Related papers: Concentration inequalities for log-concave sequenc…
The relative log-concavity ordering $\leq_{\mathrm{lc}}$ between probability mass functions (pmf's) on non-negative integers is studied. Given three pmf's $f,g,h$ that satisfy $f\leq_{\mathrm{lc}}g\leq_{\mathrm{lc}}h$, we present a pair of…
If a random variable is not exponentially integrable, it is known that no concentration inequality holds for an infinite sequence of independent copies. Under mild conditions, we establish concentration inequalities for finite sequences of…
We give concentration bounds for martingales that are uniform over finite times and extend classical Hoeffding and Bernstein inequalities. We also demonstrate our concentration bounds to be optimal with a matching anti-concentration…
We investigate the rate of convergence in the central limit theorem for convex sets. We obtain bounds with a power-law dependence on the dimension. These bounds are asymptotically better than the logarithmic estimates which follow from the…
We study the problem of estimating multivariate log-concave probability density functions. We prove the first sample complexity upper bound for learning log-concave densities on $\mathbb{R}^d$, for all $d \geq 1$. Prior to our work, no…
The phenomenon of entropy concentration provides strong support for the maximum entropy method, MaxEnt, for inferring a probability vector from information in the form of constraints. Here we extend this phenomenon, in a discrete setting,…
Quantile aggregation with dependence uncertainty has a long history in probability theory with wide applications in finance, risk management, statistics, and operations research. Using a recent result on inf-convolution of quantile-based…
Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. Closely related maximum…
We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these…
We prove a log-Sobolev inequality for a certain class of log-concave measures in high dimension. These are the probability measures supported on the unit cube in R^n whose density takes the form exp(-H) where the function H is assumed to be…
New Vapnik and Chervonenkis type concentration inequalities are derived for the empirical distribution of an independent random sample. Focus is on the maximal deviation over classes of Borel sets within a low probability region. The…
We investigate 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 analogue of this geometric inequality…
We investigate concentration inequalities for Dirichlet and Multinomial random variables.
Menon's proof of the preservation of log-concavity of sequences under convolution becomes simpler when adapted to 2-sided infinite sequences. Under assumption of log-concavity of two 2-sided infinite sequences, the existence of the…
A concentration result for quadratic form of independent subgaussian random variables is derived. If the moments of the random variables satisfy a "Bernstein condition", then the variance term of the Hanson-Wright inequality can be…
A novel computational approach to log-concave density estimation is proposed. Previous approaches utilize the piecewise-affine parametrization of the density induced by the given sample set. The number of parameters as well as non-smooth…
In recent years, log-concave density estimation via maximum likelihood estimation has emerged as a fascinating alternative to traditional nonparametric smoothing techniques, such as kernel density estimation, which require the choice of one…
A broad set of sufficient conditions that guarantees the existence of the maximum entropy (maxent) distribution consistent with specified bounds on certain generalized moments is derived. Most results in the literature are either focused on…
We consider log-convex sequences that satisfy an additional constraint imposed on their rate of growth. We call such sequences log-balanced. It is shown that all such sequences satisfy a pair of double inequalities. Sufficient conditions…
We show that the uniform distribution minimizes entropy among all one-dimensional symmetric log-concave distributions with fixed variance, as well as various generalizations of this fact to R\'enyi entropies of orders less than 1 and with…