# Probability Mini-Conference @ Melbourne Uni on Friday, 27 March 2012

• 27 March 2012,  The University of Melbourne
• Organiser: Kostya Borovkov

Speakers and talks

Abstracts

Andrew BarbourThe asymptotics of the Aldous gossip process.

As a model for the spread of gossip, Aldous used the (discrete) 2-D torus to represent space, with gossip spreading between neighbours, but also occasionally at long range. The development of a continuous space version was shown by Durrett and Chatterjee to have some randomness at the start, but thereafter to run an almost deterministic course, described by a mysterious function $h$, that also appears in Aldous’s paper. Here, we explain the asymptotics, and identify $h$, entirely in terms of branching processes. Our arguments remain valid for the spread on quite general, locally homogeneous manifolds, in any number of dimensions.

Daniel Dufresne. Gram-Charlier Distributions.

Not the greatest model for stock returns, Gram-Charlier distributions still have some interest. Will talk about their history and properties, maybe option pricing.

Fima Klebaner
. Law of Large Numbers for the age distribution in population dependent branching process.

We consider a population of particles evolving in continuous time. If the lifespans are not exponential such process (the Belman-Harris process) is not Markov. However, considered as a collection of ages, as a finite counting measure $A=\sum_a \delta_a,$ it is Markov. We give the generator of such process, and its semimartingale decomposition. Assuming further that the parameters (lifespans, offspring distributions) depend on the population composition, in such a way that it is supercritical below some threshold K and subcritical above it. We prove Law of Large Numbers for the measure valued process $\frac{1}{K}A_t^K$ as K ⭢ ∞, and describe the measure-valued limit $A_t$. Convergence is shown in the space of trajectories, the Skorohod space $D(R^+,M(R^+))$, where $M(R^+))$ is the space of measures on $R^+$ metrizable by weak convergence. This is joint work with K. Hamza (Monash) and P. Jagers (Chalmers).