Related papers: Sharp bounds for cumulative distribution functions
We investigate the multivariate central limit theorem for nonlinear statistics by means of Stein's method and Slepian's smart path interpolation method. Based on certain difference operators in theory of concentration inequalities, we…
We calculate some infinite sums containing the digamma function in closed-form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as…
We exploit qualitative probabilistic relationships among variables for computing bounds of conditional probability distributions of interest in Bayesian networks. Using the signs of qualitative relationships, we can implement abstraction…
The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp boundsare obtained for both the fractional integral operators and the…
An integral over the interval $(0,\pi)$ is given for the cumulative distribution function of a sum of independent gamma random variables with different scale and shape parameters. The cumulative distribution function of a positive definite…
We introduce methods to bound the mean of a discrete distribution (or finite population) based on sample data, for random variables with a known set of possible values. In particular, the methods can be applied to categorical data with…
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…
A bound for functional $\Delta(F)=\sup_{x\in\mathbb R}|F(x)-\Phi(x)|$ is obtained, which is uniform for all distribution functions $F$ of random variables with zero mean-value and unity variance. Moreover, a two-point distribution is found,…
Causal inference from observational data is crucial for many disciplines such as medicine and economics. However, sharp bounds for causal effects under relaxations of the unconfoundedness assumption (causal sensitivity analysis) are subject…
For a variant of the algorithm in [Pit19] (arXiv:1903.10816) to compute the approximate density or distribution function of a linear mixture of independent random variables known by a finite sample, it is presented a proof of the functional…
We obtain Rosenthal-type inequalities with sharp constants for moments of sums of independent random variables which are mixtures of a fixed distribution. We also identify extremisers in log-concave settings when the moments of summands are…
We establish sharp $L^p$ integral mean estimates for $(\alpha,\beta)$-harmonic functions on the unit disk. Explicit bounds for the functions and their partial derivatives are obtained in terms of boundary data, by means of the associated…
We obtain functional central limit theorems for both discrete time expressions of the form $1/\sqrt{N}\sum_{n=1}^{[Nt]}(F(X(q_1(n)),\ldots, X(q_{\ell}(n)))-\bar{F})$ and similar expressions in the continuous time where the sum is replaced…
Quantitative multivariate central limit theorems for general functionals of possibly non-symmetric and non-homogeneous infinite Rademacher sequences are proved by combining discrete Malliavin calculus with the smart path method for normal…
The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability of some random variables to a constant and a weak convergence…
This work develops moment bounds for the controlled rough path norm of the solution of semilinear rough partial differential equations.~The novel aspects are two-fold: first we consider rough paths of low time regularity…
We construct a non - improved exponential bounds for distribution of normed sums of i.,i.d. random variables with random numbers of summand.
In this work, we derive sharp non-asymptotic deviation bounds for weighted sums of Dirichlet random variables. These bounds are based on a novel integral representation of the density of a weighted Dirichlet sum. This representation allows…
This work obtains sharp closed-form exponential concentration inequalities of Bernstein type for the ubiquitous beta distribution, improving upon sub-gaussian and sub-gamma bounds previously studied in this context. The proof leverages a…
One reason why standard formulations of the central limit theorems are not applicable in high-dimensional and non-stationary regimes is the lack of a suitable limit object. Instead, suitable distributional approximations can be used, where…