Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.21070

Exact Endpoint Asymptotics and Almost-Sure Limits for Uniform Empirical Measures in Two Dimensions

Weitao Huang·Yichen Wu·Renzhen Wang·Deyu Meng
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 $c_*=κ_1/(2\sqrtπ)\in(0,1/4]$, where $κ_1$ 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.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Weitao Huang, Yichen Wu, Renzhen Wang, Deyu Meng. 2026-10-15. Exact Endpoint Asymptotics and Almost-Sure Limits for Uniform Empirical Measures in Two Dimensions. https://arxiv.org/abs/2610.21070

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Dark Subspaces and Invariant Measures of Quantum Trajectories

Quantum trajectories are Markov processes describing the evolution of a quantum system subject to indirect measurements. They can be viewed as place dependent iterated function systems or the result of products of dependent and non identically distributed random matrices. In this article, we establish a complete classification of their invariant measures. The classification is done in two steps. First, we prove a Markov process on some linear subspaces called dark subspaces, defined in (Maassen, Kümmerer 2006), admits a unique invariant measure. Second, we study the process inside the dark subspaces. Using a notion of minimal family of isometries from a reference space to dark subspaces, we prove a set of measures indexed by orbits of a unitary group is the set of ergodic measures of quantum trajectories.

math.PR↗
arXiv2026

Expected Signature Kernels for Rough Paths

The expected signature kernel is a similarity measure of probability laws on path space and enters directly into signature maximum mean discrepancies. Building on [P. K. Friz, P. P. Hager, and N. Tapia, Forum Math. Sigma, 10 (2022), e42] and [M. Lemercier, T. Lyons, and C. Salvi, SIAM J. Numer. Anal., 64 (2026), pp. 1095-1125], we characterize the expected signature kernel of group-valued inhomogeneous Lévy processes, aka Lévy rough paths, as the scalar component of the solution to an extended Goursat-type system. We give an original description, first in the continuous case where the finite-dimensional system is exact. In the presence of jumps, this system is inherently infinite-dimensional; we prove its well-posedness and derive finite-dimensional truncations with explicit error control. This permits direct evaluation of the reference-dependent terms in one-sample signature-MMD tests without an auxiliary reference sample. Finally, for conditionally Gaussian martingales, conditioning the usual unbiased empirical squared signature-MMD on the sampled quadratic variations yields an unbiased statistic for the same quantity with no larger variance. We derive computable finite-grid versions and illustrate the resulting variance reduction and gain in power.

math.PR↗
arXiv2026

A central limit theorem for the range of random walk indexed by an infinite tree

We are interested in the second-order fluctuations of the size of the range of a random walk indexed by an infinite tree in ${\mathbb Z}^d$. Consider the Kesten tree with geometric offspring, we view the vertices on the left of its spine (the unique infinite path) as the past, and those on the right as the future. We show that the number of new sites in the future given its past has linear variance when $d>8$, and satisfies a central limit theorem (CLT) with Gaussian limiting distribution when $d>16$. The proof relies on the stationarity of the model under depth-first exploration, a general CLT by Dedecker and Merlevède [7], a truncation technique exploiting the local independence of tree structures, and a recursion argument for moment bounds.

math.PR↗