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Aihua Xia

Publications and source records attributed to Aihua Xia.

At least 19 recordsLinked to original sources
arXiv2026

Quantitative CLTs for Geometric Statistics of Dependent Marked Point Processes

Given a geometric statistic expressible as a sum of scores which depend on local data, \citet{BYY19} established central limit theorems for centered and normalized versions of these statistics, subject to the underlying point process having fast decay of correlations and also subject to a variance growth condition. Building on this, \citet{CX23} derived rates of normal approximation, as measured by the Wasserstein distance, for statistics of point processes exhibiting fast decay of dependence. Here we go further and establish weak mixing conditions yielding rates of normal convergence for geometric statistics of marked point processes. The mixing conditions, which are in terms of verifiable geometric criteria, provide rates of normal approximation in the Wasserstein distance for statistics of a large class of point processes with dependent marks. Examples include statistics of determinantal point processes, unevenly spaced time series, continuum percolation, interacting diffusions on spatial random graphs, as well as local $U$-statistics.

math.PR↗
arXiv2026

Performance Analysis for Wireless Localization with Random Sensor Network

Accurate wireless localization underpins applications from autonomous systems to smart infrastructure. We study the mean-squared error (MSE) and conditional MSE (CMSE) of a practical fusion-based estimator in d-dimensional, stationary isotropic (translation- and rotation-invariant) random sensor networks, where a central processor combines received-signal-strength (RSS) and angle-of-arrival (AOA) measurements to infer a target's position. Our contributions are twofold. First, we establish an approximation theorem: when measurement noise is sufficiently large, the joint law of RSS and AOA observations under a broad class of stationary isotropic deployments is, in distribution, indistinguishable from that induced by a homogeneous Poisson point process (PPP). Second, leveraging this equivalence, we investigate a homogeneous PPP-based sensor network. We propose a fusion-based estimator in which a central processor aggregates RSS and AOA measurements from a set of spatially distributed sensors to infer the target position. For this PPP deployment within a finite observation region, we derive tractable analytical upper bounds for both the MSE and CMSE, establishing explicit scaling laws with respect to sensor density, observation radius, and noise variance. The approximation theorem then certifies these PPP-based bounds as reasonable proxies for non-Poisson deployments in noisy regimes. Overall, the results translate deployment and sensing parameters into achievable accuracy targets and provide robust, cost-aware guidance for the design of next-generation location-aware wireless networks.

eess.SP↗
arXiv2023

On the rate of normal approximation for Poisson continuum percolation

It is known that the number of points in the largest cluster of a percolating Poisson process restricted to a large finite box is asymptotically normal. In this note, we establish a rate of convergence for the statement. As each point in the largest cluster is determined by points as far as the diameter of the box, known results in the literature of normal approximation for Poisson functionals cannot be directly applied. To disentangle the long-range dependence of the largest cluster, we use the fact that the second largest cluster has comparatively shorter range of dependence to restrict the range of dependence, apply a recently established result in Chen, Röllin and Xia (2021) to obtain a Berry-Esseen type bound for the normal approximation of the number of points belonging to clusters that have a restricted range of dependence, and then estimate the gap between this quantity and the number of points in the largest cluster.

math.PR↗
arXiv2022

The saturated pairwise interaction Gibbs point process as a joint species distribution model

In an effort to effectively model observed patterns in the spatial configuration of individuals of multiple species in nature, we introduce the saturated pairwise interaction Gibbs point process. Its main strength lies in its ability to model both attraction and repulsion within and between species, over different scales. As such, it is particularly well-suited to the study of associations in complex ecosystems. Based on the existing literature, we provide an easy to implement fitting procedure as well as a technique to make inference for the model parameters. We also prove that under certain hypotheses the point process is locally stable, which allows us to use the well-known `coupling from the past' algorithm to draw samples from the model. Different numerical experiments show the robustness of the model. We study three different ecological datasets, demonstrating in each one that our model helps disentangle competing ecological effects on species' distribution.

stat.ME↗
arXiv2022

Convergence rate for geometric statistics of point processes with fast decay dependence

[Błaszczyszyn, Yogeshwaran and Yukich (2019)] established central limit theorems for geometric statistics of point processes having fast decay dependence. As limit theorems are of limited use unless we understand their errors involved in the approximation, in this paper, we consider the rates of a normal approximation in terms of the Wasserstein distance for statistics of point processes on $\mathbb{R}^d$ satisfying fast decay dependence. We demonstrate the use of the theorems for statistics arising from two families of point processes: the rarified Gibbs point processes and the determinantal point processes with fast decay kernels.

math.PR↗
arXiv2021

Geometric sums, size biasing and zero biasing

The geometric sum plays a significant role in risk theory and reliability theory \cite{Kala97} and a prototypical example of the geometric sum is Rényi's theorem~\cite{Renyi56} saying a sequence of suitably parameterised geometric sums converges to the exponential distribution. There is extensive study of the accuracy of exponential distribution approximation to the geometric sum \cite{Sugakova95,Kala97,PekozRollin11} but there is little study on its natural counterpart of gamma distribution approximation to negative binomial sums. In this note, we show that a nonnegative random variable follows a gamma distribution if and only if its size biasing equals its zero biasing. We combine this characterisation with Stein's method to establish simple bounds for gamma distribution approximation to the sum of nonnegative independent random variables, a class of compound Poisson distributions and the negative binomial sum of random variables.

math.PR↗
arXiv2020

Normal approximation in total variation for statistics in geometric probability

We use Stein's method to establish the rates of normal approximation in terms of the total variation distance for a large class of sums of score functions of marked Poisson point processes on $\mathbb{R}^d$. As in the study under the weaker Kolmogorov distance, the score functions are assumed to satisfy stabilizing and moment conditions. At the cost of an additional non-singularity condition for score functions, we show that the rates are in line with those under the Kolmogorov distance. We demonstrate the use of the theorems in four applications: Voronoi tessellation, $k$-nearest neighbours, timber volume and maximal layers.

math.PR↗
arXiv2020

When Is the Conway-Maxwell-Poisson Distribution Infinitely Divisible?

An essential character for a distribution to play a central role in the limit theory is infinite divisibility. In this note, we prove that the Conway-Maxwell-Poisson (CMP) distribution is infinitely divisible iff it is the Poisson or geometric distribution. This explains that, despite its applications in a wide range of fields, there is no theoretical foundation for the CMP distribution to be a natural candidate for the law of small numbers.

math.PR↗
arXiv2020

Palm theory, random measures and Stein couplings

We establish a general Berry-Esseen type bound which gives optimal bounds in many situations under suitable moment assumptions. By combining the general bound with Palm theory, we deduce a new error bound for assessing the accuracy of normal approximation to statistics arising from random measures, including stochastic geometry. We illustrate the use of the bound in four examples: completely random measures, excursion random measure of a locally dependent random process, and the total edge length of Ginibre-Voronoi tessellations and of Poisson-Voronoi tessellations. Moreover, we apply the general bound to Stein couplings and discuss the special cases of local dependence and additive functionals in occupancy problems.

math.PR↗
arXiv2020

On moderate deviations in Poisson approximation

In this paper, we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of Poisson distribution than {those} of normal distribution. We then show the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in \cite{CFS}. Our estimates contain no unspecified constants and are easy to apply. We illustrate the use of the theorems in six applications: Poisson-binomial distribution, matching problem, occupancy problem, birthday problem, random graphs and 2-runs. The paper complements the works of \cite{CC92,BCC95,CFS}.

math.PR↗
arXiv2019

A large sample property in approximating the superposition of i.i.d. point processes

One of the main differences between the central limit theorem and the Poisson law of small numbers is that the former possesses the large sample property (LSP), i.e., the error of normal approximation to the sum of $n$ independent identically distributed (i.i.d.) random variables is a decreasing function of $n$. Since 1980's, considerable effort has been devoted to recovering the LSP for the law of small numbers in discrete random variable approximation. In this paper, we aim to establish the LSP for the superposition of i.i.d. point processes.

math.PR↗
arXiv2018

Divergence from, and Convergence to, Uniformity of Probability Density Quantiles

The probability density quantile (pdQ) carries essential information regarding shape and tail behavior of a location-scale family. Convergence of repeated applications of the pdQ mapping to the uniform distribution is investigated and new fixed point theorems are established. The Kullback-Leibler divergences from uniformity of these pdQs are mapped and found to be ingredients in power functions of optimal tests for uniformity against alternative shapes.

math.ST↗
arXiv2016

Multivariate approximation in total variation, I: equilibrium distributions of Markov jump processes

For integer valued random variables, the translated Poisson distributions form a flexible family for approximation in total variation, in much the same way that the normal family is used for approximation in Kolmogorov distance. Using the Stein--Chen method, approximation can often be achieved with error bounds of the same order as those for the CLT. In this paper, an analogous theory, again based on Stein's method, is developed in the multivariate context. The approximating family consists of the equilibrium distributions of a collection of Markov jump processes, whose analogues in one dimension are the immigration--death processes with Poisson distributions as equilibria. The method is illustrated by providing total variation error bounds for the approximation of the equilibrium distribution of one Markov jump process by that of another. In a companion paper, it is shown how to use the method for discrete normal approximation in ${\mathbb Z}^d$.

math.PR↗
arXiv2016

Multivariate approximation in total variation, II: discrete normal approximation

The paper applies the theory developed in Part I to the discrete normal approximation in total variation of random vectors in ${\mathbb Z}^d$. We illustrate the use of the method for sums of independent integer valued random vectors, and for random vectors exhibiting an exchangeable pair. We conclude with an application to random colourings of regular graphs.

math.PR↗
arXiv2016

Stronger wireless signals appear more Poisson

Keeler, Ross and Xia (2016) recently derived approximation and convergence results, which imply that the point process formed from the signal strengths received by an observer in a wireless network under a general statistical propagation model can be modelled by an inhomogeneous Poisson point process on the positive real line. The basic requirement for the results to apply is that there must be a large number of transmitters with different locations and random propagation effects.The aim of this note is to apply some of the main results of Keeler, Ross and Xia (2016) in a less general but more easily applicable form to illustrate how the results can be applied in practice. New results are derived that show that it is the strongest signals, after being weakened by random propagation effects, that behave like a Poisson process, which supports recent experimental work.

cs.NI↗
arXiv2016

A dichotomy for CLT in total variation

Let $η_i$, $i\ge 1$, be a sequence of independent and identically distributed random variables with finite third moment, and let $Δ_n$ be the total variation distance between the distribution of $S_n:=\sum_{i=1}^nη_i$ and the normal distribution with the same mean and variance. In this note, we show the dichotomy that either $Δ_n=1$ for all $n$ or $Δ_n=O\left(n^{-1/2}\right)$.

math.PR↗
arXiv2016

A marked renewal process model for the size of a honey bee colony

Many areas of agriculture rely on honey bees to provide pollination services and any decline in honey bee numbers can impact on global food security. In order to understand the dynamics of honey bee colonies we present a discrete time marked renewal process model for the size of a colony. We demonstrate that under mild conditions this attains a stationary distribution that depends on the distribution of the numbers of eggs per batch, the probability an egg hatches and the distributions of the times between batches and bee lifetime. This allows an analytic examination of the effect of changing these quantities. We then extend this model to cyclic annual effects where for example the numbers of eggs per batch and {the probability an egg hatches} may vary over the year.

math.ST↗