arXiv subjects

Aihua Xia

Publications and source records attributed to Aihua Xia.

31 records · Page 2Linked to original sources

arXiv2014

When do wireless network signals appear Poisson?

We consider the point process of signal strengths from transmitters in a wireless network observed from a fixed position under models with general signal path loss and random propagation effects. We show via coupling arguments that under general conditions this point process of signal strengths can be well-approximated by an inhomogeneous Poisson or a Cox point processes on the positive real line. We also provide some bounds on the total variation distance between the laws of these point processes and both Poisson and Cox point processes. Under appropriate conditions, these results support the use of a spatial Poisson point process for the underlying positioning of transmitters in models of wireless networks, even if in reality the positioning does not appear Poisson. We apply the results to a number of models with popular choices for positioning of transmitters, path loss functions, and distributions of propagation effects.

math.PR↗

arXiv2014

Normal approximation of Gibbsian sums in geometric probability

This paper concerns the asymptotic behavior of a random variable $W_λ$ resulting from the summation of the functionals of a Gibbsian spatial point process over windows $Q_λ\uparrow R^d$. We establish conditions ensuring that $W_λ$ has volume order fluctuations, that is they coincide with the fluctuations of functionals of Poisson spatial point processes. We combine this result with Stein's method to deduce rates of normal approximation for $W_λ$, as $λ\to\infty$. Our general results establish variance asymptotics and central limit theorems for statistics of random geometric and related Euclidean graphs on Gibbsian input. We also establish similar limit theory for claim sizes of insurance models with Gibbsian input, the number of maximal points of a Gibbsian sample, and the size of spatial birth-growth models with Gibbsian input.

math.PR↗

arXiv2011

Clubbed Binomial Approximation for the Lightbulb Process

In the so called lightbulb process, on days r=1,..,n, out of n lightbulbs, all initially off, exactly r bulbs selected uniformly and independent of the past have their status changed from off to on, or vice versa. With W_n the number of bulbs on at the terminal time n and C_n a suitable clubbed binomial distribution, d_{TV}(W_n,C_n) \le 2.7314 \sqrt{n} e^{-(n+1)/3} for all n \ge 1. The result is shown using Stein's method.

math.PR↗

arXiv2011

Poisson process approximation for dependent superposition of point processes

Although the study of weak convergence of superpositions of point processes to the Poisson process dates back to the work of Grigelionis in 1963, it was only recently that Schuhmacher [Stochastic Process. Appl. 115 (2005) 1819--1837] obtained error bounds for the weak convergence. Schuhmacher considered dependent superposition, truncated the individual point processes to 0--1 point processes and then applied Stein's method to the latter. In this paper, we adopt a different approach to the problem by using Palm theory and Stein's method, thereby expressing the error bounds in terms of the mean measures of the individual point processes, which is not possible with Schuhmacher's approach. We consider locally dependent superposition as a generalization of the locally dependent point process introduced in Chen and Xia [Ann. Probab. 32 (2004) 2545--2569] and apply the main theorem to the superposition of thinned point processes and of renewal processes.

math.PR↗

arXiv2011

Stein's method and locally dependent point process approximation

Random events in space and time often exhibit a locally dependent structure. When the events are very rare and dependent structure is not too complicated, various studies in the literature have shown that Poisson and compound Poisson processes can provide adequate approximations. However, the accuracy of approximations does not improve or may even deteriorate when the mean number of events increases. In this paper, we investigate an alternative family of approximating point processes and establish Stein's method for their approximations. We prove two theorems to accommodate respectively the positively and negatively related dependent structures. Three examples are given to illustrate that our approach can circumvent the technical difficulties encountered in compound Poisson process approximation [see Barbour & Månsson (2002)] and our approximation error bound decreases when the mean number of the random events increases, in contrast to increasing bounds for compound Poisson process approximation.

math.PR↗

arXiv2010

On customer flows in Jackson queuing networks

Melamed's theorem states that for a Jackson queuing network, the equilibrium flow along a link follows Poisson distribution if and only if no customers can travel along the link more than once. Barbour \& Brown~(1996) considered the Poisson approximate version of Melamed's theorem by allowing the customers a small probability $p$ of travelling along the link more than once. In this paper, we prove that the customer flow process is a Poisson cluster process and then establish a general approximate version of Melamed's theorem accommodating all possible cases of $0\le p<1$.

math.PR↗

arXiv2010

On approximation of Markov binomial distributions

For a Markov chain $\mathbf{X}=\{X_i,i=1,2,...,n\}$ with the state space $\{0,1\}$, the random variable $S:=\sum_{i=1}^nX_i$ is said to follow a Markov binomial distribution. The exact distribution of $S$, denoted $\mathcal{L}S$, is very computationally intensive for large $n$ (see Gabriel [Biometrika 46 (1959) 454--460] and Bhat and Lal [Adv. in Appl. Probab. 20 (1988) 677--680]) and this paper concerns suitable approximate distributions for $\mathcal{L}S$ when $\mathbf{X}$ is stationary. We conclude that the negative binomial and binomial distributions are appropriate approximations for $\mathcal{L}S$ when $\operatorname {Var}S$ is greater than and less than $\mathbb{E}S$, respectively. Also, due to the unique structure of the distribution, we are able to derive explicit error estimates for these approximations.

math.ST↗

arXiv2008

Polynomial birth-death distribution approximation in Wasserstein distance

The polynomial birth-death distribution (abbr. as PBD) on $\ci=\{0,1,2, >...\}$ or $\ci=\{0,1,2, ..., m\}$ for some finite $m$ introduced in Brown & Xia (2001) is the equilibrium distribution of the birth-death process with birth rates $\{α_i\}$ and death rates $\{β_i\}$, where $\a_i\ge0$ and $\b_i\ge0$ are polynomial functions of $i\in\ci$. The family includes Poisson, negative binomial, binomial and hypergeometric distributions. In this paper, we give probabilistic proofs of various Stein's factors for the PBD approximation with $\a_i=a$ and $\b_i=i+bi(i-1)$ in terms of the Wasserstein distance. The paper complements the work of Brown & Xia (2001) and generalizes the work of Barbour & Xia (2006) where Poisson approximation ($b=0$) in the Wasserstein distance is investigated. As an application, we establish an upper bound for the Wasserstein distance between the PBD and Poisson binomial distribution and show that the PBD approximation to the Poisson binomial distribution is much more precise than the approximation by the Poisson or shifted Poisson distributions.

math.PR↗

arXiv2007

A new metric between distributions of point processes

Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric $\bar{d}_1$ that combines positional differences of points under a closest match with the relative difference in total mass in a way that fixes this flaw. A comprehensive collection of theoretical results about $\bar{d}_1$ and its induced Wasserstein metric $\bar{d}_2$ for point process distributions are given, including examples of useful $\bar{d}_1$-Lipschitz continuous functions, $\bar{d}_2$ upper bounds for Poisson process approximation, and $\bar{d}_2$ upper and lower bounds between distributions of point processes of i.i.d. points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential of $\bar{d}_1$ in applications.

math.PR↗

arXiv2007

Poisson process approximation: From Palm theory to Stein's method

This exposition explains the basic ideas of Stein's method for Poisson random variable approximation and Poisson process approximation from the point of view of the immigration-death process and Palm theory. The latter approach also enables us to define local dependence of point processes [Chen and Xia (2004)] and use it to study Poisson process approximation for locally dependent point processes and for dependent superposition of point processes.

math.PR↗

arXiv2007

On Stein's method and perturbations

Stein's (1972) method is a very general tool for assessing the quality of approximation of the distribution of a random element by another, often simpler, distribution. In applications of Stein's method, one needs to establish a Stein identity for the approximating distribution, solve the Stein equation and estimate the behaviour of the solutions in terms of the metrics under study. For some Stein equations, solutions with good properties are known; for others, this is not the case. Barbour and Xia (1999) introduced a perturbation method for Poisson approximation, in which Stein identities for a large class of compound Poisson and translated Poisson distributions are viewed as perturbations of a Poisson distribution. In this paper, it is shown that the method can be extended to very general settings, including perturbations of normal, Poisson, compound Poisson, binomial and Poisson process approximations in terms of various metrics such as the Kolmogorov, Wasserstein and total variation metrics. Examples are provided to illustrate how the general perturbation method can be applied.

math.PR↗

arXiv2006

Zero biasing and a discrete central limit theorem

We introduce a new family of distributions to approximate $\mathbb {P}(W\in A)$ for $A\subset\{...,-2,-1,0,1,2,...\}$ and $W$ a sum of independent integer-valued random variables $ξ_1$, $ξ_2$, $...,$ $ξ_n$ with finite second moments, where, with large probability, $W$ is not concentrated on a lattice of span greater than 1. The well-known Berry--Esseen theorem states that, for $Z$ a normal random variable with mean $\mathbb {E}(W)$ and variance $\operatorname {Var}(W)$, $\mathbb {P}(Z\in A)$ provides a good approximation to $\mathbb {P}(W\in A)$ for $A$ of the form $(-\infty,x]$. However, for more general $A$, such as the set of all even numbers, the normal approximation becomes unsatisfactory and it is desirable to have an appropriate discrete, nonnormal distribution which approximates $W$ in total variation, and a discrete version of the Berry--Esseen theorem to bound the error. In this paper, using the concept of zero biasing for discrete random variables (cf. Goldstein and Reinert [J. Theoret. Probab. 18 (2005) 237--260]), we introduce a new family of discrete distributions and provide a discrete version of the Berry--Esseen theorem showing how members of the family approximate the distribution of a sum $W$ of integer-valued variables in total variation.

math.PR↗

arXiv2004

Stein's method, Palm theory and Poisson process approximation

The framework of Stein's method for Poisson process approximation is presented from the point of view of Palm theory, which is used to construct Stein identities and define local dependence. A general result (Theorem \refimportantproposition) in Poisson process approximation is proved by taking the local approach. It is obtained without reference to any particular metric, thereby allowing wider applicability. A Wasserstein pseudometric is introduced for measuring the accuracy of point process approximation. The pseudometric provides a generalization of many metrics used so far, including the total variation distance for random variables and the Wasserstein metric for processes as in Barbour and Brown [Stochastic Process. Appl. 43 (1992) 9-31]. Also, through the pseudometric, approximation for certain point processes on a given carrier space is carried out by lifting it to one on a larger space, extending an idea of Arratia, Goldstein and Gordon [Statist. Sci. 5 (1990) 403-434]. The error bound in the general result is similar in form to that for Poisson approximation. As it yields the Stein factor 1/λas in Poisson approximation, it provides good approximation, particularly in cases where λis large. The general result is applied to a number of problems including Poisson process modeling of rare words in a DNA sequence.

math.PR↗