Related papers: Probabilistic discrepancy bound for Monte Carlo po…

We introduce a new quality measure to assess randomized low-discrepancy point sets of finite size $n$. This new quality measure, which we call "pairwise sampling dependence index", is based on the concept of negative dependence. A negative…

Let $ (\bx(n))_{n \geq 1} $ be an $s-$dimensional Niederreiter-Xing sequence in base $b$. Let $D((\bx(n))_{n = 1}^{N})$ be the discrepancy of the sequence $ (\bx(n))_{n = 1}^{N} $. It is known that $N D((\bx(n))_{n = 1}^{N}) =O(\ln^s N)$ as…

In this article, we consider a stationary array $(X_{j,n})_{1 \leq j \leq n, n \geq 1}$ of random variables with values in $\bR \verb2\2 \{0\}$ (which satisfy some asymptotic dependence conditions), and the corresponding sequence…

Using proper motion data for 894 stars in the Orion Nebula Cluster (ONC) compiled by Jones & Walker in 1988, we search for binaries with apparent separations in the range 1000-5000 AU, and find an upper limit of three. Using a Monte Carlo…

The following anticoncentration property is proved. The probability that the $k$-order statistic of an arbitrarily correlated jointly Gaussian random vector $X$ with unit variance components lies within an interval of length $\varepsilon$…

We consider a certain equidistributed sequence of rational numbers constructed from the primes. In particular, we determine the sharp convergence rate for the star discrepancy of said sequence. Our arguments are based on well-known…

For any $i,j \ge 0$ with $i+j =1$, let $\bad(i,j)$ denote the set of points $(x,y) \in \R^2$ for which $ \max \{\|qx\|^{1/i}, \|qy\|^{1/j} \} > c/q $ for all $ q \in \N $. Here $c = c(x,y)$ is a positive constant. Our main result implies…

Given a finite set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can we find? We make progress on three instances of subset selection problems in planar point…

Let $\{X_{k,i};i\geq 1,k\geq 1\}$ be an array of i.i.d. random variables and let $\{p_n;n\geq 1\}$ be a sequence of positive integers such that $n/p_n$ is bounded away from 0 and $\infty$. For $W_n=\max_{1\leq i<j\leq…

We study the $L_p$-discrepancy of random point sets in high dimensions, with emphasis on small values of $p$. Although the classical $L_p$-discrepancy suffers from the curse of dimensionality for all $p \in (1,\infty)$, the gap between…

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,…

We derive a consistency result, in the $L_1$-sense, for incomplete U-statistics in the non-standard case where the kernel at hand has infinite second-order moments. Assuming that the kernel has finite moments of order $p(\geq 1)$, we obtain…

Many star bodies have convex subsets with approximately the same Gaussian measure (of the complement). Inspired by this phenomenon, and in connection with the randomized Dvoretzky theorem for Lorentz spaces, we derive bounds on the…

Consider $S$, a set of $n$ points chosen uniformly at random and independently from the unit hypercube of dimension $t>2$. Order $S$ by using the Cartesian product of the $t$ standard orders of $[0,1]$. We determine a constant $\bar x(t)<e$…

For irrational $\alpha$, $\{n\alpha\}$ is uniformly distributed mod 1 in the Weyl sense, and the asymptotic behavior of its discrepancy is completely known. In contrast, very few precise results exist for the discrepancy of subsequences…

Let $X_1,\dots,X_n$ be i.i.d. log-concave random vectors in $\mathbb R^d$ with mean 0 and covariance matrix $\Sigma$. We study the problem of quantifying the normal approximation error for $W=n^{-1/2}\sum_{i=1}^nX_i$ with explicit…

How fast can you test whether a constellation of stars appears in the night sky? This question can be modeled as the computational problem of testing whether a set of points $P$ can be moved into (or close to) another set $Q$ under some…

The aims of this paper are twofold. First, it discusses the Littlewood conjecture and its variants with respect to uniformly distributed sequences. The second aim is to determine the exact order of the discrepancy of the van der…

We give exponential upper bounds for $P(S \le k)$, in particular $P(S=0)$, where $S$ is a sum of indicator random variables that are positively associated. These bounds allow, in particular, a comparison with the independent case. We give…

We extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let $\mathbf{\Omega}=(\Omega_1,\ldots,\Omega_N)$ be a partition of $[0,1]^d$ and let the $i$th point in $\mathcal{P}$ be…