For the empirical measure of the first $n$ points of one infinite sequence of independent uniform samples in the unit square, we identify the leading constant in the bounded-Lipschitz distance to area measure, equivalently in $W_1$, both in expectation and almost surely along all sample sizes. The constant is $c_*=\kappa_1/(2\sqrt\pi)\in(0,1/4]$, where $\kappa_1$ is the linear-growth Gaussian variational constant of Armegioiu--Goldman--Grotto--Trevisan. This answers the two-dimensional bounded-Lipschitz convergence question posed by Yukich in 1989. We give the complete endpoint argument, square--torus comparison, empirical reduction, and full-sequence probability argument.
We study Gaussian approximation for probability measures obtained by normalizing one-dimensional profile functions, with one-dimensional marginals of Euclidean balls as the principal example. We first establish quantitative concentration estimates near the set of maximizers. For profiles with a unique nondegenerate max...
Let $Y_1,\ldots,Y_n$ be independent symmetric random variables with log-concave tails. We give a dimension-free characterization of the expected supremum of the canonical process $X_x=\sum_{i=1}^n x_iY_i$ without any $\Delta_2$ or regular-growth assumption on the coordinate tails. The characterization is governed by sc...
In this paper, we prove that the uniform spanning tree of $\mathbb Z^4$, seen from the origin, converges locally to an infinite spatial Brownian Kesten tree: a Brownian spine onto which independent spatial Brownian trees are grafted at a constant rate. The convergence holds jointly for the intrinsic metric, the volume...
We prove finite-sample risk bounds for empirical quadratic and power Fr\'echet means in geodesic metric spaces with curvature bounded above for observations with arbitrary joint law. We compare the sample with exchangeable copies in which one observation is redrawn independently of the original sample and the others mo...
We establish a uniform two-dimensional local limit theorem for the joint distribution of $\Omega(n)$ and $\Omega(n+1)$. More precisely, along a set of scales of logarithmic density one, the normalized joint counts are uniformly approximated by the density of two independent Gaussian random variables, each with mean and...
We prove that the full discrete Hardy-Littlewood maximal operator associated with Euclidean balls satisfies dimension-free bounds on $\ell^p(\mathbb Z^d)$ for every $1<p<\infty$. We also establish analogous dimension-free bounds for the full discrete spherical maximal operator when $d\geq 5$ and $2\leq p<\infty$. The m...
M. Hormozi, Jakub Niksiński, Bla.zej Wr'obel· 1 citation· ⚡1
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