedge n}\tilde{\sigma }_{k}(\tilde{u}_{k} \otimes \tilde{ v}_{k})$ with singular values $\tilde{\sigma }_{1} \geq \ldots \geq \tilde{\sigma }_{m\wedge n}$ and singular vectors $\tilde{u}_{k},\tilde{v}_{k},k = 1,\ldots,m \wedge n.$ The goal of this paper is to develop sharp concentration bounds for linear forms $\langle \tilde{u}_{k},x\rangle,x \in \mathbb{R}^{m}$ and $\langle \tilde{v}_{k},y\rangle,y \in \mathbb{R}^{n}$ of the perturbed (empirical) singular vectors in the case when the singular values of A are distinct and, more generally, concentration bounds for bilinear forms of projection operators associated with SVD. In particular, the results imply upper bounds of the order $O{\biggl (\sqrt{\frac{\log (m+n)} {m\vee n}} \biggr )}$ (holding with a high probability) on

$$\displaystyle{\max _{1\leq i\leq m}\big\vert \big <\tilde{ u}_{k} -\sqrt{1 + b_{k}}u_{k},e_{i}^{m}\big >\big \vert \ \ \mathrm{and}\ \ \max _{ 1\leq j\leq n}\big\vert \big <\tilde{ v}_{k} -\sqrt{1 + b_{k}}v_{k},e_{j}^{n}\big >\big \vert,}$$

where b _k are properly chosen constants characterizing the bias of empirical singular vectors $\tilde{u}_{k},\tilde{v}_{k}$ and {e _i ^m, i = 1, …, m}, {e _j ⁿ, j = 1, …, n} are the canonical bases of $\mathbb{R}^{m}, \mathbb{R}^{n},$ respectively.

Vladimir Koltchinskii, Dong Xia

Optimal Kernel Selection for Density Estimation

Abstract

We provide new general kernel selection rules thanks to penalized least-squares criteria. We derive optimal oracle inequalities using adequate concentration tools. We also investigate the problem of minimal penalty as described in Birgé and Massart (2007, Probab. Theory Relat. Fields, 138(1–2):33–73).

Matthieu Lerasle, Nelo Molter Magalhães, Patricia Reynaud-Bouret

Backmatter

Titel: High Dimensional Probability VII
herausgegeben von: Christian Houdré
David M. Mason
Patricia Reynaud-Bouret
Jan Rosiński
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-40519-3
Print ISBN: 978-3-319-40517-9
DOI: https://doi.org/10.1007/978-3-319-40519-3