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Register a new userGeometric and functional inequalities for log-concave probability sequences
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We investigate various geometric and functional inequalities for the class of log-concave probability sequences. We prove dilation inequalities for log-concave probability measures on the integers. A functional analog of this geometric inequality is derived, giving large and small deviation inequalities from a median, in terms of a modulus of regularity. Our methods are of independent interest, we find that log-affine sequences are the extreme points of the set of log-concave sequences belonging to a half-space slice of the simplex. This amounts to a discrete analog of the localization lemma of Lovasz and Simonovits. Further applications of this lemma are used to produce a discrete version of the Prekopa-Leindler inequality, large deviation inequalities for log-concave measures about their mean, and provide insight on the stability of generalized log-concavity under convolution.
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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).
We study functional inequalities (Poincare, Cheeger, log-Sobolev) for probability measures obtained as perturbations. Several explicit results for general measures as well as log-concave distributions are given.The initial goal of this work was to obtain explicit bounds on the constants in view of statistical applications for instance. These results are then applied to the Langevin Monte-Carlo method used in statistics in order to compute Bayesian estimators.
Log-concave distributions include some important distributions such as normal distribution, exponential distribution and so on. In this note, we show inequalities between two Lp-norms for log-concave distributions on the Euclidean space. These inequalities are the generalizations of the upper and lower bound of the differential entropy and are also interpreted as a kind of expansion of the inequality between two Lp-norms on the measurable set with finite measure.
Two-sided bounds are explored for concentration functions and Renyi entropies in the class of discrete log-concave probability distributions. They are used to derive certain variants of the entropy power inequalities.
Let ${P_{theta}:theta in {mathbb R}^d}$ be a log-concave location family with $P_{theta}(dx)=e^{-V(x-theta)}dx,$ where $V:{mathbb R}^dmapsto {mathbb R}$ is a known convex function and let $X_1,dots, X_n$ be i.i.d. r.v. sampled from distribution $P_{theta}$ with an unknown location parameter $theta.$ The goal is to estimate the value $f(theta)$ of a smooth functional $f:{mathbb R}^dmapsto {mathbb R}$ based on observations $X_1,dots, X_n.$ In the case when $V$ is sufficiently smooth and $f$ is a functional from a ball in a Holder space $C^s,$ we develop estimators of $f(theta)$ with minimax optimal error rates measured by the $L_2({mathbb P}_{theta})$-distance as well as by more general Orlicz norm distances. Moreover, we show that if $dleq n^{alpha}$ and $s>frac{1}{1-alpha},$ then the resulting estimators are asymptotically efficient in Hajek-LeCam sense with the convergence rate $sqrt{n}.$ This generalizes earlier results on estimation of smooth functionals in Gaussian shift models. The estimators have the form $f_k(hat theta),$ where $hat theta$ is the maximum likelihood estimator and $f_k: {mathbb R}^dmapsto {mathbb R}$ (with $k$ depending on $s$) are functionals defined in terms of $f$ and designed to provide a higher order bias reduction in functional estimation problem. The method of bias reduction is based on iterative parametric bootstrap and it has been successfully used before in the case of Gaussian models.
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