R. Ibragimov, Sh. Sharakhmetov
In this paper, we prove analogues of Khintchine and Rosenthal's moment inequalities for symmetric statistics (U-statistics) of arbitrary order. An example that shows significance of each term in the analogues of Rosenthal's bounds for symmetric statistics is constructed as well.
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We show that shift Harnack type inequalities (in the sense of F.-Y.~Wang \cite{Wan14}) are preserved under Bochner's subordination. The proofs are based on two types of moment estimates for subordinators. As a by-product we establish moment estimates for general Lévy processes.
We establish a sharp moment comparison inequality between an arbitrary negative moment and the second moment for sums of independent uniform random variables, which extends Ball's cube slicing inequality.
We prove that in the context of general Markov semigroups Beckner inequalities with constants separated from zero as $p\to 1^+$ are equivalent to the modified log Sobolev inequality (previously only one implication was known to hold in this generality). Further, by adapting an argument by Boucheron et al. we derive Sobolev type moment estimates which hold under these functional inequalities. We illustrate our results with applications to concentration of measure estimates (also of higher order, beyond the case of Lipschitz functions) for various stochastic models, including random permutations…
On Moment-Entropy inequalities in the space of matrices
本文将Lutwak等人建立的仿射信息论推广至矩阵空间，研究了由具备欧氏结构的$\mathbb{R}^n$与带伪范数的$\mathbb{R}^m$相互作用诱导的$n\times m$矩阵空间上的矩-熵不等式。
We study a class of logarithmic Sobolev inequalities with a general form of the energy functional. The class generalizes various examples of modified logarithmic Sobolev inequalities considered previously in the literature. Refining a method of Aida and Stroock for the classical logarithmic Sobolev inequality, we prove that if a measure on $\mathbb{R}^n$ satisfies a modified logarithmic Sobolev inequality then it satisfies a family of $L^p$-Sobolev-type inequalities with non-Euclidean norms of gradients (and dimension-independent constants). The latter are shown to yield various concentration-…
In this paper non-asymptotic exponential estimates are derived for the tail distribution of polynomial martingale differences in terms unconditional tails distributions of summands. Applications are considered in the theory of polynomials on independent random variables, to the theory of U-statistics, multiply martingale series and in the theory of weak compactness measures on the Banach spaces. Partially supported by the Israel Ministry of Absorbtion Mathematics Subject Classification (2000): 47A45, 47B10, 60F10, 60G42.
We derive the tail inequalities between two random variables starting from inequalities between its moment, or more generally between its Lebesgue-Riesz norms, which holds true on certain sets of parameters. We consider some applications into the martingale theory.
Probability and moment inequalities for additive functionals of geometrically ergodic Markov chains
该研究针对几何遍历马尔可夫链的加性泛函，建立了适用范围更广的矩与Bernstein型不等式，推广了独立随机变量的相关结果，且不等式显式依赖条件中的常数。
We present Rosenthal-type moment inequalities for matrix-valued U-statistics of order 2. As a corollary, we obtain new matrix concentration inequalities for U-statistics. One of our main technical tools, a version of the non-commutative Khintchine inequality for the spectral norm of the Rademacher chaos, could be of independent interest.
We study the exact constants in the moment inequalities for sums of centered independent random variables: improve their asymptotics, low and upper bounds, calculate more exact asymptotics, elaborate the numerical algorithm for their calculation, study the class of smoothing etc.
In this paper we obtain the non-asymptotic exact moment and tails estimates for polynomial on martingale differences. We give also some examples on order to show the exactness of obtained results.
This paper is devoted to the study of $Φ$-moment inequalities for noncommutative martingales. In particular, we prove the noncommutative $Φ$-moment analogues of martingale transformations, Stein's inequalities, Khintchine's inequalities for Rademacher's random variables, and Burkholder-Gundy's inequalities. The key ingredient is a noncommutative version of Marcinkiewicz type interpolation theorem for Orlicz spaces which we establish in this paper.
A Bernstein-type exponential inequality for (generalized) canonical U-statistics of order 2 is obtained and the Rosenthal and Hoffmann-Jørgensen inequalities for sums of independent random variables are extended to (generalized) U-statistics of any order whose kernels are either nonnegative or canonical
In this work, we will generalize the moment generating function to Riesz spaces. We will derive some of its properties and use it to prove concentration inequalities on Riesz spaces.
The Hoeffding-type-inequalities are obtained for the distribution tails of canonical (degenerate) U- and V-statistics of an arbitrary order based on samples from a stationary sequence of observations satisfying $ρ$-mixing.
We present order of magnitude estimates for the quantiles of non-negative linear combinations of non-negative random variables, as well as deviation inequalities for general linear combinations of independent random variables, under the assumption that all random variables satisfy the same power-type tail bound on $\mathbb{P}\{\left\vert X_i\right\vert>t\}$ of the form $t^{-q}$, $t^{-q/2}$ or $t^{-q/2}(\ln t)^{q/2}$, for $q>2$. The third type is applicable in the nonlinear setting. In the situations we consider, these results improve on classical estimates of Nagaev.
We prove decoupling inequalities for random polynomials in independent random variables with coefficients in vector space. We use various means of comparison, including rearrangement invariant norms (e.g., Orlicz and Lorentz norms), tail distributions, tightness, hypercontractivity, etc.
Hoeffding-type exponential inequalities are obtained for the distribution tails of canonical von Mises' statistics of arbitrary order based on samples from a stationary sequence of random variables satisfying the φ-mixing condition. The present paper weakens the restrictions on the coefficient φ() which are contained in the paper "Exponential inequalities for the distributions of canonical U- and V-statistics of dependent observations" by Borisov and Volodko (2009). At the same time, we correct the corresponding proof in this paper.
We present moment inequalities for completely degenerate Banach space valued (generalized) U-statistics of arbitrary order. The estimates involve suprema of empirical processes which, in the real-valued case, can be replaced by simpler norms of the kernel matrix (i.e., norms of some multilinear operators associated with the kernel matrix). As a corollary, we derive tail inequalities for U-statistics with bounded kernels and for some multiple stochastic integrals.
Khinchin inequalities for uniforms on spheres with a deficit
本文改进了球面上均匀分布随机向量的矩比较不等式，给出了具有最优高维性能的缺陷项。
A general method for obtaining moment inequalities for functions of independent random variables is presented. It is a generalization of the entropy method which has been used to derive concentration inequalities for such functions [Boucheron, Lugosi and Massart Ann. Probab. 31 (2003) 1583-1614], and is based on a generalized tensorization inequality due to Latala and Oleszkiewicz [Lecture Notes in Math. 1745 (2000) 147-168]. The new inequalities prove to be a versatile tool in a wide range of applications. We illustrate the power of the method by showing how it can be used to effortlessly re-…
Normal comparison lemma and Slepian's inequality are essential tools in the study of Gaussian processes. In this paper we extend normal comparison lemma and derive various related comparison inequalities including Slepian's inequality for order statistics of two Gaussian arrays.The derived results can be applied in numerous problems related to the study of the supremum of order statistics of Gaussian processes. In order to illustrate the range of possible applications, we analyze the lower tail behaviour of order statistics of self-similar Gaussian processes and derive mixed Gumbel limit theor…
In this paper we give refinements of some convex and log-convex moment inequalities of the first and second order using a special kind of positive semi-definite form. An open problem concerning eight parameter refinement of second order is also stated with some applications in Information Theory.
We derive explicit central moment inequalities for random variables that admit a Stein coupling, such as exchangeable pairs, size--bias couplings or local dependence, among others. The bounds are in terms of moments (not necessarily central) of variables in the Stein coupling, which are typically local in some sense, and therefore easier to bound. In cases where the Stein couplings have the kind of behaviour leading to good normal approximation, the central moments are closely bounded by those of a normal. We show how the bounds can be used to produce concentration inequalities, and compare th…
We obtain bivariate forms of Gumbel's, Fréchet's and Chung's linear inequalities for $P(S\ge u, T\ge v)$ in terms of the bivariate binomial moments $\{S_{i,j}\}$, $1\le i\le k, 1\le j\le l$ of the joint distribution of $(S,T)$. At $u=v=1$, the Gumbel and Fréchet bounds improve monotonically with non-decreasing $(k,l)$. The method of proof uses combinatorial identities, and reveals a multiplicative structure before taking expectation over sample points.
Moment estimates, exponential integrability, concentration inequalities and exit times estimates on evolving manifolds
该研究在带C¹族完备黎曼度量与C^{1,∞}族向量场的非紧光滑演化流形上，证明了扩散过程的矩估计、指数可积性、集中不等式及首离时估计，拓展了演化流形上扩散过程的分析结果。
We prove a new family of inequalities involving squares of random variables belonging to the Wiener chaos associated with a given Gaussian field. Our result provides a substantial generalisation, as well as a new analytical proof, of an estimate by Frenkel (2007), and also constitute a natural real counterpart to an inequality established by Arias-de-Reyna (1998) in the framework of complex Gaussian vectors. We further show that our estimates can be used to deduce new lower bounds on homogeneous polynomials, thus partially improving results by Pinasco (2012), as well as to obtain a novel proba…
Stability of Khintchine inequalities with optimal constants between the second and the $p$-th moment for $p \ge 3$
本文研究了p≥3时第二矩与第p矩之间Khintchine不等式在最优常数下的稳定性，通过引入依赖于Rademacher和系数向量的亏量项，给出了经典不等式的一个加强版本。
Concentration inequalities using higher moments information
针对已知随机变量前 p 阶矩的情形，本文提出更紧且具闭式形式的推广 Hoeffding 与 Bennett 不等式，并通过矩母函数新界及凸性分析加以证明。
Some notes on moment inequalities for heavy-tailed distributions
本文研究重尾（尤其 Pareto 型）分布的矩与尾部关系，在正则性条件下验证结论锐性，并给出任意阶 $d$ 多项式混沌的集中界。
Moment inequalities for higher-order (inverse) stochastic dominance
本文研究了高阶逆随机支配的矩不等式条件，并探讨了背景风险变量对高阶随机支配的影响，扩展了之前的研究成果。
In 1983 Cox and Kemperman proved that $\E f(ξ)+ \E f(η) \le \E f(ξ+η)$ for all functions $f$, such that $f(0)=0$ and the second derivative $f''(y)$ is convex, and all independent centered random variables $ξ$ and $η$ satisfying certain moment restrictions. We show that the minimal moment restrictions are sufficient for the inequality to be valid, and write out a less restrictive condition on $f$ for the inequality to hold. Besides, Cox and Kemperman (1983) found out the optimal constants $A_ρ$ and $B_ρ$ for the inequalities $A_ρ(\E |ξ|^ρ+ \E |η|^ρ) \le \E |ξ+ η|^ρ\le B_ρ(\E |ξ|^ρ+ \E |η|^ρ) $,…
Concentration and moment inequalities for sums of independent heavy-tailed random matrices
本文针对仅存在低阶有限矩的独立重尾随机矩阵和，证明了依赖有效秩等内在维度的Fuk-Nagaev型与Rosenthal型不等式，并通过多个应用验证了结果优势。
Second order mixed moment inequalities based on Gram matrices
将Walker不等式推广到任意随机向量Gram矩阵的一般二阶混合矩不等式族，并讨论其对有偏估计Cramer-Rao下界的影响。
We establish several optimal moment comparison inequalities (Khinchin-type inequalities) for weighted sums of independent identically distributed symmetric discrete random variables which are uniform on sets of consecutive integers. Specifically, we obtain sharp constants for the second moment and any moment of order at least 3 (using convex dominance by Gaussian random variables). In the case of only 3 atoms, we also establish a Schur-convexity result. For moments of order less than 2, we get sharp constants in two cases by exploiting Haagerup's arguments for random signs.
Absolute moment inequalities under quadratic-form positivity
本文证明了Zhuang和Hu在Remark 3.1中提出的开放问题，并更一般地考虑了对称联合概率质量函数和联合密度，其关联二次型非负，得到了绝对矩不等式成立的充要条件。
Let $(ξ_i)_{i=1,...,n}$ be a sequence of independent and symmetric random variables. We consider the upper bounds on tail probabilities of self-normalized deviations $$ \mathbf{P} \Big( \max_{1\leq k \leq n} \sum_{i=1}^{k} |ξ_i|\big/ \big(\sum_{i=1}^{n} |ξ_i|^β\big)^{1/β} \geq x \Big) $$ for $x>0$ and $β>1.$ Our bound is the best that can be obtained from the Bernstein inequality under the present assumption. An application to Student's $t$-statistics is also given.
We obtain moment and Gaussian bounds for general Lipschitz functions evaluated along the sample path of a Markov chain. We treat Markov chains on general (possibly unbounded) state spaces via a coupling method. If the first moment of the coupling time exists, then we obtain a variance inequality. If a moment of order 1+epsilon of the coupling time exists, then depending on the behavior of the stationary distribution, we obtain higher moment bounds. This immediately implies polynomial concentration inequalities. In the case that a moment of order 1+epsilon is finite uniformly in the starting po…
This paper is devoted to the study of $Φ$-moments of sums of independent/freely independent random variables. More precisely, let $(f_k)_{k=1}^n$ be a sequence of positive (symmetrically distributed) independent random variables and let $Φ$ be an Orlicz function with $Δ_2$-condition. We provide an equivalent expression for the quantity $\mathbb{E}(Φ(\sum_{k=1}^n f_k))$ in term of the sum of disjoint copies of the sequence $(f_k)_{k=1}^n.$ We also prove an analogous result in the setting of free probability. Furthermore, we provide an equivalent characterization of $τ(Φ(\sup^+_{1\leq k\leq n}x_…
In this work we design a general method for proving moment inequalities for polynomials of independent random variables. Our method works for a wide range of random variables including Gaussian, Boolean, exponential, Poisson and many others. We apply our method to derive general concentration inequalities for polynomials of independent random variables. We show that our method implies concentration inequalities for some previously open problems, e.g. permanent of a random symmetric matrices. We show that our concentration inequality is stronger than the well-known concentration inequality due …
Deviation and moment inequalities for Banach-valued $U$-statistics
本文在可分 Banach 空间光滑性假设下建立了独立数据 U 统计量的偏差不等式，并将其用于大数定律速率、Hölder 型泛函中心极限定理和不完全 U 统计量的矩不等式。
We establish exponential inequalities for a class of V-statistics under strong mixing conditions. Our theory is developed via a novel kernel expansion based on random Fourier features and the use of a probabilistic method. This type of expansion is new and useful for handling many notorious classes of kernels.
Exponential inequalities and laws of the iterated logarithm for multiple Poisson--Wiener integrals and Poisson $U$-statistics
本文针对泊松空间上的多重随机积分和泊松U-统计量建立尾概率与矩不等式，并据此在泊松过程强度趋于无穷时证明其重对数律，进而改进和推广了几何随机图、泊松k-平面交、Ornstein-Uhlenbeck Lévy过程二次泛函及标记过程U-统计量等经典泛函的测度集中结果。
Constructive approaches to concentration inequalities with independent random variables
本文提出系统性凸优化方法研究独立随机变量浓度不等式，通过变分方法扩展经典矩母函数至一阶矩条件，并开发基于平方和近似的多项式方法处理高阶矩条件，进而改进Hoeffding、Bennett和Bernstein不等式，提供更优的最坏情况保证。
Concentration properties of functionals of general Poisson processes are studied. Using a modified $Φ$-Sobolev inequality a recursion scheme for moments is established, which is of independent interest. This is applied to derive moment and concentration inequalities for functionals on abstract Poisson spaces. Applications of the general results in stochastic geometry, namely Poisson cylinder models and Poisson random polytopes, are presented as well.
We show that the probability that a multilinear polynomial $f$ of independent random variables exceeds its mean by $λ$ is at most $e^{-λ^2 / (R^q Var(f))}$ for sufficiently small $λ$, where $R$ is an absolute constant. This matches (up to constants in the exponent) what one would expect from the central limit theorem. Our methods handle a variety of types of random variables including Gaussian, Boolean, exponential, and Poisson. Previous work by Kim-Vu and Schudy-Sviridenko gave bounds of the same form that involved less natural parameters in place of the variance.
We show sharpened forms of the concentration of measure phenomenon centered at first order stochastic expansions. The bound are based on second order difference operators and second order derivatives. Applications to functions on the discrete cube and stochastic Hoeffding type expansions in mathematical statistics are studied as well as linear eigenvalue statistics in random matrix theory.
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。
Spherical symmetry arguments are used to produce a general device to convert identities and inequalities for the $p$th absolute moments of real-valued random variables into the corresponding identities and inequalities for the $p$th moments of the norms of random vectors in Hilbert spaces. Particular results include the following: (i) an expression of the $p$th moment of the norm of such a random vector $X$ in terms of the characteristic functional of $X$; (ii) an extension of a previously obtained von~Bahr--Esseen-type inequality for real-valued random variables with the best possible constan…
Multivariate Second-Order $p$-Poincaré Inequalities
本文在最小矩假设下建立了多元 Poisson 泛函正态逼近的新界，将所需矩条件从4阶降低到(2+ε)阶，并首次给出了随机几何图两个多元泛函的定量中心极限定理。
This note describes non-asymptotic variance and tail bounds for order statistics of samples of independent identically distributed random variables. Those bounds are checked to be asymptotically tight when the sampling distribution belongs to a maximum domain of attraction. If the sampling distribution has non-decreasing hazard rate (this includes the Gaussian distribution), we derive an exponential Efron-Stein inequality for order statistics: an inequality connecting the logarithmic moment generating function of centered order statistics with exponential moments of Efron-Stein (jackknife) est…
We consider a class of non-homogeneous Markov chains, that contains many natural examples. Next, using martingale methods, we establish some deviation and moment inequalities for separately Lipschitz functions of such a chain, under moment conditions on some dominating random variables.
Certain previously known upper bounds on the moments of the norm of martingales in 2-smooth Banach spaces are improved. Some of these improvements hold even for sums of independent real-valued random variables. Applications to concentration of measure on product spaces for separately Lipschitz functions are presented, including ones concerning the central moments of the norm of the sums of independent random vectors in any separable Banach space.
Concentration inequalities using approximate zero bias couplings with applications to Hoeffding's statistic under the Ewens distribution
本文利用近似零偏耦合技术，推导了Hoeffding统计量在Ewens分布下的集中不等式，并通过模拟验证了其在θ≤1时的有效性。
Motivated by problems arising in random sampling of trigonometric polynomials, we derive exponential inequalities for the operator norm of the difference between the sample second moment matrix $n^{-1}U^*U$ and its expectation where $U$ is a complex random $n\times D$ matrix with independent rows. These results immediately imply deviation inequalities for the largest (smallest) eigenvalues of the sample second moment matrix, which in turn lead to results on the condition number of the sample second moment matrix. We also show that trigonometric polynomials in several variables can be learned f…
Matrix Moment and Concentration Inequalities for Martingales and Ergodic Markov Chains with Applications in Statistical Learning
本文建立了适用于几何遍历马尔可夫链等的新型矩阵矩与集中不等式，将其推广至无限维空间并应用于马尔可夫数据的协方差估计与主成分分析，获得了更优的界。
We propose a new approach for deriving probabilistic inequalities based on bounding likelihood ratios. We demonstrate that this approach is more general and powerful than the classical method frequently used for deriving concentration inequalities such as Chernoff bounds. We discover that the proposed approach is inherently related to statistical concepts such as monotone likelihood ratio, maximum likelihood, and the method of moments for parameter estimation. A connection between the proposed approach and the large deviation theory is also established. We show that, without using moment gener…
Rosenthal-type inequalities for linear statistics of Markov chains
本文为几何遍历马尔可夫链的可加泛函建立了类似Rosenthal不等式的集中不等式，显式界依赖于混合时间，并采用基于泊松分解的新证明方法。
Moments Comparison Inequalities for Critical 2d Stochastic Heat Flow, Polymers and GMC
本文利用高斯凸不等式和自举论证，为临界二维随机热流等对象建立了矩比较不等式，证明局部平均SHF的p阶矩与其二阶矩的p(p-1)/2次幂在乘法常数下互有界。
We introduce a version of Stein's method for proving concentration and moment inequalities in problems with dependence. Simple illustrative examples from combinatorics, physics, and mathematical statistics are provided.
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