New Zealand Probability Workshop (Mon-Tue)

and

Australia and New Zealand Applied Probability
Workshop (Wed-Fri)

The University of Auckland, January 23-27, 2012.

The University of Auckland, January 23-27, 2012.

Monday | Tuesday |
Wednesday |
Thursday | Friday | |||||

9:05 | opening remarks | 9:15 | F. Camia | 9:10 | Opening remarks | 9:20 | J. Ross | 9:20 | J. Hunter |

9:15 | R. van der Hofstad | 10:00 | A. Sakai | 9:20 | Y. Nazarathy | 10:00 | A. Black | 10:00 | Y. Salomon |

10:00 | J. Goodman | 10:50 | Morning tea | 10:00 | Z. Palmowski | 10:40 | Morning tea | 10:40 | Morning tea |

10:50 | Morning tea | 11:10 | R. Sun | 10:40 | Morning tea | 11:00 | M. O'Reilly | 11:00 | R. Costa |

11:10 | C. Newman | 11:55 | P. Greenwood | 11:00 | A. Xia | 11:40 | G. Decroueux | 11:40 | Z. Lu |

11:55 | D. Stein | 12:45 | Lunch | 11:40 | S. Hautphenne | 12:20 | Lunch | 12:20 | Lunch |

12:45 | Lunch | 14:00 | T. Salisbury | 12:20 | Lunch followed by excursion | 14:00 | D. Daley | 14:00 | C. Pearce |

14:00 | V. Jones | 14:45 | L. Addario-Berry | 14:40 | A. Eshragh | 14:40 | N. Bean | ||

14:45 | C. Cotar | 15:30 | afternoon tea | 15:20 | P. Taylor | 15:20 | Afternoon tea | ||

15:30 | Afternoon tea | 16:00 | Afternoon tea | ||||||

16:00 | Mini-excursion | 18:00 | Bus pickup | ||||||

19:00 | Conference dinner |

The scaling limit of the minimum spanning tree of the complete graph

Let K_n be the complete graph on n vertices,
and assign iid uniform [0,1] weights to its edges. Write T_n for the (almost
surely unique) minimum weight spanning tree of K_n with these weights. The
random tree T_n comes equipped with an intrinsic distance (graph distance)
and a mass measure (each vertex has mass one). We show that that after a
suitable rescaling of distances and masses, T_n converges in distribution to
a limiting random measured real tree T, in the Gromov--Hausdorff--Prokhorov
sense, and establish some basic properties of the limiting tree T.

Joint work with Nicolas Broutin, Christina Goldschmidt, Gregory Miermont.

Joint work with Nicolas Broutin, Christina Goldschmidt, Gregory Miermont.

Nigel Bean

QBDs, Rational Arrival Processes and Richard Tweedie.

In this talk I will introduce you to Rational Arrival Processes, which are the natural process extension of the family of Matrix Exponential distributions. The family of Matrix Exponential distributions are exactly the family of distributions with rational Laplace-Stieltjes transform, and are a natural generalization of the family of Phase-Type distributions. We use such processes in a queue and so generalize the standard quasi-birth-and-death (QBD) process into a richer environment. Such a process is then an example of a Markov chain on an uncountable state-space with an \emph{operator}-geometric stationary distribution, as studied by Richard Tweedie in 1982. We apply Richard's results and then exploit a peculiar linearity property to reduce the resulting calculations to simple matrix calculations. We further show that any relevant algorithm from the matrix-analytic methods literature can be applied to this richer class of models. This approach provides general results, replacing the laborious derivation of specific results that we had performed earlier.

Andrew
Black

Modelling household disease dynamics and control.

We present a modelling framework that allows us to incorporate accurate
household structure in a stochastic epidemic model while still being
computationally feasible to investigate. We assume that within-household
dynamics are governed by a continuous-time Markov chain, and consider the
initial stages of disease emergence, during which an assumption of
between-household mixing resulting in novel households being infected is
justified. This allows us to calculate a number of population level quantities,
such as the early growth rate and offspring distribution, for a heterogeneous
distribution of household sizes. This methodology also allows us to easily asses
the impact of different intervention strategies such as the use of antivirals.
We will illustrate the methodology by using census data from several countries
and evaluating the impact of delays in antiviral prophylaxis.

Renato Costa

Option Pricing under a
Nonlinear and Nonnormal GARCH

We investigate the pricing of options in a class of discrete-time Flexible
Coefficient Generalized Autoregressive Conditional Heteroskedastic
(FC-GARCH) models with nonnormal innovations. A conditional Esscher
transform was used to select a price kernel for valuation in the incomplete
market. This choice of the price kernel can be justified by an economic
equilibrium argument based on maximizing the expected power utility. We
provide a numerical study on the pricing and risk management results when
the GARCH and the FCGARCH innovations have a normal distribution or a
shifted-Gamma distribution and identify some key features of the pricing
results. Joint work with Alvaro Veiga and Tak Kuen Siu.

In this talk I explain a promising and previously unnoticed
link between electronic structure of molecules and optimal transportation, and
first results. The `exact' mathematical model for electronic structure, the
many-electron Schroedinger equation, becomes computationally unfeasible for more
than a dozen or so electrons. For larger systems, the standard model underlying
a huge literature in computational physics/chemistry/materials science is
density functional theory (DFT). In DFT, one only computes the single-particle
density instead of the full many-particle wave function. In order to obtain a
closed equation, one needs a closure assumption which expresses the pair density
in terms of the single-particle density rho.

We show that in the
semiclassical limit, there holds an exact closure relation, namely the pair
density is the solution to a optimal transport problem with Coulomb cost. We
prove that this problem has a unique solution given by an optimal map; moreover
we derive an explicit formula for the optimal map in the case when $\rho$ is
radially symmetric (note: atomic ground state densities are radially symmetric
for many atoms such as He, Li, N, Ne, Na, Mg, Cu).

In my talk I focus on how
to deal with its main mathematical novelties (cost decreases with distance; cost
has a singularity on the diagonal). I also discus the derivation of the
Coulombic OT problem from the many-electron Schroedinger equation for the case
with N electrons. Joint work with Gero Friesecke (TU Munich) and Claudia
Klueppelberg (TU Munich)

Daryl
Daley

BRAVO
effect in M/M/k/K queueing systems

In its stationary state, the departure process N_dep of
a M/M/k/K queueing system with large buffer capacity K is approximately a
Poisson process at rate min(rho, 1) where rho is the arrival rate (subject to
suitable choice of scale). For the
case k=1 the asymptotic variance behaviour is then

var N_dep(0,t] ~ min(rho,1) * E N_dep(0,t] = min(rho, 1) * t as t goes to infinity

except when rho=1, in which case the constant is replaced by 2/3.

Nazarathy and Weiss (2008) demonstrated this (to them)
surprising fact, and dubbed it the BRAVO effect (``Balancing [i.e. rho = 1]
Reduces Asymptotic Variance of Outputs).
We have now shown that for general k, a BRAVO effect persists but the
factor 2/3
is replaced by one that depends on how k grows (or not) with K.
Indeed, more is known: there is nontrivial asymptotic behaviour for \rho
= 1 - beta/sqrt{k} as k, K go to infinity subject to K= sqrt{k}*alpha for
positive alpha and finite beta.
Thus, while there is a `continuous' transition in stochastic behaviour as rho
increases through the critical value rho=1, higher order moments need not be
continuous.

Geoffrey Decrouez

On the Ergodicity and the
Stationary distribution of a Stochastic Neuron Network

We present a stochastic network model for real-life neuron
networks that takes into account Hebbian learning. The system can be
represented by a Markov process whose state space is an infite hierarchy of
finite-dimensional simplices. nder broad assumptions, the process is shown
to be ergodic and have a continuously differentiable density w.r.t. the sum
of Lebesgue measures of the simplices. We also demonstrate that the
stationary distribution of the network can be approximated by a
finite-dimensional one corresponding to a similar Markov process on a
truncated version of the state space and that the convergence rate is
super-exponential rate. (Joint work with Kostya Borovkov and Matthieu
Gilson).

Ali Eshragh

Optimal Experimental Design for a Growing Population

Our goal is to estimate the rate of growth, lambd), of a population governed by
a simple stochastic model. We may choose (n) time points at which to count the
number of individuals present, but due to detection difficulties, or constraints
on resources, we are able only to observe each individual with fixed probability
(p). We discuss the optimal times at which to make our n observations in order
to approximately maximize the accuracy of our estimate of ambda. For
computational and analytical reasons which will be discussed, we specifically
focus on the cases n=1 and n=2, presenting both theoretical and numerical
findings.

Long paths for first passage percolation on the complete graph

First passage percolation on the complete graph -- also known as the
stochastic mean-field model of distance -- describes the flows and optimal
routing strategies for a network with variable link costs. To each edge of
the complete graph, associate an i.i.d. positive edge cost X_e. The cost
of a path is the sum of its edge costs, and the optimal path between two
vertices is the path of lowest cost.

When the edge weights are
exponential, optimal paths contain of order log n edges. Using branching
process and coupling techniques, we show how distributions with a heavy tail
at 0 produce optimal paths that are much longer, and show how to identify
the growth rate explicitly in terms of the tail of the distribution. This
allows us to exhibit a smooth transition between path lengths of order log
n (as in super-critical random graphs) and path lengths of order n^{1/3}
(as in the critical Erd\"os-R\'enyi random graph).

Sustained oscillations for density dependent Markov processes

Simulations of models of epidemics, biochemical systems, and other bio-systems show that when deterministic models yield damped oscillations, stochastic counterparts show sustained oscillations at an amplitude well above the expected noise level. A characterization of damped oscillations in terms of the local linear structure of the associated dynamics is well known, but in general there remains the problem of identifying the stochastic process which is observed in stochastic simulations. Here we show that in a general limiting sense the stochastic path describes a circular motion modulated by a slowly varying Ornstein-Uhlenbeck process. Numerical examples are shown for the Volterra predator-prey model, Sel'kov's model for glycolysis, and a damped linear oscillator. (Joint work with Peter H. Baxendale).

Extinction probability of branching processes with inﬁnitely many types

We consider multitype branching processes with inﬁnitely
many types. We emphasize the differences with the finite-type case in some
asymptotic properties and in the extinction criteria. We propose converging
sequences to the extinction probability vector and give them a probabilistic
interpretation. Joint work with Guy Latouche and Giang Nguyen.

Hypercube percolation

Consider bond percolation on the hypercube
{0,1}^n at the critical edge probability p_c defined such that the expected
cluster size equals 2^{n/3}, where 2^{n/3} acts as the cube root of the
number of vertices of the n-dimensional hypercube. Percolation on the
hypercube was proposed by Erdos and Spencer (1979), and has proved to be
substantially harder than percolation on the complete graph due to the
non-trivial geometry of the hypercube.

In this talk, I will describe the
phase transition for percolation on the hypercube, and show that it shares many
features with that on the complete graph.

In previous work, we have
identified the subcritical and critical regimes of percolation on the hypercube.
In particular, we know that for p=p_c(1+O(2^{-n/3})), the largest connected
component is of size roughly 2^{2n/3} and that this quantity is
non-concentrated. So far, we were missing an analysis of the behavior of the
largest connected component just above the critical value. In this work, we
identify the supercritical behavior of percolation on the hypercube by showing
that, for any sequence epsilon_n tending to zero, but epsilon_n being much
larger than 2^{-n/3}, percolation on the hypercube with edge probability
p=p_c(1+\epsilon_n) has, with high probability, a unique giant component of
size (2+o(1))\epsilon_n 2^n. A main tool is the use of non-backtracking
random walk, which we use to show that long percolation paths have endpoints
that are almost uniform on the hypercube.

This is joint work with Asaf
Nachmias, building on previous work with Markus Heydenreich, Gordon Slade,
Christian Borgs, Jennifer Chayes and Joel Spencer.

Jeffrey J. Hunter

The Role of Kemeny's Constant in Properties of Markov Chains

In a finite
m-state
irreducible Markov chain with stationary probabilities
pi
and mean first passage times *m*ij (mean
recurrence time when i = j)
it was first shown, by Kemeny and Snell, that sum_{j=1}^m
pi_j m_ij
is
a constant,
K, not
depending on i.
This constant has since become known as Kemenys
constant. We consider a variety of techniques for finding expressions for
K,
derive some bounds for
K, and explore
various applications and interpretations of these results.
Interpretations include the expected number of links that a surfer on the World
Wide Web located on a random page needs to follow before reaching a desired
location, as well as the expected time to mixing in a Markov chain. Various
applications have been considered including some
perturbation results, mixing on directed graphs and its relation to the Kirchhoff
index of regular graphs.

Asymptotic distribution and uniform convergences for local linear fitting under stochastic processes of generalised dependence

Local linear fitting is a popular nonparametric method in nonlinear
statistical and econometric modelling; see, for example, Fan and Gijbels (1996),
Fan and Yao (2003) and Li and Racine (2007). Under various mixing stochastic
processes (in particular alpha--mixing, i.e., strong mixing, which includes many
other mixings such as phi-mixing, beta-mixing, as special cases), this technique
of local linear fitting has been well studied in the literature by many
researchers, c.f., Liebscher (1996), Masry (1996), Bosq (1998), Fan and Yao
(2003), Hansen (2008), Kristensen (2009), among others. However, from a
practical point of view, the mixing (e.g., alpha—mixing) processes suffer from
many undesirable features. For example, for a lot of popular processes in
econometrics such as an ARMA process mixed with ARCH or GARCH errors, it is
still difficult to show whether they are alpha--mixing or not except in some
very special cases. Even for a very simple linear AR(1) model with innovation
being independent symmetric Bernoulli random variables taking on values of -1
and 1, the stationary solution to the model is not alpha--mixing (Andrew 1984).

In this talk, I will first review some of the extensions of the stochastic
processes from the mixing ones, in particular a class of generalised stable
processes from mixings, or called near epoch dependence, which covers a variety
of interesting stochastic processes in time series econometric modelling. Then I
will report some recent developments on the local linear fitting technique under
this kind of generalised dependent processes. I will particularly introduce some
results that my co-authors and I have recently established in this regard,
including the pointwise asymptotic distributions for the probability density
estimation and local linear estimator of a nonparametric regression function as
well as the uniform strong and weak consistencies with convergence rates for the
local linear fitting under the condition of near epoch dependence. These results
are of wide potential interest in time series semiparametric modelling.

Yoni Nazarathy

Scaling limits of cyclically varying birth-death
processes

Fluid limits of stochastic queueing systems have received considerable attention in recent years. The general idea is to scale space, time and/or system parameters as to obtain a simpler, yet accurate description of the system. A basic example is the single server queue with time speeded up and space scaled down at the same rate. A second well known example is the Markovian infinite server queue with the arrival rate speeded up and space scaled down at the same rate. Such scalings and their network generalizations are often useful for obtaining stability conditions and approximating optimal control policies. In this talk we consider birth-death processes with general transition rates and obtain an asymptotic scaling result, generalizing the Markovian single server and infinite server cases. We apply our results to the steady-state analysis of queueing systems with cyclic or time varying behaviour. Examples are systems governed by deterministic cycles, queues with hysteresis control and queues with Markov-modulated arrival or service rates. The unifying property of such systems, is that if they are properly scaled, the resulting trajectories follow a cyclic or piece-wise deterministic behavior which is determined by the asymptotic scaling. This yields simple a approximation for the stationary distribution which is shown to be asymptotically exact. Joint work with Matthieu Jonckheere.

Short-Range Spin Glasses: an Introduction

Spin glasses are a prototype disordered system
whose successful analysis could fill a gap in our understanding of condensed
matter physics. They present a mathematical problem of both depth and
complexity. Parisi's replica symmetry breaking (RSB) solution of the mean-field
spin glass exhibits a new kind of broken symmetry with many novel features.
However, more realistic short-range models remain unsolved, and a controversy
exists as to whether RSB is valid for them. But work of recent years has
clarified and restricted the types of ordering short-range spin glasses can
exhibit.

In this talk I will review spin glasses and their history and
why they are of interest to both physicists and mathematicians. I will
briefly discuss old and newer results about short-range spin glasses at low and
zero temperature. Our focus is on low-temperature equilibrium properties of
short-range spin glasses, including: numbers of pure and ground states in
various dimensions; the question of whether a phase transition and broken
spin-flip symmetry occur; the use of metastates to analyze the organization of
the (presumed) low-temperature phase; and whether mean-field spin glasses can
shed light on short-range models in moderate dimensions.

This talk will be mostly heuristic. In an accompanying talk, Dan Stein will discuss a rigorous result on spin glass ground states obtained by us and our collaborators Louis-Pierre Arguin and Michael Damron.

Infinitesimal Generators of Markovian Stochastic Fluid Flow Models

Recently there has been considerable
interest in Markovian stochastic fluid flow models. A number of authors
have used different methods to calculate quantities of interest. We discuss
methods which allow for the direct analysis of the stochastic fluid flow models,
without the need for transformation to other equivalent models. The foundation
of the analysis is the derivation of the infinitesimal generator of the model.
The advantage of having established the infinitesimal generator is that the
expressions for various performance measures can be written in terms of that
generator.

First, we discuss our methods in the context of the traditional
one-dimensional stochastic fluid flow models [1]. Next, we consider the
stochastic two-dimensional fluid flow model [2] which consists of two stochastic
fluid flows, one of which is unconstrained, driven by the same underlying Markov
chain.

Finally, we consider the stochastic fluid-fluid model [3], which is
a stochastic fluid model driven by an uncountable-state process, which is a
stochastic fluid model itself. This work is the first direct analysis of a
stochastic fluid model that is Markovian on a continuous state space.

References:

[1] Bean, N. G., O'Reilly, M. M. and Taylor, P. G. (2005). Hitting
probabilities and hitting times for stochastic fluid flows. Stoch. Proc.
Appl.115, 1530-1556.

[2] Bean, N. G. and O'Reilly M. M. (2011). Stochastic 2-dimensional
fluid model. Submitted for publication.

[3] Bean, N. G. and O'Reilly M. M. (2012). A Stochastic Fluid Model
driven by an uncountable-state process, which is a Stochastic Fluid Model itself:
Stochastic Fluid-Fluid model. Submitted for publication

Zbigniew Palmowski

Forward-backward extrema of Levy processes and fluid queues

For a Levy process X and fixed S>t (possibly S=+\infty) the future up-down process is defined by: U^*_{t,S} = \sup_{t <= u < t+S} (X_u - X_t).

The fluctuations of U^*_{t,S} are described by the running supremum and running infimum

\overline U ^*_{T,S} = sup_{0 <= t <= T} U^*_{t,S}
and
\underline U^*_{T,S} = inf_{0 <= t <= T} U^*_{t,S}.

The random variables \overline U^*_{T,S} and
\underline U^*_{T,S} are path-dependent performance measures of fluid queues:
U^*_{t,S} describe the buffer content of fluid queues observed at time t when
queue already has been already running S-t units of time before 0 and \overline
U ^*_{T,S} and \underline U^*_{T,S} are the maximal and minimal such contents
for t ranging over [0,T].

Charles Pearce

Duration problem with multiple exhanges and the
classical secretary problem

The multiple-choice duration problem has, as objective, maximizing the time of posession of relatively best objects. We show that for the m-choice problem there exists a sequence (s_1, s_2, ... ,s_m) of critical numbers such that, when there are k choices still to be made, the optimal strategy selects a relatively best object if it appears at or after time s_k. We exhibit an equivalence with the classical secretary problem.

Invasion of an infectious disease in a finite, homogeneous population

A novel strain of an infectious disease starts life in a single individual. To become established in a population it must be successfully transmitted to at least one other individual in the population. Perhaps the most well-known quantity related to this process is R_0 (the basic reproduction number) which is the expected number of secondary infections arising directly from the initial individual. R_0 is typically evaluated under the assumption of an infinite population size.

Despite the importance of R_0 in infectious disease modelling and
management, little attention has been paid to the impact of finite
population size, and, as far I know, no one has investigated the probability
mass function of secondary infections. We will explore the impact of
population size along with the type of infectious period distribution on the
distribution and expectation of number of secondary infections.

Asymptotic behavior of the critical two-point functions for statistical-mechanical models with power-law decaying potentials.

We consider self-avoiding walk, percolation and the Ising model on the d-dimensional integer lattice mathbb{Z}^d that are defined by power-law decaying pair potentials of the form D(x)\approx|x|^{-d-\alpha} for some alpha>0. These models are known to exhibit critical behavior as the parameter $p$, such as the inverse temperature for the Ising model, tends to its critical value p_c.

Let d_c denote the upper-critical dimension: 2(\alpha\wedge 2) for self-avoiding walk and the Ising model and 3(\alpha\wedge2) for percolation. I will explain the result of joint work in progress with Lung-Chi Chen that, if \alpha\ne2, d>d_c and the spread-out parameter L is sufficiently large, then the critical two-point function G_{p_c}(x) for each model, such as the spin-spin correlation function for the critical Ising model, is asymptotically a multiple of |x|^{\alpha\wedge2-d}. This is a key element for the so-called 1-arm exponent to take on its mean-field value: 1/2 for percolation and 1 for the Ising model. The proof is based on application of the lace expansion.

Random Walk in Degenerate Random Environment

I will describe results about random walk in random environments, which
degenerate in the sense that standard ellipticity assumptions do not hold. In
other words, a walker on the square 2-dimensional lattice uses walk
probabilities in which certain nearest neighbour transitions are forbidden.
These questions lead naturally into studies of percolation on directed random
graphs. This is work with Mark Holmes (Auckland).

Unimodal density estimation using cross entropy minimisation

In a range of applications it is often required to
estimate the probability density associated with some random phenomena under
study, or uncertain parameters. In some cases, the density is known to be
unimodal. Other information regarding the density, such as some of its
moments or percentiles, may also be known. If the explicit parametric form
of the density cannot be determined from physical considerations alone then
parametric methods may prove unsatisfactory. Since only summary statistics,
rather than raw data, is available, nonparametric kernel estimation is not
applicable. However, if a standard Cross Entropy approach is used, the
resulting density may not be unimodal. Using a characterisation of unimodal
random variables due to Shepp following the work of Khinchin, we demonstrate
a method for incorporating unimodality in cross entropy density estimation.
We illustrate our method by applying it in the context of expert
elicitation.

Ground States of the Two-Dimensional Spin Glass

This is joint
work with Louis-Pierre Arguin, Michael Damron and Chuck Newman (Commun.
Math. Phys. 300 (2010) 641-657).

It is an open problem to determine
the number of infinite-volume ground states in the Edwards-Anderson (nearest
neighbor) spin glass model on \Z^d for d \geq 2 (with, say, mean zero
Gaussian couplings). This is a limiting case of the problem of determining
the number of extremal Gibbs states at low temperature. In both cases, there
are competing conjectures for d \geq 3, but no complete results even for
d=2. I report on results which go some way toward proving that (with zero
external field, so that ground states come in pairs, related by a global
spin flip) there is only a single ground state pair (GSP).

Our result is weaker in two ways: First, it applies not to the full plane \Z^2, but to a half-plane. Second, rather than showing that a.s. (with respect to the quenched random coupling realization J) there is a single GSP, we show that there is a natural joint distribution on J and GSP's such that for a.e. J, the conditional distribution on GSP's given J is supported on only a single GSP. The methods used combine percolation-like geometric arguments with translation invariance (in one of the two coordinate directions of the half-plane) and uses as a main tool the ``excitation metastate'' which is a probability measure on GSP's and on how they change as one or more individual couplings vary.

Symmetric Rearrangements Around Infinity with Applications to Levy Processes

We prove a new rearrangement inequality for multiple integrals, which partly generalizes a result of Friedberg and Luttinger (1976) and can be interpreted as involving symmetric rearrangements of domains around infinity. As applications, we prove two comparison results for general Levy processes and their symmetric rearrangements. The first application concerns the survival probability of a point particle in a Poisson field of moving traps following independent Levy motions. We show that the survival probability can only increase if the point particle does not move, and the traps and the Levy motions are symmetrically rearranged. This essentially generalizes an isoperimetric inequality of Peres and Sousi (2011) for the Wiener sausage. In the second application, we show that the $q$-capacity of a Borel measurable set for a Levy process can only increase if the set and the Levy process are symmetrically rearranged. This result generalizes an inequality obtained by Watanabe (1983) for symmetric Levy processes. Joint work with Alex Drewitz (ETH) and Perla Sousi (Cambridge).

Queueing Theory and `Ideal Hospital Occupancy'

I'm getting more and more interested in the
contribution that the applied probability community can make to decision-making
at the political and administrative levels. The relationship between these areas
is not easy.

I shall illustrate some points using the concept of `ideal hospital occupancy'.
The paper that has had the most influence on the dimensioning of hospitals, at
least in the UK, was written in 1999 by three health economists Bagust, Place
and Posnett. They used an Excel simulation to make the point there is trade-off
between availability and utilisation, something that is taught in undergraduate
stochastic modelling courses. From a mathematical point of view, their paper has
many problems, and yet it had a greater effect on decision-making in the UK
heath system than any of the huge number of queueing theory papers that have
been written before and since that make essentially the same point.

In this talk, I shall discuss some recent contributions by myself and others on
the subject of `ideal hospital occupancy'. I shall also pose the question of how
best the applied probability community can contribute to the political debate in
all sorts of areas where there is a need to make decisions under conditions of
uncertainty.

On asymptotics of locally dependent point processes

We investigate a family of approximating processes that can capture the asymptotic behaviour of locally dependent point processes. We prove two theorems to accommodate respectively the positively and negatively related dependent structures. Three examples are given to illustrate that our approximating processes are doing a better job than compound Poisson processes when the mean number of random events increases. This talk is based on a joint work with Fuxi Zhang.