This paper investigates the discrepancy of a family of random sampling methods obtained by perturbing the grid $\frac{1}{M}\mathbb{Z}^{d}\cap\left[ -1/2,1/2\right)^{d}$, where $M$ is a large positive integer. Parameterized by an arbitrary probability measure $\mu$, this family encompasses several classical methods for evaluating the quality of an $N$-point set in $\mathbb{T}^{d}$, where $N=M^{d}$. We show that all probability measures, except for Dirac measures, behave like the Lebesgue measure in the Monte Carlo discrepancy. This represents a limiting case where the measure $\mu$ depends on $M$. In this latter context, we prove that, up to a constant, the lowest possible discrepancy is achieved when the support of $\mu$ has diameter $\leq c/M$, and that this upper bound is sharp.
Let $\nu$ be a compactly supported Borel probability measure on $\mathbb R^{d}$ with $\nu(B(x,r))\leq Cr^{a}$ for some $a>0$. Refine a dyadic cube exactly when its mass is at least $t$, and let $\mathcal{L}_{\nu}(t)$ be the mean depth at which this refinement stops. We show that the lower and upper geometric-mean quant...
Let $\pi(\mathrm{d} x)\propto e^{-U(x)}\, \mathrm{d} x$ on $\mathbb{R}^d$, where $U$ is continuously differentiable and $m$-strongly convex with a globally $L$-Lipschitz gradient, $0<m\leq L<\infty$, and $\kappa=L/m$. Fixed-step Metropolis-adjusted Langevin algorithm (MALA) has known warm-start mixing-time upper bounds...
We develop Gaussian approximation bounds in higher-order Wasserstein distance $W_p$, $p\geq2$, for sums of multivariate martingale differences generated by a uniformly ergodic Markov chain. Under an $L^{(2+\eta)p}$-moment condition with $\eta>0$, we establish the explicit bound $$ O\left( p^3 \|A\|_4^2 + pd^{1/4}\|A\|_...
Let $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in $\mathbb{Z}^d$. We prove that the full maximal operator $f\mapsto\sup_{t\geq0}\lvert M_t f\rvert$ is bounded on $\ell^p(\mathbb{Z}^d)$, for every $1<p\leq\infty$, with a constant independent of the dimension. In parti...
In this work, we prove several equivalent characterizations of the extreme points of convex sets of probability measures of the form $\mathcal{M}=\mathcal{P} \cap H$, where $\mathcal{P}$ denotes the set of all probability measures on an arbitrary measurable space $(\Omega,\mathcal{F})$ and $H$ is an affine set of signe...
Gerrit Bauch, Max Nendel, Alessandro Sgarabottolo· 0 citations
Let $W=n^{-1/2}\sum_{i=1}^n X_i$, where the $X_i$ are independent centered random vectors in ${\mathbb R}^p$ with $|X_{ij}|\le B$ almost surely. Suppose that $\text{Cov}(W)$ has unit diagonal and smallest eigenvalue at least $b^2>0$. We prove that the distance between $W$ and a Gaussian vector with the same covariance,...
P. M. Aronow, Patrick Lopatto· 0 citations
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