Sergey G. Bobkov, Arnaud Marsiglietti, James Melbourne
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.
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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).
One-sided concentration near the mean of log-concave distributions
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。
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。
Order statistics and concentration of l_r norms for log-concave vectors
本文建立了各向同性对数凹向量的顺序统计量尾部的上界，并应用它们导出了这类向量的l_r范数的集中。
Concentration of the information in data with log-concave distributions
证明了当随机向量X在$\mathbb{R}^n$上具有对数凹密度f时，泛函${-}\log f(X)$的浓度性质.这种集中性特别意味着Shannon-McMillan-Breiman强遍历定理在具有对数凹边缘的离散时间随机过程类中的推广。
Concentration inequalities for $s$-concave measures of dilations of Borel sets and applications
利用$s$ -凹概率证明了Bobkov关于$\mathbb{R}^n$中Borel集的扩张测度的一个尖锐不等式.我们的结果给出了Nazarov，Sodin和Volberg的一个不等式和Guédon的一个浓度不等式的一般推广.将该不等式应用于满足Remez型不等式的函数的水平集，我们得到了这类函数关于任意$s$ -凹概率的无量纲分布不等式和具有正负指数的Kahane-Khintchine型不等式.
An example of a non-log-concave distribution where the difference has a log-concave density
本文通过具体例子证明，非对数凹分布的差值不一定具有对数凹密度，推翻了Prékopa-Leindler不等式的逆向结论。
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。
We prove a sharp moment inequality for a log-concave or a log-convex function, on Gaussian random vectors. As an application we take a stability result for the classical logarithmic Sobolev inequality of L. Gross in the case where the function is log-concave.
A reverse entropy power inequality for log-concave random vectors
证明了对数凹随机向量的一维投影熵的指数定义了1/5-熵。给出了对数凹情形下的两个逆熵幂不等式，并讨论了几个例子。
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。
A Note on Statistical Distances for Discrete Log-Concave Measures
研究离散对数凹分布上总变差、Wasserstein与$f$-散度等标准统计距离的等价性。
An invariant variational proof for canonical processes with independent symmetric log-concave tails
本文通过不变变分方法，证明了无需加倍假设的独立对称对数凹尾典范过程的指定指数律比较。
Moments of unconditional logarithmically concave vectors
本文导出了无条件对数凹向量坐标的线性组合的矩的双边界。我们还调查如何以及这样的组合的时刻可以近似高斯随机变量的时刻。
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.
Geometric and Functional Inequalities for Log-concave Probability Sequences
该研究针对对数凹概率序列，证明了整数集上的伸缩不等式及对应的泛函偏差不等式，揭示对数仿射序列为相关集合极点，借此简化已有卷积不等式证明并得到离散Prékopa-Leindler不等式。
Spectral gap for some invariant log-concave probability measures
我们证明了Kannan，Lovász和Simonovits关于凸体和对数凹测度的等周性质的猜想，对于$\mathbb{R}^n$上的$ρ(|x|_B)dx$和$\mathbb{R}^{1+n}$上的$ρ(t,|x|_B) dx$形式的对数凹测度是成立的，其中$|x|_B$是与已经满足该猜想的任何凸体$B$相关的范数.特别地，该猜想对旋转的凸体成立。
We show that a measure on the real line that is the image of a log-concave measure under a polynomial of degree $d$ possesses a density from the Nikol'skii--Besov class of fractional order $1/d$. This result is used to prove an estimate of the total variation distance between such measures in terms of the Fortet--Mourier distance.
We show that for every mean zero log-concave real random variable $X$ one has $\|X\|_p \leq \frac{p}{q} \|X\|_q$ for $p \geq q \geq 1$, going beyond the well-known case of symmetric random variables. We also prove that in the class of arbitrary log-concave real random variables for $p>q > 0$ the quantity $\|X\|_p / \|X\|_q$ is maximized for some shifted exponential distribution. Building upon this we derive the bound $\|X\|_p \leq C_0 \frac{p}{q} \|X\|_q$ for arbitrary log-concave $X$, with best possible absolute constant $C_0=e^{W(1/e)} \approx 1.3211$ in front of $\frac{p}{q}$, where $W$ sta…
Extending results of Harg{é} and Hu for the Gaussian measure, we prove inequalities for the covariance Cov$_μ(f, g)$ where $μ$ is a general product probability measure on $\mathbb{R}^d$ and $f,g: \mathbb{R}^d \to \mathbb{R}$ satisfy some convexity or log-concavity assumptions, with possibly some symmetries.
Sharp Rosenthal-type inequalities for mixtures and log-concave variables
本文为固定分布混合的独立随机变量和矩建立了带尖锐常数的 Rosenthal 型不等式，并在对数凹情形下确定了极值元。
Optimal Concentration of Information Content For Log-Concave Densities
给出了具有对数凹密度的随机向量的变熵以及信息量偏离其均值的精确界的一个初等证明。这些界限显著改进了Bobkov和Madiman（{ \it Ann. Probab.}，39（4）：1528- 1543，2011）。
A note on the Littlewood-Offord problem for discrete log-concave distributions
本文扩展了Littlewood-Offord问题，研究离散对数凹分布下的熵功率不等式，并探讨了算术级数和熵版本的问题。
A note on the maximum probability of ultra log-concave distributions
本文指出，超对数凹分布的最大概率不小于同均值泊松分布最大概率的结论，在均值大于1时并非普遍成立。
The first step to study lower bounds for a stochastic process is to prove a special property - Sudakov minoration. The property means that if a certain number of points from the index set are well separated then we can provide an optimal type lower bound for the mean value of the supremum of the process. Together with the generic chaining argument the property can be used to fully characterize the mean value of the supremum of the stochastic process. In this article we prove the property for canonical processes based on radial-type log concave measures.
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 uniform bounds on the low-order derivatives of Stein equation solutions for a broad class of multivariate, strongly log-concave target distributions. These "Stein factor" bounds deliver control over Wasserstein and related smooth function distances and are well-suited to analyzing the computable Stein discrepancy measures of Gorham and Mackey. Our arguments of proof are probabilistic and feature the synchronous coupling of multiple overdamped Langevin diffusions.
From Triangular Array Progression to Near-Log-Concave State Transitions
本文提出一种生成三角阵列的无限整数序列，该阵列具有近对数凹性等特性，可用于物理建模电子跃迁系统，还能生成满足特定状态转移特征的右随机矩阵族。
On a Conjecture of Feige for Discrete Log-Concave Distributions
本文针对Feige猜想，在离散对数凹分布类上证明了其加强版本，即独立离散对数凹随机变量之和小于其期望加1的概率至少为1/e，且对任意期望均成立。
Isoperimetry for spherically symmetric log-concave probability measures
证明了概率测度$μ$在$\mathbb{R}^n$上的等周不等式，其中$μ$是$\mathbb{R}^n$上的欧氏范数，$ϕ$是非减凸函数.当$ϕ(x)=x^α$与$α\ge1$配合使用时，尤其适用。在对$ϕ$的温和假设下，如果选择$λ$使得$μ$的协方差是恒等式，则不等式是无量纲的。
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…
Limiting Root Distribution of Random Log-concave Polynomials
研究随机对数凹多项式零点在复平面上的渐进行为，提出统一模型和贝塔模型，揭示其零点分布的极限特性。
Transport proofs of weighted Poincaré inequalities for log-concave distributions
利用最优输运工具证明了对数凹随机向量满足中心条件的加权Poincar不等式.通过这种方法，我们恢复了Klartag和Barthe-Cordero-Erausquin对具有对称性的对数凹随机向量的类似结果。此外，我们还证明了方差猜想对对数凹鞅的增量是成立的。
Stability version of the Prekopa-Leindler inequality for log-concave functions on the n-dimensional Euclidean space is established.
We discuss interplays between log-concave functions and log-concave sequences. We prove a Bernstein-type theorem, which characterizes the Laplace transform of log-concave measures on the half-line in terms of log-concavity of the alternating Taylor coefficients. We establish concavity inequalities for sequences inspired by the Prékopa-Leindler and the Walkup theorems. One of our main tools is a stochastic variational formula for the Poisson average.
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…
The paper provides an estimate of the total variation distance between distributions of polynomials defined on a space equipped with a logarithmically concave measure in terms of the $L^2$-distance between these polynomials.
Log-concave measures
我们研究了对数凹测度，通过Prékopa-Leindler性质刻画了它们，并定义了一个子集，其元素称为超对数凹测度，这些超对数凹测度具有满足对数Sobolev不等式的性质.我们给出了它们的稳定性的一些结果。并指出了Monge-Kantorovitch方程和Monge-Ampére方程与测度输运的某些关系及其应用。
Logarithmic Sobolev inequality for log-concave measure from Prekopa-Leindler inequality
本文对S. Bobkov和M. Ledoux in GAFA（2000）.利用Prekopa-Leindler定理证明了一个适用于$\dR^n$上所有对数凹测度的最优修正对数Sobolev不等式.这个不等式包含了Bobkov和Ledoux证明的结果，推广了Euclidean Logarithmic Sobolev不等式，也包含了一些关于大熵的凸对数Sobolev不等式.
Sudakov-type minoration for log-concave vectors
本文提出并讨论了对数凹向量范数的一个下界猜想，推广了经典的Sudakov降阶原理。我们证明了该猜想对某些特殊的对数凹测度类成立，并且在一般情况下，该猜想的某些较弱形式得到了满足。我们还提出了一些应用程序的基础上链接技术。
We review some simple techniques based on monotone mass transport that allow to obtain transport-type inequalities for any log-concave probability measure, and for more general measures as well. We discuss quantitative forms of these inequalities, with application to the Brascamp-Lieb variance inequality.
We give a self-contained proof of the strongest version of Mason's conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave.
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…
A Chaining Approach to Canonical Processes with Log-Concave Tails
本文证明了无需Δ2条件的对数凹尾典型过程的分解定理，采用基于可容许划分和切割映射的经典链式方法，并恢复了控制测度刻画。
Solution of Monge-Ampère Equation on Wiener Space for log-concave measures: General case
在不作任何正则性假设的情况下，证明了对数凹测度下Monge-Ampere方程的强解的存在性.特别是密度的形式$\exp -f$的措施，其中$f$是一个$H$凸函数，可以采取值$\infty$的一组积极的措施。
Minimum entropy of a log-concave variable for fixed variance
本文证明固定方差下对数凹变量的最小熵由指数分布达到，并据此推导加性噪声信道容量上界及改进反向熵幂不等式常数。
Bi-log-concavity of probability measures is a univariate extension of the notion of log-concavity that has been recently proposed in a statistical literature. Among other things, it has the nice property from a modelisation perspective to admit some multimodal distributions, while preserving some nice features of log-concave measures. We compute the isoperimetric constant for a bi-log-concave measure, extending a property available for log-concave measures. This implies that bi-log-concave measures have exponentially decreasing tails. Then we show that the convolution of a bi-log-concave measu…
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.
A note on the spectral gap for log-concave probability measures on convex bodies
本文研究了凸体上对数凹概率测度的谱隙问题，提出了基于分布参数、维度、域几何特征和原点位置等量的显式下界，并通过经典与非经典例子验证了结果。
An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies
证明了一致凸体上一致测度和我们引入的一类一致对数凹测度的一个等周不等式。这些不等式包含Bobkov-Ledoux证明的log-Sobolev不等式以及Bakry-Ledoux和Bobkov-Zegarlinski证明的等周不等式。我们还恢复了一致凸体的浓度不等式，类似于Gromov-Milman证明。
Weak and strong moments of l_r-norms of log-concave vectors
证明了对于$p\geq 1$和$r\geq 1$，对数凹随机向量的$l_r$ -范数的$p$阶矩与一阶矩和弱$p$阶矩之和相当，直到与$r$成比例的常数.这扩展了Paouris关于欧几里得范数的先前结果。
We discuss situations where perturbing a probability measure on $\mathbb{R}^n$ does not deteriorate its Poincaré constant by much. A particular example is the symmetric exponential measure in $\mathbb{R}^n$, even log-concave perturbations of which have Poincaré constants that grow at most logarithmically with the dimension. This leads to estimates for the Poincaré constants of $(n/2)$-dimensional sections of the unit ball of $\ell_p^n$ for $1 \leq p \leq 2$, which are optimal up to logarithmic factors. We also consider symmetry properties of the eigenspace of the Laplace-type operator associat…
Mixture and separation of log-concave measures
研究双组分对数凹混合模型，探讨给定样本时各组分分布参数的精确恢复程度，识别出混合分布的两种情况，提出用二次分类器分离组分分布的方法。
On the equivalence of modes of convergence for log-concave measures
在最近的工作中，在渐近几何分析的一个重要主题是，许多经典的影响之间的不同类型的几何或功能不等式可以扭转凸性假设的存在。在这篇文章中，我们探讨了在对数凹分布中，不同的概率测度之间的距离概念在多大程度上是可比的。结果表明，当极限测度为高斯时，各向同性对数凹分布的弱收敛等价于全变差收敛，并进一步等价于相对熵收敛.
Convolution of a symmetric log-concave distribution and a symmetric bimodal distribution can have any number of modes
本文证明了离散及光滑情形下，对称对数凹分布与对称双峰分布的卷积可产生任意正数量的众数。
A dimension-free bound for isoperimetry of unconditional log-concave measures
本文证明无条件对数凹测度的无维数Poincaré不等式，从而在无条件情形下解决KLS猜想，方法结合Dunkl算子理论与测度变换及谱分析，核心思想由AI辅助生成。
The Brunn-Minkowski and Prékopa-Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions so it seems that convexity has no special signficance. On the other hand, it was recently shown that the Brunn-Minkowski inequality, specialized to convex sets, follows from a local stochastic dominance for naturally associated random polytopes. We show that for the subclass of $\log$-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Prékopa-L…
Entropy-variance inequalities for discrete log-concave random variables via degree of freedom
本文利用离散自由度概念证明整数值对数凹随机变量的精确最小熵-方差不等式，证明几何分布能最小化最小熵，并由此推导出改进的离散 Rényi 熵幂不等式。
Geometry of log-concave Ensembles of random matrices and approximate reconstruction
研究了具有独立迷向对数凹行的随机矩阵$Γ$的限制等距性。为此，我们引入了一个参数$Γ_{k,m}$，它统一控制子矩阵的操作符范数与$k$行和$m$列。利用顺序统计量的新尾估计和各向同性对数凹向量投影范数的偏差不等式估计了该参数。
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