BURA Collection:

Covering One Point Process with Another

2025-04-29T00:00:00Z

Title: Covering One Point Process with Another Authors: Higgs, F; Penrose, MD; Yang, X Abstract: Let X1,X2,… and Y1,Y2,… be i.i.d. random uniform points in a bounded domain A⊂R2 with smooth or polygonal boundary. Given n,m,k∈N, define the two-sample k-coverage thresholdRn,m,k to be the smallest r such that each point of {Y1,…,Ym} is covered at least k times by the disks of radius r centred on X1,…,Xn. We obtain the limiting distribution of Rn,m,k as n→∞ with m=m(n)∼τn for some constant τ>0, with k fixed. If A has unit area, then nπRn,m(n),12-logn is asymptotically Gumbel distributed with scale parameter 1 and location parameter logτ. For k>2, we find that nπRn,m(n),k2-logn-(2k-3)loglogn is asymptotically Gumbel with scale parameter 2 and a more complicated location parameter involving the perimeter of A; boundary effects dominate when k>2. For k=2 the limiting cdf is a two-component extreme value distribution with scale parameters 1 and 2. We also give analogous results for higher dimensions, where the boundary effects dominate for all k. Description: Data Availability The code for the simulations discussed in Section 6 is available at https://github.com/frankiehiggs/CovXY and the samples generated by that code are available at https://researchdata.bath.ac.uk/id/eprint/1359.; A preprint version is available at arXiv:2401.03832v2 [math.PR] (https://arxiv.org/abs/2401.03832) under a CC BY license. It has not been certified by peer review.; Mathematics Subject Classification (2010): 60D05; 60F05; 60F15

On k-clusters of high-intensity random geometric graphs

2026-01-11T00:00:00Z

Title: On k-clusters of high-intensity random geometric graphs Authors: Penrose, MD; Yang, X Abstract: Let k, d be positive integers. We determine a sequence of constants that are asymptotic to the probability that the cluster at the origin in a d-dimensional Poisson Boolean model with balls of fixed radius is of order k, as the intensity becomes large. Using this, we determine the asymptotics of the mean of the number of components of order k, denoted S_n,k in a random geometric graph on n uniformly distributed vertices in a smoothly bounded compact region of d-dimensional Euclidean space, with distance parameter r(n) chosen so that the expected degree grows slowly as n becomes large (the so-called mildly dense limiting regime). We also show that the variance of S_n,k is asymptotic to its mean, and prove Poisson and normal approximation results for S_n,k in this limiting regime. We provide analogous results for the corresponding Poisson process (i.e. with a Poisson number of points). We also give similar results in the so-called mildly sparse limiting regime where r(n) is chosen so the expected degree decays slowly to zero as n becomes large. Description: A preprint version of the article is available at arXiv:2209.14758v4 [math.PR (https://arxiv.org/abs/2209.14758) under a CC BY license. It is not certified by peer review.

Fluctuations of the connectivity threshold and largest nearest-neighbour link

2025-12-01T00:00:00Z

Title: Fluctuations of the connectivity threshold and largest nearest-neighbour link Authors: Penrose, MD; Yang, X Abstract: Consider a random uniform sample χ_n of n points in a compact region A of Euclidean d-space, d ≥ 2, with a smooth or (when d = 2) polygonal boundary. Fix k ∈ N. Let M_k (χ_n) be the threshold r at which the geometric graph on these n vertices with distance parameter r becomes k-connected. We show that if d = 2 then n (π/|A)M₁(χ_n² − log n is asymptotically standard Gumbel. For (d,k) ≠ (2,1), it is n(θ_d/|A|)M_{k<.sub>(χ_n)^d − (2 − 2/d) log n − (4 − 2_k −2/d) log log n
that converges in distribution to a nondegenerate limit, where θ_d is the volume of the unit ball. The limit is Gumbel with scale parameter 2 except when (d,k) = (2,2) where the limit is two component extreme value distributed. The different cases reflect the fact that boundary effects are more important in some cases than others. We also give similar results for the largest k-nearest neighbour link L_k(χ_n) in the sample, and show M_k(χ_n) = L_k(χ_n) with high probability. We provide estimates on rates of convergence and give similar results for Poisson samples in A. Finally, we give similar results even for nonuniform samples, with a less explicit sequence of centring constants.
Description: Subjects:
Primary: 60D05 , 60F05.
Secondary: 05C80 , 60G70.; A preprint version of the article is available at arXiv:2406.00647v3 [math.PR] (https://arxiv.org/abs/2406.00647) under a CC BY license. It has not been certified by peer review,}

Convergent Lifted Lasserre Hierarchy of SDPs for Minimizing Expectation of Piecewise Polynomial Loss over Wasserstein Balls

2026-05-08T00:00:00Z

Title: Convergent Lifted Lasserre Hierarchy of SDPs for Minimizing Expectation of Piecewise Polynomial Loss over Wasserstein Balls Authors: Dizon, NDV; Huang, QY; Chuong, TD; Li, G; Jeyakumar, V Abstract: This paper investigates the minimization of the expectation of piecewise polynomial loss functions over Wasserstein balls. This optimization problem often appears as a key sub-problem of distributionally robust optimization problems. We establish the asymptotic convergence of a hierarchy of semi-definite programming (SDP) relaxations, providing a framework for approximating the optimal values of these inherently infinite-dimensional optimization problems. A central foundational contribution is the development of a new lifted positivity certificate: we demonstrate that piecewise polynomials positive over Archimedean basic semi-algebraic sets admit a structured system of sum-of-squares (SOS) representations. Furthermore, we prove that the proposed hierarchy achieves finite convergence under suitable conditions when the defining polynomials are convex. The practical utility and versatility of this approach are demonstrated via numerical experiments in revenue estimation and portfolio optimization. Description: Data Availability: All data generated and analyzed in this study are provided within the article. The data used in the numerical experiments were generated randomly, and we have clearly described the procedure for reproducing them.