Related papers: Exact lower bounds on the exponential moments of W…
We consider upper exponential bounds for the probability of the event that an absolute deviation of sample mean from mathematical expectation p is bigger comparing with some ordered level epsilon. These bounds include 2 coefficients {alpha,…
Known Bernstein-type upper bounds on the tail probabilities for sums of independent zero-mean sub-exponential random variables are improved in several ways at once. The new upper bounds have a certain optimality property.
Simulation from the truncated multivariate normal distribution in high dimensions is a recurrent problem in statistical computing, and is typically only feasible using approximate MCMC sampling. In this article we propose a minimax tilting…
We consider the least singular value of a large random matrix with real or complex i.i.d. Gaussian entries shifted by a constant $z\in\mathbb{C}$. We prove an optimal lower tail estimate on this singular value in the critical regime where…
We provide moment bounds for expressions of the type $(X^{(1)} \otimes \dots \otimes X^{(d)})^T A (X^{(1)} \otimes \dots \otimes X^{(d)})$ where $\otimes$ denotes the Kronecker product and $X^{(1)}, \dots, X^{(d)}$ are random vectors with…
We consider the problem of choosing design parameters to minimize the probability of an undesired rare event that is described through the average of $n$ iid random variables. Since the probability of interest for near optimal design…
We provide an elementary proof of the lower bound for the variance of continuous unimodal distributions and obtain analogous bounds for the higher order central moments. A lower bound for the rth central moment of discrete distribution is…
A central limit theorem is proved for some strictly stationary sequences of random variables that satisfy certain mixing conditions and are subjected to the "shrinking operators" $U_r(x):=[\max\{|x|-r,0\}]\cdot x/|x|,\ r \ge 0$. For…
We give a simple proof of the recent remarkable exponential improvement for Ramsey lower bounds, obtained by Ma, Shen and Xie. Our key ingredient is an alternative construction based on Gaussian random graphs, which allows us to simplify…
In this brief note, we consider estimation of the bitwise combination $x_1 \lor \dots \lor x_n = \max_i x_i$ observing a set of noisy bits $\tilde x_i \in \{0, 1\}$ that represent the true, unobserved bits $x_i \in \{0, 1\}$ under…
Spectral properties of Gram matrices are central to high dimensional asymptotic analyses of statistical estimators in regression and covariance estimation. These properties, in turn, depend critically on the extreme singular values and…
We give the proof of a tight lower bound on the probability that a binomial random variable exceeds its expected value. The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in…
We derive a novel variational expectation maximization approach based on truncated posterior distributions. Truncated distributions are proportional to exact posteriors within subsets of a discrete state space and equal zero otherwise. The…
The bounds for absolute moments of order statistics are established. Let $X_1,\dots ,X_n$ be independent identically distributed real-valued random variables and let $X_{1:n}\le \dots \le X_{n:n}$ be the corresponding order statistics. The…
We propose an approach to compute the conditional moments of fat-tailed phenomena that, only looking at data, could be mistakenly considered as having infinite mean. This type of problems manifests itself when a random variable Y has a…
We establish nonuniform Berry-Esseen bounds for martingales under the conditional Bernstein condition. These bounds imply Cram\'er type large deviations for moderate $x$'s, and are of exponential decay rate as de la Pe\~na's inequality when…
We develop novel empirical Bernstein inequalities for the variance of bounded random variables. Our inequalities hold under constant conditional variance and mean, without further assumptions like independence or identical distribution of…
We find the exponential exact two-terms non-asymptotic expression for the maximum and minimum distribution of a non-Gaussian, in general case, random vector.
We show, how the classical Berry-Esseen theorem for normal approximation may be used to derive rates of convergence for random sums of centerd, real-valued random variables with respect to a certain class of probability metrics, including…
We provide a lower bound on the probability that a binomial random variable is exceeding its mean. Our proof employs estimates on the mean absolute deviation and the tail conditional expectation of binomial random variables.