arXiv subjects

Xiaochuan Yang

Publications and source records attributed to Xiaochuan Yang.

At least 19 recordsLinked to original sources

arXiv2026

Random coverage of a manifold with boundary

Let $A$ be a compact $d$-dimensional $C^2$ Riemannian manifold with boundary, embedded in ${\bf R}^m$ where $m \geq d \geq 2$, and let $B$ be a nice subset of $A$ (possibly $B=A$). Let $X_1,X_2, \ldots $ be independent random uniform points in $A$. Define the coverage threshold $R_n$ to be the smallest $r$ such that $B$ is covered by the geodetic balls of radius $r$ centred on $X_1,\ldots,X_n$. We obtain the limiting distribution of $R_n$ and also a strong law of large numbers for $R_n$ in the large-$n$ limit. For example, if $A$ has Riemannian volume 1 and its boundary has surface measure $|\partial A|$, and $B=A$, then if $d=3$ then ${\bf P}[nπR_n^3 - \log n - 2 \log (\log n) \leq x]$ converges to $\exp(-2^{-4}π^{5/3} |\partial A| e^{-2 x/3})$ and $(n πR_n^3)/(\log n) \to 1$ almost surely, while if $d=2$ then ${\bf P}[n πR_n^2 - \log n - \log (\log n) \leq x]$ converges to $\exp(- e^{-x}- |\partial A|π^{-1/2} e^{-x/2})$. We generalize to allow for multiple coverage. For the strong laws of large numbers, we can relax the requirement that the underlying density on $A$ be uniform. For the limiting distribution, we have a similar result for Poisson samples. Our results still hold if we use Euclidean rather than geodetic balls.

math.PR↗

arXiv2026

On the components of random geometric graphs in the dense limit

Consider the geometric graph on $n$ independent uniform random points in a connected compact region $A$ of ${\bf R}^d, d \geq 2$, with $C^2$ boundary, or in the unit square, with distance parameter $r_n$. Let $K_n$ be the number of components of this graph, and $R_n$ the number of vertices not in the giant component. Let $S_n$ be the number of isolated vertices. We show that if $r_n$ is chosen so that $nr_n^d$ tends to infinity but slowly enough that ${\bf E}[S_n]$ also tends to infinity, then $K_n$, $R_n$ and $S_n$ are all asymptotic to $μ_n$ in probability as $n \to \infty$ where (with $|A|$, $θ_d$ and $|\partial A|$ denoting the volume of $A$, of the unit $d$-ball, and the perimeter of $A$ respectively) $μ_n := ne^{-πn (r_n)^d/|A|}$ if $d=2$ and $μ_n := ne^{-θ_d n (r_n)^d/|A|} + θ_{d-1}^{-1} |\partial A| (r_n)^{1-d} e^{- θ_d n (r_n)^d/(2|A|)}$ if $d\geq 3$. We also give variance asymptotics and central limit theorems for $K_n$ and $R_n$ in this limiting regime when $d \geq 3$, and for Poisson input with $d \geq 2$. We extend these results (substituting ${\bf E}[S_n]$ for $μ_n$) to a class of non-uniform distributions on $A$.

math.PR↗

arXiv2025

Coverage of the unit cube by dynamic Boolean models

Motivated by peer-to-peer telecommunication, we study a dynamic Boolean model. We define a Poisson number of random lines through the $(d-1)$-dimensional base of a $d$-dimensional unit cube and dilate them to define cylinders. Letting $ρ$ be the expected number of cylinders, the random variable of interest is the coverage radius $R_ρ$, which is the cylinder radius required to cover the $d$-dimensional unit cube. We show that $R_ρ^{d-1}$ is of the order $\log ρ/ ρ$ with high probability as $ρ$ tends to infinity. We also consider alternative dynamics resulting in generalized cylinders that are generated by dilating the trajectories of stochastic processes, in particular Brownian motions. This leads to a coverage radius of the same order.

math.PR↗

arXiv2025

On the rate of convergence in the Hall-Janson coverage theorem

Consider a spherical Poisson Boolean model $Z$ in Euclidean $d$-space with $d \geq 2$, with Poisson intensity $t$ and radii distributed like $rY$ with $r \geq 0$ a scaling parameter and $Y$ a fixed nonnegative random variable with finite $(2d-2)$-nd moment (or if $d=2$, a finite $(2 + \varepsilon)$-moment condition for some $\varepsilon >0$). Let $A \subset {\bf R}^d$ be compact with a nice boundary. Let $α$ be the expected volume of a ball of radius $Y$, and suppose $r=r(t)$ is chosen so that $αt r^d - \log t - (d-1) \log \log t$ is a constant independent of $t$. A classical result of Hall and of Janson determines the (non-trivial) large-$t$ limit of the probability that $A$ is fully covered by $Z$. In this paper we provide an $O((\log \log t)/\log t)$ bound on the rate of convergence in that result. With a slight adjustment to $r(t)$, this can be improved to $O(1/\log t)$.

math.PR↗

arXiv2025

On $k$-clusters of high-intensity random geometric graphs

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 $R^d$, 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.

math.PR↗

arXiv2025

Approximation of Smooth Numbers for Harmonic Samples A Stein method Approach

We present a de Bruijn type approximation for quantifying the content of m smooth numbers, derived from samples obtained through a probability measure over the set of integers less than or equal to n, with point mass function at k inversely proportional to k. Our analysis is based on a stochastic representation of the measure of interest, utilizing weighted independent geometric random variables. This representation is analyzed through the lens of Stein method for the Dickman distribution. A pivotal element of our arguments relies on precise estimations concerning the regularity properties of the solution to the Dickman Stein equation for heaviside functions, recently developed by Bhattacharjee and Schulte. Remarkably, our arguments remain mostly in the realm of probability theory, with Mertens first and third theorems standing as the only number theory estimations required.

math.PR↗

arXiv2025

Covering one point process with another

Let $X_1,X_2, \ldots $ and $Y_1, Y_2, \ldots$ be i.i.d. random uniform points in a bounded domain $A \subset \mathbb{R}^2$ with smooth or polygonal boundary. Given $n,m,k \in \mathbb{N}$, define the {\em two-sample $k$-coverage threshold} $R_{n,m,k}$ to be the smallest $r$ such that each point of $ \{Y_1,\ldots,Y_m\}$ is covered at least $k$ times by the disks of radius $r$ centred on $X_1,\ldots,X_n$. We obtain the limiting distribution of $R_{n,m,k}$ as $n \to \infty$ with $m= m(n) \sim τn$ for some constant $τ>0$, with $k $ fixed. If $A$ has unit area, then $n πR_{n,m(n),1}^2 - \log n$ is asymptotically Gumbel distributed with scale parameter $1$ and location parameter $\log τ$. For $k >2$, we find that $n πR_{n,m(n),k}^2 - \log n - (2k-3) \log \log n$ 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 $k$.

math.PR↗

arXiv2024

Fluctuations of the connectivity threshold and largest nearest-neighbour link

Consider a random uniform sample of $n$ points in a compact region $A$ of Euclidean $d$-space, $d \geq 2$, with a smooth or (when $d=2$) polygonal boundary. Fix $k \in {\bf N}$. Let $T_{n,k}$ 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 $n (π/|A|) T_{n,1}^2 - \log n$ is asymptotically standard Gumbel. For $(d,k) \neq (2,1)$, it is $n (θ_d/|A|) T_{n,k}^d - (2-2/d) \log n - (4-2k-2/d) \log \log n$ 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 more important in some cases than others. We also give similar results for the largest $k$-nearest neighbour link $U_{n,k}$ in the sample, and show $T_{n,k}=U_{n,k}$ 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 non-uniform samples, with a less explicit sequence of centring constants.

math.PR↗

arXiv2023

Multivariate stable approximation in Wasserstein distance by Stein's method

By a delicate analysis for the Stein's equation associated to the $α$-stable law approximation with $α\in (0,2)$, we prove a quantitative stable central limit theorem in Wasserstein type distance, which generalizes the results in the series of work \cite{Xu19, CNX19+,CNXYZ19+} from the univariate case to the multiple variate case. From an explicit computation for Pareto's distribution, we see that the rate of our approximation is sharp. The analysis of the Stein's equation is new and has independent interest.

math.PR↗

arXiv2023

Largest nearest-neighbour link and connectivity threshold in a polytopal random sample

Let $X_1,X_2, \ldots $ be independent identically distributed random points in a convex polytopal domain $A \subset \mathbb{R}^d$. Define the largest nearest neighbour link $L_n$ to be the smallest $r$ such that every point of $\mathcal X_n:=\{X_1,\ldots,X_n\}$ has another such point within distance $r$. We obtain a strong law of large numbers for $L_n$ in the large-$n$ limit. A related threshold, the connectivity threshold $M_n$, is the smallest $r$ such that the random geometric graph $G(\mathcal X_n, r)$ is connected. We show that as $n \to \infty$, almost surely $nL_n^d/\log n$ tends to a limit that depends on the geometry of $A$, and $nM_n^d/\log n$ tends to the same limit.

math.PR↗

arXiv2022

Fractal Gaussian Networks: A sparse random graph model based on Gaussian Multiplicative Chaos

We propose a novel stochastic network model, called Fractal Gaussian Network (FGN), that embodies well-defined and analytically tractable fractal structures. Such fractal structures have been empirically observed in diverse applications. FGNs interpolate continuously between the popular purely random geometric graphs (a.k.a. the Poisson Boolean network), and random graphs with increasingly fractal behavior. In fact, they form a parametric family of sparse random geometric graphs that are parametrized by a fractality parameter which governs the strength of the fractal structure. FGNs are driven by the latent spatial geometry of Gaussian Multiplicative Chaos (GMC), a canonical model of fractality in its own right. We asymptotically characterize the expected number of edges, triangles, cliques and hub-and-spoke motifs in FGNs, unveiling a distinct pattern in their scaling with the size parameter of the network. We then examine the natural question of detecting the presence of fractality and the problem of parameter estimation based on observed network data, in addition to fundamental properties of the FGN as a random graph model. We also explore fractality in community structures by unveiling a natural stochastic block model in the setting of FGNs. Finally, we substantiate our results with phenomenological analysis of the FGN in the context of available scientific literature for fractality in networks, including applications to real-world massive network data.

stat.ML↗

arXiv2021

A generalized Kubilius-Barban-Vinogradov bound for prime multiplicities

We present an assessment of the distance in total variation of \textit{arbitrary} collection of prime factor multiplicities of a random number in $[n]=\{1,\dots, n\}$ and a collection of independent geometric random variables. More precisely, we impose mild conditions on the probability law of the random sample and the aforementioned collection of prime multiplicities, for which a fast decaying bound on the distance towards a tuple of geometric variables holds. Our results generalize and complement those from Kubilius et al. which consider the particular case of uniform samples in $[n]$ and collection of "small primes". As applications, we show a generalized version of the celebrated Erdös Kac theorem for not necessarily uniform samples of numbers.

math.PR↗

arXiv2021

Multivariate normal approximation on the Wiener space: new bounds in the convex distance

We establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field, and that of a normal vector with a positive definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin, Peccati and Réveillac (2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener-Itô integrals, and (ii) we characterise the rate of convergence for the finite-dimensional distributions in the functional Breuer-Major theorem.

math.PR↗

arXiv2021

Malliavin-Stein Method: a Survey of Recent Developments

Initiated around the year 2007, the Malliavin-Stein approach to probabilistic approximations combines Stein's method with infinite-dimensional integration by parts formulae based on the use of Malliavin-type operators. In the last decade, Malliavin-Stein techniques have allowed researchers to establish new quantitative limit theorems in a variety of domains of theoretical and applied stochastic analysis. The aim of this survey is to illustrate some of the latest developments of the Malliavin-Stein method, with specific emphasis on extensions and generalisations in the framework of Markov semigroups and of random point measures.

math.PR↗

arXiv2021

A probabilistic approach to the Erdös-Kac theorem for additive functions

We present a new perspective of assessing the rates of convergence to the Gaussian and Poisson distributions in the Erdös-Kac theorem for additive arithmetic functions $ψ$ of a random integer $J_n$ uniformly distributed over $\{1,...,n\}$. Our approach is probabilistic, working directly on spaces of random variables without any use of Fourier analytic methods, and our $ψ$ is more general than those considered in the literature. Our main results are (i) bounds on the Kolmogorov distance and Wasserstein distance between the distribution of the normalized $ψ(J_n)$ and the standard Gaussian distribution, and (ii) bounds on the Kolmogorov distance and total variation distance between the distribution of $ψ(J_n)$ and a Poisson distribution under mild additional assumptions on $ψ$. Our results generalize the existing ones in the literature.

math.PR↗

arXiv2020

Quantitative two-scale stabilization on the Poisson space

We establish inequalities for assessing the distance between the distribution of a (possibly multidimensional) functional of a Poisson random measure and that of a Gaussian element. Our bounds only involve add-one cost operators at the order one - that we evaluate and compare at two different scales - and are specifically tailored for studying the Gaussian fluctuations of sequences of geometric functionals displaying a form of weak stabilization - see Penrose and Yukich (2001) and Penrose (2005). Our main bounds extend the estimates recently exploited by Chatterjee and Sen (2017) in the proof of a quantitative version of the central limit theorem (CLT) for the length of the Poisson-based Euclidean minimal spanning tree (MST). We develop in full detail three applications of our bounds, namely: (i) to a quantitative multidimensional spatial CLT for functionals of the on-line nearest neighbor graph, (ii) to a quantitative multidimensional CLT involving functionals of the empirical measure associated with the edge-length of the Euclidean MST, and (iii) to a collection of multidimensional CLTs for geometric functionals of the excursion set of heavy-tailed shot noise random fields. Application (i) is based on a collection of general probabilistic approximations for strongly stabilizing functionals, that is of independent interest.

math.PR↗

arXiv2019

Uniform dimension results for the inverse images of symmetric Lévy processes

We prove uniform Hausdorff and packing dimension results for the inverse images of a large class of real-valued symmetric Lévy processes. Our main result for the Hausdorff dimension extends that of Kaufman (1985) for Brownian motion and that of Song, Xiao, and Yang (2018) for $α$-stable Lévy processes with $1<α<2$. Along the way, we also prove an upper bound for the uniform modulus of continuity of the local times of these processes.

math.PR↗

arXiv2019

Berry-Esseen bounds in the Breuer-Major CLT and Gebelein's inequality

We derive explicit Berry-Esseen bounds in the total variation distance for the Breuer-Major central limit theorem, in the case of a subordinating function $φ$ satisfying minimal regularity assumptions. Our approach is based on the combination of the Malliavin-Stein approach for normal approximations with Gebelein's inequality, bounding the covariance of functionals of Gaussian fields in terms of maximal correlation coefficients.

math.PR↗