Gaussian approximation of moments of sums of independent symmetric random variables with logarithmically concave tails
Rafał Latała
本文研究了具有几何凹尾的独立对称随机变量和的矩如何用高斯随机变量的矩来近似。
We study how well moments of sums of independent symmetric random variables with logarithmically concave tails may be approximated by moments of Gaussian random variables.
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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。
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.
Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails
本文给出了由具有对数凹尾的独立对称随机变量产生的至多三阶多项式混沌的矩和尾的双边估计，以及由独立对称指数变量产生的任意阶混沌的矩和尾的双边估计.估计只涉及确定性的数量和最佳的常数只取决于混沌变量的顺序。
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.
We study two topologies $τ_{KR}$ and $τ_K$ on the space of measures on a completely regular space generated by Kantorovich--Rubinshtein and Kantorovich seminorms analogous to their classical norms in the case of a metric space. The Kantorovich--Rubinshtein topology $τ_{KR}$ coincides with the weak topology on nonnegative measures and on bounded uniformly tight sets of measures. A~sufficient condition is given for the compactness in the Kantorovich topology. We show that for logarithmically concave measures and stable measures weak convergence implies convergence in the Kantorovich topology. We…
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.
Tail estimates for norms of sums of log-concave random vectors
我们建立了顺序统计量和各向同性对数凹随机向量投影的欧氏范数的新的尾估计。更一般地说，我们证明了尾部估计的独立的对数凹随机向量的总和的投影的规范，和统一的版本，这些形式的尾部估计的运营商规范的矩阵及其子矩阵的设置的对数凹合奏。这是用来研究一个数量$A_{k,m}$，控制均匀的子矩阵的运营商范数与$k$行和$m$列的矩阵$A$与独立的各向同性对数凹随机行。我们应用我们的尾部估计的$A_{k,m}$的限制等距性质的研究中起着重要作用的压缩感知理论。
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。
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。
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.
Let $C$ and $K$ be centrally symmetric convex bodies of volume $1$ in ${\mathbb R}^n$. We provide upper bounds for the multi-integral expression \begin{equation*}\|{\bf t}\|_{C^s,K}=\int_{C}\cdots\int_{C}\Big\|\sum_{j=1}^st_jx_j\Big\|_K\,dx_1\cdots dx_s\end{equation*} in the case where $C$ is isotropic. Our approach provides an alternative proof of the sharp lower bound, due to Gluskin and V. Milman, for this quantity. We also present some applications to "randomized" vector balancing problems.
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.
Approximate Discrete Entropy Monotonicity for Log-Concave Sums
本文证明Tao关于对数凹随机变量和的离散熵单调性猜想，通过连续化方法将问题转化为微分熵，并给出误差项的显式界。
For every given real value of the ratio $μ:=A_X/G_X>1$ of the arithmetic and geometric means of a positive random variable $X$ and every real $v>0$, exact upper bounds on the right- and left-tail probabilities $\mathsf{P}(X/G_X\ge v)$ and $\mathsf{P}(X/G_X\le v)$ are obtained, in terms of $μ$ and $v$. In particular, these bounds imply that $X/G_X\to1$ in probability as $A_X/G_X\downarrow1$. Such a result may be viewed as a converse to a reverse Jensen inequality for the strictly concave function $f=\ln$, whereas the well-known Cantelli and Chebyshev inequalities may be viewed as converses to a…
Some stochastic inequalities for weighted sums
我们比较i.i.d.正随机变量根据通常的随机顺序。主要的不等式是在一定的对数假设下使用优化技术得出的。具体而言，令$Y_i$为i.i.d.。$\mathbf{R}_+$上的随机变量。假设$\log Y_i$具有对数凹密度，证明了如果$(\log a_1,...,\log a_n)$被$(\log b_1,...,\log b_n)$优化，则$\sum a_iY_i$随机小于$\sum b_iY_i$.另一方面，假设$Y_i^p$有一个对数凹密度的一些$p>1$，我们表明，$\sum a_iY_i$是随机大于$\sum b_iY_i$，如果$(a_1^q,...,a_n^q)$是优由$(b_1^q,...,b_n^q)$，其中$p^{-1}+q^{-1}=1$。这些统一的几个随机排序结果为特定的分布。特别地，证明了Hitczenko [Sankhybrid A 60（1998）171- 175]关于威布尔变量的一个猜想.在可靠性和无线通信的潜在应用中提到。
This article investigates sharp comparison of moments for various classes of random variables appearing in a geometric context. In the first part of our work we find the optimal constants in the Khintchine inequality for random vectors uniformly distributed on the unit ball of the space $\ell_q^n$ for $q\in(2,\infty)$, complementing past works that treated $q\in(0,2]\cup\{\infty\}$. As a byproduct of this result, we prove an extremal property for weighted sums of symmetric uniform distributions among all symmetric unimodal distributions. In the second part we provide a one-to-one correspondenc…
We consider the deviation inequalities for the sums of independent $d$ by $d$ random matrices, as well as rank one random tensors. Our focus is on the non-isotropic case and the bounds that do not depend explicitly on the dimension $d$, but rather on the effective rank. In an elementary and unified manner, we show the following results: 1) A deviation bound for the sums of independent positive-semi-definite matrices. This result complements the dimension-free bound of Koltchinskii and Lounici [Bernoulli, 2017] on the sample covariance matrix in the sub-Gaussian case. 2) A new bound for truncat…
What does a random contingency table look like?
设R=（r_1，...，r_m）和C=（c_1，...，c_n）是正整数向量，使得r_1 +. + r_m=c_1 +. + c_n。本文考虑行和为R、列和为C的非负m × n整数矩阵（列联表）的集合Sigma（R，C）是具有一致测度的有限概率空间.我们证明了Sigma（R，C）中的随机表D以高概率接近于如下定义的特定矩阵（“典型表”）Z。对于非负矩阵x，设g（x）=（x +1）ln（x+1）-xln x，对于非负矩阵X=（x_ij），设g（X）=sum_ij g（x_ij）.则g（X）是严格凹的，并且在非负mxn矩阵X的多胞形上达到最大值，该多胞形的行和R和列和C在唯一点处，我们称之为典型表Z。
An approximation algorithm for counting contingency tables
本文给出了一个计算列联表列联表、m × n个非负整数矩阵的随机逼近算法，r_m）和列和C=（c_1，...，c_n）。我们定义了光滑边界（R，C），并证明了对于这样的边界，算法具有拟多项式N^{O（ln N）}复杂性，其中N=r_1+.+ r_m=c_1+...+ c_n。各种类别的边缘是平滑的，例如，当m=O（n），n=O（m）时，最大行和与最小行和之比以及最大列和与最小列和之比严格小于黄金分割比（1+sqrt{5}）/2 = 1.618.该算法建立在蒙特卡洛积分和抽样算法的对数凹密度，矩阵缩放算法，永久近似算法，和一个整表示的列联表的数量。
Digesting the proof of the sharp thin-shell inequality
研究高维对数凹分布薄壳定理中通用常数最优值，通过分析加权黎曼流形和蒙日 - 安培方程给出证明，确定\({\rm Var}( |X|^2 ) \leq 8 n\)且常数\(8\)最优，还给出相关张量范数界。
On general strong laws of large numbers for fields of random variables
给出了证明随机场强大数定律的一般方法。它是基于在Noszály和Tómács \cite{noszaly}和Tómács和Líbor \cite{thomas06}中提出的哈耶克-雷尼型方法。Noszály和Tómács \cite{noszaly}利用矩不等式得到了随机场的一个Hájek-Rényi型极大不等式.最近，Tómács和Líbor \cite{thomas06}对基于概率的随机序列得到了一个Hájek-Rényi型极大不等式，但对随机场没有得到.本文利用概率给出了随机场的一个Hájek-Rényi型极大不等式，它是Noszály和Tómács \cite{noszaly}主要结果的推广，用概率代替矩，并将Tómács和Líbor \cite % {thomas 06}关于随机序列的主要结果推广到随机场.我们应用我们的结果来建立一个数学加权和没有矩的假设和一般的依赖条件下的随机场。
Efficient Mass Matrix Estimation with Gaussian Cooling
提出基于高斯冷却的多阶段预处理算法，用于高效估计对数凹分布的质量矩阵，复杂度随条件数对数增长，并在一阶拒绝采样下给出查询复杂度上界及格点φ^4模型数值验证。
Exact inequalities for sums of asymmetric random variables, with applications
令$\BS_1,...,\BS_n$为独立同分布随机变量，每个随机变量具有带参数$p\in(0,1)$的标准化伯努利分布。设$m_*(p):=(1+p+2p^2)/(2\sqrt{p-p^2}+4p^2)$为$0<p\le 1/2$，$m_*(p):=1$为$1/2\le p<1$。让$m\ge m_*(p)$。设$f$是这样一个函数，使得$f$和$f''$是非减凸函数。然后证明了对所有非负数$c_1,...,c_n$都有不等式$$\E f(c_1\BS_1+...+c_n\BS_n)\le\E f(s^{(m)}\cdot(\BS_1+...+\BS_n)),$$，其中$s^{(m)}:=(\frac1n \sum_{i=1}^n c_i^{2m})^\frac1{2m}$ . $m$上的下限$m_*(p)$对于每个$p\in(0,1)$是精确的。此外，$\E f(c_1\BS_1+...+c_n\BS_n)$在$(c_1^{2m},...,c_n^{2m})$中是Schur凹的。提出了一些相关的结果，包括`` martic''案例的结果。得到了一些推论，包括有界不对称差的（上）鞅的广义矩和尾概率的上界，以及这类（上）鞅的极大函数的上界.它表明，这些结果可能是重要的，在某些统计应用。
Enumerating contingency tables via random permanents
给定m个正整数R=（r_i），n个正整数C=（c_j），使得sum r_i = sum c_j =N，并且mn个非负权重W=（w_{ij}），我们考虑总权重T=T（R，C; W）的非负整数矩阵（列联表）D=（d_{ij}），其中行和r_i，列和c_j，并且D的权重等于prod w_{ij} ^{ d_{ij} }。我们提出了一个N复杂度多项式的随机算法，该算法计算一个数T '= T'（R，C; W），使得T' < T < alpha（R，C）T'其中alpha（R，C）= min{prod r_i！r_i^{-r_i}，prod c_j！c_j^{-c_j}} N^N/N！在许多情况下，ln T'提供ln T的渐近精确估计。该算法的思想是将T表示为具有指数分布元素的N × N随机矩阵的积和式的期望，并通过R^{mn}上可有效计算的log-凹函数的积分T'来近似期望。应用计数图中的整数流也进行了讨论。
We initiate the study of the Stam region, defined as the subset of the positive orthant in $\mathbb{R}^{2^n-1}$ that arises from considering entropy powers of subset sums of $n$ independent random vectors in a Euclidean space of finite dimension. We show that the class of fractionally superadditive set functions provides an outer bound to the Stam region, resolving a conjecture of A. R. Barron and the first author. On the other hand, the entropy power of a sum of independent random vectors is not supermodular in any dimension. We also develop some qualitative properties of the Stam region, sho…
On the monotonicity of discrete entropy for log-concave random vectors on $\mathbb{Z}^d$
本文研究了在$\mathbb{Z}^d$上对称log-凹随机向量的离散熵单调性，证明了其单调性并给出了收敛速度。
Maximum Entropy of Sums of Independent Ternary Random Variables
研究三元随机变量和的熵最大化问题，证明在特定分布下熵达到最大，结合Hermite-Biehler定理等方法扩展Shepp-Olkin-Mateev定理。
The symmetric convex hull of random points that are independent and distributed according to the cone probability measure on the $\ell_p$-unit sphere of $\mathbb R^n$ for some $1\leq p < \infty$ is considered. We prove that these random polytopes have uniformly absolutely bounded isotropic constants with overwhelming probability. This generalizes the result for the Euclidean sphere ($p=2$) obtained by D. Alonso-Gutiérrez. The proof requires several different tools including a probabilistic representation of the cone measure due to G. Schechtman and J. Zinn and moment estimates for sums of inde…
Saddle Point Approximation and Central Limit Theorem for Densities in High Dimensions
本研究改进高维独立同分布随机向量和的鞍点近似误差率至O(d^2/n)，并首次在d^2/n趋于0条件下建立密度函数的局部中心极限定理，提供显式误差界。
Over the past few years, a family of interesting new inequalities for the entropies of sums and differences of random variables has been developed by Ruzsa, Tao and others, motivated by analogous results in additive combinatorics. The present work extends these earlier results to the case of random variables taking values in $\mathbb{R}^n$ or, more generally, in arbitrary locally compact and Polish abelian groups. We isolate and study a key quantity, the Ruzsa divergence between two probability distributions, and we show that its properties can be used to extend the earlier inequalities to the…
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