Related papers: Normal Approximation for Weighted Sums under a Sec…

For Kolmogorov test we find natural conditions of uniform consistency of sets of alternatives approaching to hypothesis. Sets of alternatives can be defined both in terms of distribution functions and in terms of densities.

We consider the number of crossings in a graph which is embedded randomly on a convex set of points. We give an estimate to the normal distribution in Kolmogorov distance which implies a convergence rate of order $n^{-1/2}$ for various…

We present combinatorial approximation algorithms for the weighted correlation clustering problem. In this problem, we have a set of vertices and two weight values for each pair of vertices, denoting their difference and similarity. The…

We derive new explicit bounds for the total variation distance between two convolution products of $n$ probability distributions, one of which having identical convolution factors. Approximations by finite signed measures of arbitrary order…

For rescaled additive functionals of the sine-process, upper bounds are obtained for their speed of convergence to the Gaussian distribution with respect to the Kolmogorov-Smirnov metric. Under scaling with coefficient $R$ the…

The famous results of Koml\'os, Major and Tusn\'ady (see [15] and [17]) state that it is possible to approximate almost surely the partial sums of size n of i.i.d. centered random variables in L p (p > 2) by a Wiener process with an error…

In this paper, we consider a "compensated" random sum that arises from numerical approximation of stochastic integrations and differential equations. We show that the compensated sum exhibits some surprising cancellations among its…

We derive tight non-asymptotic bounds for the Kolmogorov distance between the probabilities of two Gaussian elements to hit a ball in a Hilbert space. The key property of these bounds is that they are dimension-free and depend on the…

In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev $\zeta$-metric is used. The…

We present a concentration result concerning random weighted projections in high dimensional spaces. As applications, we prove (1) New concentration inequalities for random quadratic forms; (2) The infinity norm of most unit eigenvectors of…

In this paper, higher-order expansions for distributions and densities of powered extremes of standard normal random sequences are established under an optimal choice of normalized constants. Our findings refine the related results in Hall…

Bounds of the accuracy of the normal approximation to the distribution of a sum of independent random variables are improved under relaxed moment conditions, in particular, under the absence of moments of orders higher than the second.…

While Kolmogorov's probability axioms are widely recognized, it is less well known that in an often-overlooked 1930 note, Kolmogorov proposed an axiomatic framework for a unifying concept of the mean -- referred to as regular means. This…

Let $S_n=I_1+\cdots+I_n$ be a sum of independent indicators $I_i$, with $p_i=\Pr(I_i=1)=1-\Pr(I_i=0)$, $i=1,\ldots,n$. It is well-known that the total variation distance between $S_n$ and $Z_\lambda$, where $Z_\lambda$ has a Poisson…

Let $X_1,X_2, \ldots $ be independent random uniform points in a bounded domain $A \subset \mathbb{R}^d$ with smooth boundary. Define the coverage threshold $R_n$ to be the smallest $r$ such that $A$ is covered by the balls of radius $r$…

Let $\{X_n;n\ge 1\}$ be a sequence of independent random variables on a probability space $(\Omega, \mathcal{F}, P)$ and $S_n=\sum_{k=1}^n X_k$. It is well-known that the almost sure convergence, the convergence in probability and the…

In this paper we establish the error rate of first order asymptotic approximation for the tail probability of sums of log-elliptical risks. Our approach is motivated by extreme value theory which allows us to impose only some weak…

In this paper, we develop local expansions for the ratio of the centered matrix-variate $T$ density to the centered matrix-variate normal density with the same covariances. The approximations are used to derive upper bounds on several…

Kolmogorov's axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional…

We investigate quantitative implications of the notion of log-concavity through a probabilistic interpretation. In particular, we derive concentration inequalities, moment and entropy bounds for random variables satisfying a precise degree…