One-sided concentration near the mean of log-concave distributions
Iosif Pinelis
本文研究了对数凹分布均值附近的一侧浓度特性，得出了在EX=0和EX²=1条件下概率P(0<X<δ)的下界。
本文获得了对于所有实数δ>0和所有满足EX=0且EX²=1的随机变量X，概率P(0<X<δ)的下界。
A lower bound on the probability $P(0<X<δ)$ for all real $δ>0$ and all random variables $X$ with log-concave p.d.f.'s such that $EX=0$ and $EX^2=1$ is obtained.
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We establish two-sided bounds for expectations of order statistics ($k$-th maxima) of moduli of coordinates of centered log-concave random vectors with uncorrelated coordinates. Our bounds are exact up to multiplicative universal constants in the unconditional case for all $k$ and in the isotropic case for $k \leq n-cn^{5/6}$. We also derive two-sided estimates for expectations of sums of $k$ largest moduli of coordinates for some classes of random vectors.
We establish concentration inequalities in the class of ultra log-concave distributions. In particular, we show that ultra log-concave distributions satisfy Poisson concentration bounds. As an application, we derive concentration bounds for the intrinsic volumes of a convex body, which generalizes and improves a result of Lotz, McCoy, Nourdin, Peccati, and Tropp (2019).
Two-sided bounds are explored for concentration functions and Rényi entropies in the class of discrete log-concave probability distributions. They are used to derive certain variants of the entropy power inequalities.
We derive two-sided bounds for moments of linear combinations of coordinates od unconditional log-concave vectors. We also investigate how well moments of such combinations may be approximated by moments of Gaussian random variables.
Moments and tails of Lq-valued chaoses based on independent variables with log-concave tails
本文推导了任意巴拿赫空间中二阶随机混沌矩的下界，给出了Lq空间中该混沌矩的两个上界（一个需亚高斯假设，一个无需但非最优），并可用于形状参数≥1的威布尔随机变量生成的Lq值混沌的双侧矩估计。
Majorization, Entropy and Concentration Inequalities for Discrete $α$-Concave Random Variables
针对离散 $\alpha$-凹随机变量（$-1/3<\alpha<0$），提出最大方差不等式及凸优超原理，推广对数凹情形，并导出逆向熵幂不等式、Lévy 集中函数界及多项式型集中不等式。
We present two-sided estimates of moments and tails of polynomial chaoses of order at most three generated by independent symmetric random variables with log-concave tails as well as for chaoses of arbitrary order generated by independent symmetric exponential variables. The estimates involve only deterministic quantities and are optimal up to constants depending only on the order of the chaos variable.
We establish upper bounds for tails of order statistics of isotropic log-concave vectors and apply them to derive a concentration of l_r norms of such vectors.
A concentration property of the functional ${-}\log f(X)$ is demonstrated, when a random vector X has a log-concave density f on $\mathbb{R}^n$. This concentration property implies in particular an extension of the Shannon-McMillan-Breiman strong ergodic theorem to the class of discrete-time stochastic processes with log-concave marginals.
We show that for any $α>0$ the Rényi entropy of order $α$ is minimized, among all symmetric log-concave random variables with fixed variance, either for a uniform distribution or for a two sided exponential distribution. The first case occurs for $α\in (0,α^*]$ and the second case for $α\in [α^*,\infty)$, where $α^*$ satisfies the equation $\frac{1}{α^*-1}\log α^*= \frac12 \log 6$, that is $α^* \approx 1.241$. Using those results, we prove that one-sided exponential distribution minimizes Rényi entropy of order $α\geq 2$ among all log-concave random variables with fixed variance.
We prove a sharp inequality conjectured by Bobkov on the measure of dilations of Borel sets in $\mathbb{R}^n$ by a $s$-concave probability. Our result gives a common generalization of an inequality of Nazarov, Sodin and Volberg and a concentration inequality of Guédon. Applying our inequality to the level sets of functions satisfying a Remez type inequality, we deduce, as it is classical, that these functions enjoy dimension free distribution inequalities and Kahane-Khintchine type inequalities with positive and negative exponent, with respect to an arbitrary $s$-concave probability.
Anti-concentration inequalities for log-concave variables on the real line
本文针对实直线上的对数凹随机变量，在离散与连续情形下证明了精确的反集中不等式，采用初等优化方法恢复并推广了已有结果。
Let $μ$ be a probability measure on $\rr^n$ ($n \geq 2$) with Lebesgue density proportional to $e^{-V (\Vert x\Vert )}$, where $V : \rr_+ \to \rr$ is a smooth convex potential. We show that the associated spectral gap in $L^2 (μ)$ lies between $(n-1) / \int_{\rr^n} \Vert x\Vert ^2 μ(dx)$ and $n / \int_{\rr^n} \Vert x\Vert ^2 μ(dx)$, improving a well-known two-sided estimate due to Bobkov. Our Markovian approach is remarkably simple and is sufficiently robust to be extended beyond the log-concave case, at the price of potentially modifying the underlying dynamics in the energy, leading to weigh…
Concentration inequalities for log-concave sequences
本文通过凸序中的主次关系，为满足精确对数凹度的随机变量推导出集中不等式、矩界和熵界，并改进了现有文献中的结果。
Threshold-type counts based on multivariate occupancy models with log concave marginals admit bounded size biased couplings under weak conditions, leading to new concentration of measure results for random graphs, germ-grain models in stochastic geometry, multinomial allocation models and multivariate hypergeometric sampling. The work generalizes and improves upon previous results in a number of directions.
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 ({\it Ann. Probab.}, 39(4):1528--1543, 2011).
On measures strongly log-concave on a subspace
研究仅在方向子空间上弯曲的对数凹测度的集中性质，采用适配的随机局部化过程进行证明。
We derive two concentration inequalities for linear functions of log-concave distributions: an enhanced version of the classical Brascamp--Lieb concentration inequality, and an inequality quantifying log-concavity of marginals in a manner suitable for obtaining variance and tail probability bounds. These inequalities are applied to the statistical mechanics problem of estimating the fluctuations of random surfaces of the $\nablaφ$ type. The classical Brascamp--Lieb inequality bounds the fluctuations whenever the interaction potential is uniformly convex. We extend these bounds to the case of c…
We derive two-sided estimates for random multilinear forms (random chaoses) generated by independent symmetric random variables with logarithmically concave tails. Estimates are exact up to multiplicative constants depending only on the order of chaos.
Chernoff's Density Is Strongly Log-Concave
研究切尔诺夫随机变量密度性质，核心方法是利用双边布朗运动相关理论，主要贡献是证明了该密度是强对数凹的，且证明由GPT - 5.6 Sol完成。
We establish sharp exponential deviation estimates of the information content as well as a sharp bound on the varentropy for the class of convex measures on Euclidean spaces. This generalizes a similar development for log-concave measures in the recent work of Fradelizi, Madiman and Wang (2016). In particular, our results imply that convex measures in high dimensions are concentrated in an annulus between two convex sets (as in the log-concave case) despite their possibly having much heavier tails. Various tools and consequences are developed, including a sharp comparison result for Rényi entr…
We prove logarithmic Sobolev inequalities and concentration results for convex functions and a class of product random vectors. The results are used to derive tail and moment inequalities for chaos variables (in spirit of Talagrand and Arcones, Gine). We also show that the same proof may be used for chaoses generated by log-concave random variables, recovering results by Lochowski and present an application to exponential integrability of Rademacher chaos.
The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincar{é} constant. First we revisit E. Milman's result [40] on the link between weak (Poincar{é} or concentration) inequalities and Cheeger's inequality in the logconcave cases, in particular extending localization ideas and a result of Latala, as well as providing a simpler proof of the nice Poincar{é} (dimensional) bound in the inconditional case. Then we prove alternative transference principle by concentration or using various distances (total variation, Wasserstein…
Matrix Concentration Inequalities and Free Probability II. Two-sided Bounds and Applications
本文针对矩阵集中不等式与自由概率研究，为谱边缘获得匹配下界以完善理论，其双侧界可解决经典方法无法处理的谱边缘首阶精确确定问题，并通过两个结果解释非齐次随机矩阵谱异常值相变，阐明多领域应用中的相变现象。
We show that the modified log-Sobolev constant for a natural Markov chain which converges to an $r$-homogeneous strongly log-concave distribution is at least $1/r$. Applications include a sharp mixing time bound for the bases-exchange walk for matroids, and a concentration bound for Lipschitz functions over these distributions.
We prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log-concave measures (that we introduce). These inequalities imply (up to universal constants) the log-Sobolev inequalities proved by Bobkov--Ledoux as well as the isoperimetric inequalities due to Bakry-Ledoux and Bobkov--Zegarlinski. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov--Milman.
Log-concavity and concentration bounds for a single gap between GUE eigenvalues
研究GUE特征值间间隙的对数凹性，并通过半圆归一化证明改进的集中界。
Brascamp--Lieb-type, weighted Poincaré-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general $κ$-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinite-dimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration.
Chevet-type inequalities for subexponential Weibull variables and estimates for norms of random matrices
本文针对形状参数 r∈[1,2] 的对称威布尔变量证明了两侧 Chevet 型不等式，并据此给出威布尔和对数凹随机矩阵及其子矩阵算子范数的两侧估计。
We develop an approach for solving one-sided optimal stopping problems in discrete time for general underlying Markov processes on the real line. The main idea is to transform the problem into an auxiliary problem for the ladder height variables. In case that the original problem has a one-sided solution and the auxiliary problem has a monotone structure, the corresponding myopic stopping time is optimal for the original problem as well. This elementary line of argument directly leads to a characterization of the optimal boundary in the original problem: The optimal threshold is given by the t…
Sudakov Minoration for Unconditional Log-Concave Vectors
本文为无条件对数凹向量提出 Sudakov 极小化原理的证明候选，通过稀疏支撑约化、阈值见证选取与指数矩估计等步骤，给出两个关于覆盖数和集中性的推论。
We study the behavior of the capital process of a continuous Bayesian mixture of fixed proportion betting strategies in the one-sided unbounded forecasting game in game-theoretic probability. We establish the relation between the rate of convergence of the strong law of large numbers in the self-normalized form and the rate of divergence to infinity of the prior density around the origin. In particular we present prior densities ensuring the validity of Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm.
Anti-concentration of polynomials: $L^{p}$ balls and symmetric measures
研究多项式在对数凹随机向量上的方差界，揭示在特定条件下方差下界与维度无关的反集中性质，拓展了经典乘积测度之外的分析范围。
Motivated by connections to random matrices, Littlewood-Richardson coefficients and tilings, we study random discrete concave functions on an equilateral lattice. We show that such functions having a periodic Hessian of a fixed average value $- s$ concentrate around a quadratic function. We consider the set of all concave functions $g$ on an equilateral lattice $\mathbb L$ that when shifted by an element of $n \mathbb L$ have a periodic discrete Hessian, with period $n \mathbb L$. We add a convex quadratic of Hessian $s$; the sum is then periodic with period $n \mathbb L$, and view this as a m…
In this paper, we give necessary and sufficient conditions for Talagrand's like transportation cost inequalities on the real line. This brings a new wide class of examples of probability measures enjoying a dimension-free concentration of measure property. Another byproduct is the characterization of modified Log-Sobolev inequalities for Log-concave probability measures on R.
We study the concentration of the norm of a random vector $Y$ uniformly sampled in the centered zero cell of two types of stationary and isotropic random mosaics in $\mathbb{R}^n$ for large dimensions $n$. For a stationary and isotropic Poisson-Voronoi mosaic, $Y$ has a radial and log-concave distribution, implying that ${|Y|}/{\mathbb{E}(|Y|^2)^{\frac{1}{2}}}$ approaches one for large $n$. Assuming the cell intensity of the random mosaic scales like $e^{n ρ_n}$, where $\lim_{n \to \infty} ρ_n = ρ$, $|Y|$ is on the order of $\sqrt{n}$ for large $n$. For the Poisson-Voronoi mosaic, we show that…
Optimal constants in concentration inequalities on the sphere and in the Gauss space
本文研究球面和高斯空间中函数偏离中位数和均值的集中不等式，给出最优常数的界，改进了现有文献中的结果。
Under correlation-type conditions, we derive an upper bound of order $(\log n)/n$ for the average Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. The result is based on improved concentration inequalities on high-dimensional Euclidean spheres. Applications are illustrated on the example of log-concave probability measures.
Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave functions on the unit simplex measure the concentration of capital, and their gradient maps define novel investment strategies. The gradient maps may also be regarded as optimal transport maps on the simplex. In this paper we construct and study probability measures supported on spaces of concave functions. These measures may serve as prior distributions in …
We consider a generic class of log-concave, possibly random, (Gibbs) measures. We prove the concentration of an infinite family of order parameters called multioverlaps. Because they completely parametrise the quenched Gibbs measure of the system, this implies a simple representation of the asymptotic Gibbs measures, as well as the decoupling of the variables in a strong sense. These results may prove themselves useful in several contexts. In particular in machine learning and high-dimensional inference, log-concave measures appear in convex empirical risk minimisation, maximum a-posteriori in…
We present precise multilevel exponential concentration inequalities for polynomials in Ising models satisfying the Dobrushin condition. The estimates have the same form as two-sided tail estimates for polynomials in Gaussian variables due to Latała. In particular, for quadratic forms we obtain a Hanson-Wright type inequality. We also prove concentration results for convex functions and estimates for nonnegative definite quadratic forms, analogous as for quadratic forms in i.i.d. Rademacher variables, for more general random vectors satisfying the approximate tensorization property for entropy…
High-Dimensional Ultra-Log-Concave Distributions
本文提出高维超对数凹分布的定量推广（$\delta$-超对数凹性），建立函数不等式、集中性与最大熵原理，并应用于多个模型。
A One-Sided McDiarmid-Type Inequality for Heavy Tails via Reverse KL Tensorization
该研究提出重尾分布下基于反向KL张量化的单边McDiarmid型不等式，适用于p∈(1,2]阶条件矩假设的独立随机变量函数，可应用于样本均值等的集中性及无界损失的风险界分析。
Hoeffding-Type Concentration Bounds for Exchangeable Random Variables
针对可交换随机变量序列，提出基于de Finetti混合测度支撑中分布的最大和最小均值的Hoeffding型浓度不等式，填补了有限样本均值与总体均值之间的空白。
It is well-known that measures whose density is the form $e^{-V}$ where $V$ is a uniformly convex potential on $\RR^n$ attain strong concentration properties. In search of a notion of log-concavity on the discrete hypercube, we consider measures on $\{-1,1\}^n$ whose multi-linear extension $f$ satisfies $\log \nabla^2 f(x) \preceq β\Id$, for $β\geq 0$, which we refer to as $β$-semi-log-concave. We prove that these measures satisfy a nontrivial concentration bound, namely, any Hamming Lipchitz test function $φ$ satisfies $\Var_ν[φ] \leq n^{2-C_β}$ for $C_β>0$. As a corollary, we prove a concent…
Sharp Frobenius-Norm Concentration for Sample Moment Tensors
本文针对样本矩张量建立无维度Sharp Frobenius范数集中不等式，发现奇数阶张量的高斯与次高斯尺度重合、偶数阶不同的奇偶效应，通过结合结构化鞅耦合定理等完成矩界证明，为相关领域提供关键理论支撑。
Given a $p$-coin that lands heads with unknown probability $p$, we wish to produce an $f(p)$-coin for a given function $f: (0,1) \rightarrow (0,1)$. This problem is commonly known as the Bernoulli Factory and results on its solvability and complexity have already been obtained. Nevertheless, generic ways to design a practical Bernoulli Factory for a given function $f$ exist only in a few special cases. We present a constructive way to build an efficient Bernoulli Factory when $f(p)$ is a rational function with coefficients in $\mathbb{R}$. Moreover, we extend the Bernoulli Factory problem to a…
We show that the convex hull of a large i.i.d. sample from an absolutely continuous log-concave distribution approximates a predetermined convex body in the logarithmic Hausdorff distance and in the Banach-Mazur distance. For log-concave distributions that decay super-exponentially, we also have approximation in the Hausdorff distance. These results are multivariate versions of the Gnedenko law of large numbers, which guarantees concentration of the maximum and minimum in the one-dimensional case. We provide quantitative bounds in terms of the number of points and the dimension of the ambient …
We establish a reversal of Lyapunov's inequality for monotone log-concave sequences, settling a conjecture of Havrilla-Tkocz and Melbourne-Tkocz. A strengthened version of the same conjecture is disproved through counter example. We also derive several information theoretic inequalities as consequences. In particular sharp bounds are derived for the varentropy, Rényi entropies, and the concentration of information of monotone log-concave random variables. Moreover, the majorization approach utilized in the proof of the main theorem, is applied to derive analogous information theoretic results …
We prove an almost constant lower bound of the isoperimetric coefficient in the KLS conjecture. The lower bound has the dimension dependency $d^{-o_d(1)}$. When the dimension is large enough, our lower bound is tighter than the previous best bound which has the dimension dependency $d^{-1/4}$. Improving the current best lower bound of the isoperimetric coefficient in the KLS conjecture has many implications, including improvements of the current best bounds in Bourgain's slicing conjecture and in the thin-shell conjecture, better concentration inequalities for Lipschitz functions of log-concav…
High-dimensional limits and extremizers for maximal functions associated with log-concave densities
研究与径向对数凹概率密度相关极大函数的高维渐近行为，通过一般转移原理，利用傅里叶乘子符号及新工具给出量化界，证明欧几里得球相关极大算子有渐近最小\(L^p\)算子范数。
Concentration of information on discrete groups
受渐近均分性启发，本文系统研究离散群上的varentropy，提出近似张量化不等式，证明共轭不变随机游走的varentropy受限于生成元数目d，并揭示其非负Bakry-Émery曲率性质。
On McDiarmid's Inequality under Dependence via Approximate Tensorization of Entropy
本文通过熵的近似张量化(ATE)推导依赖数据的McDiarmid不等式，应用于非各向同性高斯向量、强对数凹和对数光滑测度，并解决符号函数集中问题、依赖下Erdős-Rényi图及Dvoretzky-Kiefer-Wolfowitz型不等式，改进收敛速率至$1/\sqrt{n}$。
We consider high dimensional Wishart matrices $\mathbb{X} \mathbb{X}^{\top}$ where the entries of $\mathbb{X} \in {\mathbb{R}^{n \times d}}$ are i.i.d. from a log-concave distribution. We prove an information theoretic phase transition: such matrices are close in total variation distance to the corresponding Gaussian ensemble if and only if $d$ is much larger than $n^3$. Our proof is entropy-based, making use of the chain rule for relative entropy along with the recursive structure in the definition of the Wishart ensemble. The proof crucially relies on the well known relation between Fisher i…
Entropy-Wasserstein regularization, defective local concentration and a cutoff criterion beyond non-negative curvature
本文提出一种放松的Ollivier粗黎曼曲率变种，通过熵-运输正则化效应和缺陷性局部集中，推导出在负曲率设置中截断现象的出现准则。
Concentration of Empirical First-Passage Times
针对欠采样数据中首次通过统计量难以可靠推断的问题，本章研究可逆 Markov 过程经验首次通过时间的有限样本集中性，给出样本均值偏差界、非渐近置信区间及极值波动双侧界。
On sample complexity for covariance estimation via the unadjusted Langevin algorithm
本文通过未调整的兰格vin算法研究了协方差矩阵估计的样本复杂性，揭示了单链方法在样本复杂性上优于并行方法的结论。
Concentration of the Langevin Algorithm's Stationary Distribution
本文针对Langevin算法离散化后的平稳分布，利用旋转不变矩生成函数直接分析离散时间动力学，建立了与连续情形相对应的次指数与次高斯集中界，并推广到算法迭代轨迹和不精确梯度实现。
A Conditional-Distribution Framework for Validating Synthetic Multivariate Data
提出基于全条件分布的模型无关框架，通过MAP对齐统计量和最近真实相似性诊断，区分合成数据中的依赖丧失、记录重用和模式集中三种失败模式，并在多个真实数据集上验证其有效性。
A Chernoff-type distribution is a nonnormal distribution defined by the slope at zero of the greatest convex minorant of a two-sided Brownian motion with a polynomial drift. While a Chernoff-type distribution is known to appear as the distributional limit in many non-regular statistical estimation problems, the accuracy of Chernoff-type approximations has remained largely unknown. In the present paper, we tackle this problem and derive Berry-Esseen bounds for Chernoff-type limit distributions in the canonical non-regular statistical estimation problem of isotonic (or monotone) regression. The …
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