Some Moment and Exponential Inequalities for V-Statistics with Bounded Kernels

This paper provides moment and exponential inequalities for V-statistics with uniformly bounded kernels. The main results are based on moment inequalities for sub-Bernoulli functions of independent random variables. These moment inequalities lead to extensions of the Bernstein inequality to V-statistics. U-statistics are also discussed.     

About the rate function in concentration inequalities for suprema of bounded empirical processes

We provide new deviation inequalities in the large deviations bandwidth for suprema of empirical processes indexed by classes of uniformly bounded functions associated with independent and identically distributed random variables. The improvements we get concern the rate function which is, as expected, the Legendre transform of the suprema of the log-Laplace transform of the pushforward measure by the functions of the considered class (up to an additional corrective term). Our approach is based on a decomposition in martingale together with some comparison inequalities.     

On extensions of Hoeffding’s inequality for panel data

Hoeffding’s inequality provides a probability bound for the deviation between the average of n independent bounded random variables and its mean. This paper introduces two inequalities that extend Hoeffding’s inequality to panel data, which consists of several mutually independent sequences of dependent data with strong mixing or with a dependence structure being even more general than strong mixing. One is denoted as the Bosq’s Extension which is an extension of Bosq’s inequality (Bosq, 1993) to panel data and the other one is called the Triplex Extension, which extends the Triplex inequality (Jiang, 2009) to panel data. The Bosq’s Extension provides a tighter upper probability bound, while the Triplex Extension is more relaxed in assumption allowing unboundedness and more general dependence than strong mixing. We also apply these two inequalities to establish the convergence rate of empirical risk minimization for high dimensional panel data with variable selection.     

Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm

Kolmogorov’s exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov type exponential inequalities of the partial sums of independent random variables as well as negatively dependent random variables under the sub-linear expectations. As applications of the exponential inequalities, the laws of the iterated logarithm in the sense of non-additive capacities are proved for independent or negatively dependent identically distributed random variables with finite second order moments. For deriving a lower bound of an exponential inequality, a central limit theorem is also proved under the sub-linear expectation for random variables with only finite variances.     

New Bernstein's Inequalities for Dependent Observations and Applications to Learning Theory

The classical concentration inequalities deal with the deviations of functions of independent and identically distributed (i.i.d.) random variables from their expectation and these inequalities have numerous important applications in statistics and machine learning theory. In this paper we go far beyond this classical framework by establish two new Bernstein type concentration inequalities for -mixing sequence and uniformly ergodic Markov chains. As the applications of the Bernstein's inequalities, we also obtain the bounds on the rate of uniform deviations of empirical risk minimization (ERM) algorithms based on -mixing observations.     

The dual Brunn-Minkowski theory for bounded Borel sets: Dual affine quermassintegrals and inequalities

This paper develops a significant extension of E. Lutwak's dual Brunn-Minkowski theory, originally applicable only to star-shaped sets, to the class of bounded Borel sets. The focus is on expressions and inequalities involving chord-power integrals, random simplex integrals, and dual affine quermassintegrals. New inequalities obtained include those of isoperimetric and Brunn-Minkowski type. A new generalization of the well-known Busemann intersection inequality is also proved. Particular attention is given to precise equality conditions, which require results stating that a bounded Borel set, almost all of whose sections of a fixed dimension are essentially convex, is itself essentially convex.     

经验过程大偏差

本文研究了独立但不同分布的随机变量序列的经验过程大偏差原理.运用Talagrand-Ledoux偏差不等式建立了该经验过程大偏差估计的充分和必要条件.     

On classical analogues of free entropy dimension

We define a classical probability analogue of Voiculescu's free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on Rn. We show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension. First, it can be viewed as a kind of fractal dimension. Second, if one extends Bochner's inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical Bochner's inequalities, then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in Bochner's inequality for a measure regularized by convolution with the Gaussian law as the regularization is removed. We introduce a free analogue of the Bochner inequality and study the related free entropy dimension quantity. We show that it is greater or equal to the non-microstates free entropy dimension.     

Moment inequalities for the partial sums of random variables

This paper discusses the conditions under which Rosenthal type inequality is obtained from M-Z-B type inequality. And M-Z-B type inequality is proved for a wide class of random variables. Hence Rosenthal type inequalities for some classes of random variables are obtained.     

Statistical dependence through common risk factors: With applications in uncertainty analysis

A model for building statistical dependence between marginal distribution with bounded support is discussed. The model is geared towards elicitation of dependence parameters through expert judgment. The resulting joint distribution may be useful in uncertainty analyses where dependence between random variables with a bounded support is present due to common risk factors, such as, e.g., in the classical Project Evaluation and Review Technique.     

The Fundamental Blossoming Inequality in Chebyshev Spaces—I: Applications to Schur Functions

A classical theorem by Chebyshev says how to obtain the minimum and maximum values of a symmetric multiaffine function of n variables with a prescribed sum. We show that, given two functions in an Extended Chebyshev space good for design, a similar result can be stated for the minimum and maximum values of the blossom of the first function with a prescribed value for the blossom of the second one. We give a simple geometric condition on the control polygon of the planar parametric curve defined by the pair of functions ensuring the uniqueness of the solution to the corresponding optimization problem. This provides us with a fundamental blossoming inequality associated with each Extended Chebyshev space good for design. This inequality proves to be a very powerful tool to derive many classical or new interesting inequalities. For instance, applied to Müntz spaces and to rational Müntz spaces, it provides us with new inequalities involving Schur functions which generalize the classical MacLaurin’s and Newton’s inequalities. This work definitely demonstrates that, via blossoms, CAGD techniques can have important implications in other mathematical domains, e.g., combinatorics.     

Model spaces for sharp isoperimetric inequalities

We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are sharp for sets of any given measure and with respect to all parameters (curvature, dimension and diameter). Moreover, for each choice of parameters, we identify the model spaces which are extremal for the isoperimetric problem. In particular, we recover the Gromov–Lévy and Bakry–Ledoux isoperimetric inequalities, which state that whenever the curvature is strictly positively bounded from below, these model spaces are the n-sphere and Gauss space, corresponding to generalized dimension being n and ∞, respectively. In all other cases, which seem new even for the classical Riemannian-volume measure, it turns out that there is no single model space to compare to, and that a simultaneous comparison to a natural one parameter family of model spaces is required, nevertheless yielding a sharp result.     

A local probability exponential inequality for the large deviation of an empirical process indexed by an unbounded class of functions and its application

A local probability exponential inequality for the tail of large deviation of an empirical process over an unbounded class of functions is proposed and studied. A new method of truncating the original probability space and a new symmetrization method are given. Using these methods, the local probability exponential inequalities for the tails of large deviations of empirical processes with non-i.i.d. independent samples over unbounded class of functions are established. Some applications of the inequalities are discussed. As an additional result of this paper, under the conditions of Kolmogorov theorem, the strong convergence results of Kolmogorov on sums of non-i.i.d. independent random variables are extended to the cases of empirical processes indexed by unbounded classes of functions, the local probability exponential inequalities and the laws of the logarithm for the empirical processes are obtained.     

Maximal inequalities for demimartingales and a strong law of large numbers

Chow's maximal inequality for (sub)martingales is extended to the case of demi(sub)martingales introduced by Newman and Wright (Z. Wahrsch. Verw. Geb. 59 (1982) 361–371). This result serves as a “source” inequality for other inequalities such as the Hajek–Renyi inequality and Doob's maximal inequality and leads to a strong law of large numbers. The partial sum of mean zero associated random variables is a demimartingale. Therefore, maximal inequalities and a strong law of large numbers are obtained for associated random variables as special cases.     

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本文给出了上期望空间中独立随机变量部分和的最大不等式、指数 不等式、Marcinkiewicz-Zygmund不等式. 并且应用指数不等式和Marcinkiewicz-Zygmund不等式 研究了随机变量部分和序列完备收敛的性质.     

A Bennett concentration inequality and its application to suprema of empirical processesUne inégalité de concentration de type Bennett et son application aux maxima de processus empiriques

We introduce new concentration inequalities for functions on product spaces. They allow to obtain a Bennett type deviation bound for suprema of empirical processes indexed by upper bounded functions. The result is an improvement on Rio's version [6] of Talagrand's inequality [7] for equidistributed variables. To cite this article: O. Bousquet, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 495–500.     

Exact Estimates for Moments of Random Bilinear Forms

The present paper concentrates on the analogues of Rosenthal's inequalities for ordinary and decoupled bilinear forms in symmetric random variables. More specifically, we prove the exact moment inequalities for these objects in terms of moments of their individual components. As a corollary of these results we obtain the explicit expressions for the best constant in the analogues of Rosenthal's inequality for ordinary and decoupled bilinear forms in identically distributed symmetric random variables in the case of the fixed number of random variables.     

On the exponential inequalities for negatively dependent random variables

A number of exponential inequalities for identically distributed negatively dependent and negatively associated random variables have been established by many authors. The proofs use the truncation technique together with the control of the bounded terms and unbounded terms. In this paper, we improve essentially the control of bounds for the unbounded terms and obtain exponential inequalities for negatively dependent random variables which include negatively associated random variables. Our results improve on the corresponding ones in the literature.     

Isoperimetric, Sobolev, and Eigenvalue Inequalities via the Alexandroff-Bakelman-Pucci Method: A Survey

This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.     

An Inequality for the Sum of Independent Bounded Random Variables

We give a simple inequality for the sum of independent, bounded random variables. This inequality improves on the celebrated result of Hoeffding in a special case. It is optimal in the limit where the sum tends to a Poisson random variable.