Given two vectors in Euclidean space, how unlikely is it that a random vector has a larger inner product with the shorter vector than with the longer one? When the random vector has independent, identically distributed components, we conjecture that this probability is no more than a constant multiple of the ratio of the Euclidean norms of the two given vectors, up to an additive term to allow for the possibility that the longer vector has more arithmetic structure. We give some partial results to support the basic conjecture.
In this note, we obtain a Gaussian concentration inequality for a class of non-Lipschitz functions. In the one-dimensional case, our results supplement those established by Paouris and Valettas in [8].
We obtain a concentration inequality for the maximum degree of a vertex in a uniformly random dissection of a polygon. This resolves a conjecture posed by Curien and Kortchemski in 2012. Our approach is based on a bijection with dual trees and the tools of analytic combinatorics.
We present a proof of the concentration inequality for a discrete random surface model, where the underlying potential is perturbed by an additive random potential. The proof is based on annealing the random potential, and follows the method of [CEP96] and other works. Our result demonstrates the robustness of this method.
In this paper we revisit Talagrand's proof of concentration inequality for empirical processes. We give a different shorter proof of the main technical lemma that guarantees the existence of a certain kernel. Our proof provides the almost optimal value of the constant involved in the statement of this lemma.
The classical Gaussian concentration inequality for Lipschitz functions is adapted to a setting where the classical assumptions (i.e. Lipschitz and Gaussian) are not met. The theory is more direct than much of the existing theory designed to handle related generalizations. An application is presented to linear combinations of heavy tailed random variables.
Gaussian martingale inequality applies to random functions and maxima of empirical processes
得到了鞅的一个Bernstein型高斯浓度不等式.我们的不等式改进了中等偏差下的Azuma-Hoeffding不等式$x$ .继McDiarmid（1989），Talagrand（1996）和Boucheron，Lugosi and Massart（2000，2003）的工作之后，我们证明了我们的结果可以应用于随机函数的集中度，Erdös-Rényi随机图，以及经验过程的极大值.得到了几个有趣的高斯浓度不等式。
We derive concentration inequalities for empirical means $\frac{1}{t} \int_0^t f(X_s) ds$ where $X_s$ is an irreducible Markov jump process on a finite state space and $f$ some observable. Using a Feynman-Kac semigroup we first derive a general concentration inequality. Then, based on this inequality we derive further concentration inequalities. Hereby we use three different methods; perturbation theory, Poincaré inequalities and information inequalities. We also obtain a Bernstein type concentration inequality.
An inequality of K. Marton shows that the joint distribution of a Markov chain with uniformly contracting transition kernels exhibits concentration. We prove an analogous inequality for broadcast models on finite trees. We use this inequality to develop a condition for the sequence of depth-$k$ marginals of a broadcast model on a rooted infinite tree to form a normal Lévy family in terms of the Lipschitz constants of the transition kernels and the growth rate of the tree.
We give a concentration inequality based on the premise that random variables take values within a particular region. The concentration inequality guarantees that, for any sequence of correlated random variables, the difference between the sum of conditional expectations and that of the observed values takes a small value with high probability when the expected values are evaluated under the condition that the past values are known. Our inequality outperforms other well-known inequalities, e.g. the Azuma-Hoeffding inequality, especially in terms of the convergence speed when the random variabl…
This paper deduces exponential matrix concentration from a Poincaré inequality via a short, conceptual argument. Among other examples, this theory applies to matrix-valued functions of a uniformly log-concave random vector. The proof relies on the subadditivity of Poincaré inequalities and a chain rule inequality for the trace of the matrix Dirichlet form. It also uses a symmetrization technique to avoid difficulties associated with a direct extension of the classic scalar argument.
We present concentration inequalities on the multislice which are based on (modified) log-Sobolev inequalities. This includes bounds for convex functions and multilinear polynomials. As an application we show concentration results for the triangle count in the $G(n,M)$ Erdős--Rényi model resembling known bounds in the $G(n,p)$ case. Moreover, we give a proof of Talagrand's convex distance inequality for the multislice. Interpreting the multislice in a sampling without replacement context, we furthermore present concentration results for $n$ out of $N$ sampling without replacement. Based on a b…
We consider a generic modified logarithmic Sobolev inequality (mLSI) of the form $\mathrm{Ent}_μ(e^f) \le \tfracρ{2} \mathbb{E}_μe^f Γ(f)^2$ for some difference operator $Γ$, and show how it implies two-level concentration inequalities akin to the Hanson--Wright or Bernstein inequality. This can be applied to the continuous (e.\,g. the sphere or bounded perturbations of product measures) as well as discrete setting (the symmetric group, finite measures satisfying an approximate tensorization property, \ldots). Moreover, we use modified logarithmic Sobolev inequalities on the symmetric group $S…
We present a non-asymptotic concentration inequality for the random matrix product \begin{equation}\label{eq:Zn} Z_n = \left(I_d-αX_n\right)\left(I_d-αX_{n-1}\right)\cdots \left(I_d-αX_1\right), \end{equation} where $\left\{X_k \right\}_{k=1}^{+\infty}$ is a sequence of bounded independent random positive semidefinite matrices with common expectation $\mathbb{E}\left[X_k\right]=Σ$. Under these assumptions, we show that, for small enough positive $α$, $Z_n$ satisfies the concentration inequality \begin{equation}\label{eq:CTbound} \mathbb{P}\left(\left\Vert Z_n-\mathbb{E}\left[Z_n\right]\right\V…
We show that any probability measure satisfying a Matrix Poincaré inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carré du champ operator. This extends to the matrix setting a classical phenomenon in the scalar case. Moreover, the proof gives rise to new matrix trace inequalities which could be of independent interest. We then apply this general fact by establishing matrix Poincaré inequalities to derive matrix concentration inequalities for Gaussian measures, product measures and for Strong…
We give a concentration inequality for a stochastic version of the facility location problem. We show the objective $C_n = \min_{F \subseteq [0,1]^2}|F|+\sum_{x\in X}\min_{f\in F}\|x-f\|$ is concentrated in an interval of length $O(n^{1/6})$ and $\E[C_n]=Θ(n^{2/3})$ if the input $X$ consists of i.i.d. uniform points in the unit square. Our main tool is to use a geometric quantity, previously used in the design of approximation algorithms for the facility location problem, to analyze a martingale process. Many of our techniques generalize to other settings.
Recent development in high-dimensional statistical inference has necessitated concentration inequalities for a broader range of random variables. We focus on sub-Weibull random variables, which extend sub-Gaussian or sub-exponential random variables to allow heavy-tailed distributions. This paper presents concentration inequalities for independent sub-Weibull random variables with finite Generalized Bernstein-Orlicz norms, providing generalized Bernstein's inequalities and Rosenthal-type moment bounds. The tightness of the proposed bounds is shown through lower bounds of the concentration ineq…
Random union sets $Z$ associated with stationary Poisson processes of $k$-cylinders in $\mathbb{R}^d$ are considered. Under general conditions on the typical cylinder base a concentration inequality for the volume of $Z$ restricted to a compact window is derived. Assuming convexity of the typical cylinder base and isotropy of $Z$ a concentration inequality for intrinsic volumes of arbitrary order is established. A number of special cases are discussed, for example the case when the cylinder bases arise from a random rotation of a fixed convex body. Also the situation of expanding windows is st…
We show that the number of real roots of random trigonometric polynomials with i.i.d. coefficients, which are either bounded or satisfy the logarithmic Sobolev inequality, satisfies an exponential concentration of measure.
Inequalities between transportation costs and Fisher information are known to characterize certain concentration properties of Markov processes around their invariant measures. This note provides a new characterization of the quadratic transportation-information inequality $W_2I$ in terms of a dimension-free concentration property for i.i.d. (conditionally on the initial positions) copies of the underlying Markov process. This parallels Gozlan's characterization of the quadratic transportation-entropy inequality $W_2H$. The proof is based on a new Laplace-type principle for the operator norms …
We prove a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Working with bounded and $π$-canonical kernels, we show that we can recover the convergence rate of Arcones and Gin{é} who proved a concentration result for U-statistics of independent random variables and canonical kernels. Our result allows for a dependence of the kernels $h_{i,j}$ with the indexes in the sums, which prevents the use of standard blocking tools. Our proof relies on an inductive analysis where we use martingale techniques, uniform ergodicity, Nummelin splitting and Bernst…
The Entropy method provides a powerful framework for proving scalar concentration inequalities by establishing functional inequalities like Poincare and log-Sobolev inequalities. These inequalities are especially useful for deriving concentration for dependent distributions coming from stationary distributions of Markov chains. In contrast to the scalar case, the study of matrix concentration inequalities is fairly recent and there does not seem to be a general framework for proving matrix concentration inequalities using the Entropy method. In this paper, we initiate the study of Matrix value…
We explore the question whether Lipschitz functions of random variables under various forms of negative correlation satisfy concentration bounds similar to McDiarmid's inequality for independent random variables. We prove such a concentration bound for random variables satisfying the condition of negative regression, correcting an earlier proof by Dubhashi and Ranjan.
Initially motivated by the study of the non-asymptotic properties of non-parametric tests based on permutation methods, concentration inequalities for uniformly permuted sums have been largely studied in the literature. Recently, Delyon et al. proved a new Bernstein-type concentration inequality based on martingale theory. This work presents a new proof of this inequality based on the fundamental inequalities for random permutations of Talagrand. The idea is to first obtain a rough inequality for the square root of the permuted sum, and then, iterate the previous analysis and plug this first i…
We present novel martingale concentration inequalities for martingale differences with finite Orlicz-$ψ_α$ norms. Such martingale differences with weak exponential-type tails scatters in many statistical applications and can be heavier than sub-exponential distributions. In the case of one dimension, we prove in general that for a sequence of scalar-valued supermartingale difference, the tail bound depends solely on the sum of squared Orlicz-$ψ_α$ norms instead of the maximal Orlicz-$ψ_α$ norm, generalizing the results of Lesigne & Volný (2001) and Fan et al. (2012). In the multidimensional ca…
We prove concentration inequalities for $f\left( X\right) $ about its median, where $X$ is a random vector in $\mathbb{R}^n$ with independent heavy tailed coordinates of Weibull or power type, and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a locally Lipschitz function. This paper is part of a series of four papers, Part I, Part II and two supporting papers. It can be read independently of Part I.
Stability for the Brunn-Minkowski and Riesz rearrangement inequalities, with applications to Gaussian concentration and finite range non-local isoperimetry
We present an adaptation of the Kato--Temple inequality for bounding perturbations of eigenvalues with applications to statistical inference for random graphs, specifically hypothesis testing and change-point detection. We obtain explicit high-probability bounds for the individual distances between certain signal eigenvalues of a graph's adjacency matrix and the corresponding eigenvalues of the model's edge probability matrix, even when the latter eigenvalues have multiplicity. Our results extend more broadly to the perturbation of singular values in the presence of quite general random matrix…
