We derive Berry-Esseen approximation bounds for general functionals of independent random variables, based on a continuous-time integration by parts setting and discrete chaos expansions methods. Our approach improves related results obtained in discrete-time settings applies to U-statistics satisfying the weak assumption decomposability Hoeffding sense, yield Kolmogorov distance instead Wasser...

to demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the banach-zareckitheorem is presented on the basis of the radon-nikodym theoremwhich emphasizes on measure-type properties of the lebesgueintegral. the banach-zarecki theorem says that a real-valuedfunction $f$ is absolutely continuous on a finite closed intervalif and only if it is continuo...

We prove the following conjecture, due to Tomaszewski (1986): Let X=∑i=1naixi, where ∑iai2=1 and each xi is a uniformly random sign. Then Pr⁡[|X|≤1]≥1/2. Our main novel tools are local concentration inequalities an improved Berry-Esseen inequality for Rademacher sums.

This note provides a conditional Berry-Esseen bound for the sum of martingale difference sequence $\{X_i\}_{i=1}^n$ in $\mathbb{R}^d$, $d\ge 1$, adapted to filtration $\{\mathcal{F}_i\}_{i=1}^n$. We approximate distribution $S=\sum_{i=1}^n X_i$ given some $\sigma$-field $\mathcal{F}_0\subset \mathcal{F}_1$ by that mean-zero normal random vector having same variance $\mathcal{F}_0 $ as $S$. Assu...

Abstract In this paper, a new technique is introduced to obtain non-uniform Berry-Esseen bounds for normal and nonnormal approximations by unbounded exchangeable pairs. This does not rely on the concentration inequalities developed Chen Shao [4,5] can be applied quadratic forms general Curie-Weiss model.