arXiv subjects

A. D. Barbour

Publications and source records attributed to A. D. Barbour.

46 records · Page 3Linked to original sources

arXiv2009

Local limit approximations for Markov population processes

The paper is concerned with the equilibrium distribution $Π_n$ of the $n$-th element in a sequence of continuous-time density dependent Markov processes on the integers. Under a $(2+\a)$-th moment condition on the jump distributions, we establish a bound of order $O(n^{-(\a+1)/2}\sqrt{\log n})$ on the difference between the point probabilities of $Π_n$ and those of a translated Poisson distribution with the same variance. Except for the factor $\sqrt{\log n}$, the result is as good as could be obtained in the simpler setting of sums of independent integer-valued random variables. Our arguments are based on the Stein-Chen method and coupling.

math.PR↗

arXiv2009

Laws of large numbers for epidemic models with countably many types

In modeling parasitic diseases, it is natural to distinguish hosts according to the number of parasites that they carry, leading to a countably infinite type space. Proving the analogue of the deterministic equations, used in models with finitely many types as a "law of large numbers" approximation to the underlying stochastic model, has previously either been done case by case, using some special structure, or else not attempted. In this paper we prove a general theorem of this sort, and complement it with a rate of convergence in the $\ell_1$-norm.

math.PR↗

arXiv2008

Translated Poisson approximation for Markov chains

The paper is concerned with approximating the distribution of a sum W of n integer valued random variables Y_i, whose distributions depend on the state of an underlying Markov chain X. The approximation is in terms of a translated Poisson distribution, with mean and variance chosen to be close to those of W, and the error is measured with respect to the total variation norm. Error bounds comparable to those found for normal approximation with respect to the weaker Kolmogorov distance are established, provided that the distribution of the sum of the Y_i's between the successive visits of X to a reference state is aperiodic. Without this assumption, approximation in total variation cannot be expected to be good.

math.PR↗

arXiv2008

Small counts in the infinite occupancy scheme

The paper is concerned with the classical occupancy scheme with infinitely many boxes, in which $n$ balls are thrown independently into boxes $1,2,...$, with probability $p_j$ of hitting the box $j$, where $p_1\geq p_2\geq...>0$ and $\sum_{j=1}^\infty p_j=1$. We establish joint normal approximation as $n\to\infty$ for the numbers of boxes containing $r_1,r_2,...,r_m$ balls, standardized in the natural way, assuming only that the variances of these counts all tend to infinity. The proof of this approximation is based on a de-Poissonization lemma. We then review sufficient conditions for the variances to tend to infinity. Typically, the normal approximation does not mean convergence. We show that the convergence of the full vector of $r$-counts only holds under a condition of regular variation, thus giving a complete characterization of possible limit correlation structures.

math.PR↗

arXiv2007

Coupling a branching process to an infinite dimensional epidemic process

Branching process approximation to the initial stages of an epidemic process has been used since the 1950's as a technique for providing stochastic counterparts to deterministic epidemic threshold theorems. One way of describing the approximation is to construct both branching and epidemic processes on the same probability space, in such a way that their paths coincide for as long as possible. In this paper, it is shown, in the context of a Markovian model of parasitic infection, that coincidence can be achieved with asymptotically high probability until o(N^{2/3}) infections have occurred, where N denotes the total number of hosts.

math.PR↗

arXiv2006

Inter-species regression analysis

When conducting inter-species regression analyses, the phylogenetic relationships between the individual species need to be taken into account. In this paper, a procedure for conducting such analyses is proposed, which only requires the use of a measure of relationship between pairs of species, rather than a complete phylogeny, and which at the same time assesses the importance to be attached to the relationships with regard to the conclusions reached. The procedure is applied to previous data, relating testis size to mean hind tibia length, duct length and spermathecal area in 15 species of Scathophagidae.

q-bio.PE↗

arXiv2005

Asymptotic behavior of a metapopulation model

We study the behavior of an infinite system of ordinary differential equations modeling the dynamics of a metapopulation, a set of (discrete) populations subject to local catastrophes and connected via migration under a mean field rule; the local population dynamics follow a generalized logistic law. We find a threshold below which all the solutions tend to total extinction of the metapopulation, which is then the only equilibrium; above the threshold, there exists a unique equilibrium with positive population, which, under an additional assumption, is globally attractive. The proofs employ tools from the theories of Markov processes and of dynamical systems.

math.PR↗

arXiv2004

Random Combinatorial structures:the convergent case

This paper studies the distribution of the component spectrum of combinatorial structures such as uniform random forests, in which the classical generating function for the numbers of (irreducible) elements of the different sizes converges at the radius of convergence; here, this property is expressed in terms of the expectations of {\it independent} random variables $Z_j$, $j\ge1$, whose joint distribution, conditional on the event that $\sum_{j=1}^n jZ_j = n$, gives the distribution of the component spectrum for a random structure of size $n$. For a large class of such structures, we show that the component spectrum is asymptotically composed of $Z_j$ components of size $j$, $j\ge1$, with the remaining part, of size $n-\sum_{j\ge1} Z_j$, being made up of a single, giant component.

math.PR↗

arXiv2004

Small worlds

Small world models are networks consisting of many local links and fewer long range `shortcuts'. In this paper, we consider some particular instances, and rigorously investigate the distribution of their inter--point network distances. Our results are framed in terms of approximations, whose accuracy increases with the size of the network. We also give some insight into how the reduction in typical inter--point distances occasioned by the presence of shortcuts is related to the dimension of the underlying space.

cond-mat.dis-nn↗

arXiv2003

Discrete small world networks

Small world models are networks consisting of many local links and fewer long range 'shortcuts', used to model networks with a high degree of local clustering but relatively small diameter. Here, we concern ourselves with the distribution of typical inter-point network distances. We establish approximations to the distribution of the graph distance in a discrete ring network with extra random links, and compare the results to those for simpler models, in which the extra links have zero length and the ring is continuous.

cond-mat↗