Timothy C. Brown

Stein's method and point process approximation

A correction to an error in a recent paper of Barbour and Browns is detailed, and a relevant corollary stated.

Negative Binomial Approximation with Stein's Method

Bounds on the rate of convergence to the negative binomial distribution are found, where this rate is measured by the total variation distance between probability laws. For an arbitrary discrete random variable written as a sum of indicators, an upper bound of coupling form is expressed as an average of terms each of which measures the difference between the effect of particular indicator being one and the value of a geometrically distributed random variable. When a monotone coupling exists a lower bound can also be shown. Application of these results is illustrated with the example of the Po´lya distribution for which the rate of approach to the negative binomial limit is found.

Removing logarithms from Poisson process error bounds

We present a new approximation theorem for estimating the error in approximating the whole distribution of a finite-point process by a suitable Poisson process. The metric used for this purpose regards the distributions as close if there are couplings of the processes with the expected average distance between points small in the best-possible matching. In many cases, the new bounds remain constant as the mean of the process increases, in contrast to previous results which, at best, increase logarithmically with the mean. Applications are given to Bernoulli-type point processes and to networks of queues. In these applications the bounds are independent of time and space, only depending on parameters of the system under consideration. Such bounds may be important in analysing properties, such as queueing parameters which depend on the whole distribution and not just the distribution of the number of points in a particular set.

Point processes in time and stein's method

This article gives an upper bound for a Wasserstein distance between the distribution of a simple point process and that of a Poisson process on the positive half line. The bound is partly expressed in terms of their compensators, and partly in terms of the expected future effect of having a point at a given time. The argument is based on Steins method, together with a martingale approach. Some examples are provided, which illustrate the computation of the upper bound and demonstrate its accuracy

Poisson convergence in two dimensions with application to row and column exchangeable arrays

A Poisson convergence theorem is given for a sequence of simple point processes on the plane. The approach taken here is to formulate sufficient conditions for Poisson convergence in terms of the behaviour of two one-dimensional compensators. This limit theorem is then applied to obtain a functional Poisson convergence result for a sequence of row and column exchangeable arrays.

Poisson approximations for Markov-driven point processes

An asymptotically finite bound is derived for the total variation distance between the distribution of N(t) and the Poisson distribution with mean EN(t) when N is a simple point process whose interpoint times are exponential with means determined by an ergodic, finite-state Markov chain and when it is a Cox process with a stationary, irreducible, finite-state continuous-time Markov chain for intensity.

IMPROVED RESULTS ON POISSON PROCESS APPROXIMATION IN JACKSON NETWORKS

Melamed (1979) proved that for an open migration process, a necessary and sufficient condition for the equilibrium flow along a link to be Poissonian is the absence of loops: no customer can travel along the link more than once. Barbour and Brown (1996) quantified the statement by allowing the customers a small probability of travelling along the link more than once and proved Poisson process approximation theorems analogous to Melameds Theorem. Amongst the three bounds presented in Barbour and Brown (1996), the one in terms of the Wasserstein metric is of particular interest since it reveals more insightful information about the closeness between the process of flows and an approximating Poisson process, and it is small when the parameter of the system is small, except a logarithmic factor in terms of time in which the flows are considered. The bound was later improved by Brown, Weinberg and Xia (2000) who showed that the logarithmic factor in terms of time can be lifted at the cost of an extra parameter being introduced into the bound. In this paper, we present a new bound which simplifies and sharpens the bounds in the above-mentioned two papers and compare the performance of these bounds for a simple open migration process.

Estimation for a class of positive nonlinear time series models

This paper considers the a symptotic properties of an estimator of a parameter that generalizes the correlation coefficient to a class of nonlinear, non-Gaussian and positive time series models. The models considered are one step Markov chains whose innovations have an infinitely divisible distribution, as do the marginal distributions. The models and their statistical analysis do not degenerate as is the case for some linear models that have been suggested for positive time series data. The asymptotic theory derives from a point process weak convergence argument that uses a regular variation assumption on the left tail of the innovation distribution.