- 2 June, 2016, The University of Melbourne
- Organiser: Kostya Borovkov

**Speakers and talks**

**Kostya Borovkov**(University of Melbourne).*A refined version of the integro-local Stone theorem.***Andrea Collevecchio**(Monash University).*Attraction properties for general urn processes.***Daniel Dufresne**(University of Melbourne).*Discounted Sums with Renewal Times***Jie Yen Fan**(University of Melbourne).*Limit theorems for the age structure of a population.***Laurence Field**(EPF Lausanne).*SLE and conformally invariant loop measures.***Greg Markowsky**(Monash University).*The planar Brownian Green’s function, and a probabilistic proof of the Riemann Mapping Theorem.***Nathan Ross (**University of Melbourne).*Limit behavior of some Polya urn models associated to preferential attachment graphs and random trees.*

**Kostya Borovkov.*** A refined version of the integro-local Stone theorem.*

Let S_n be a random walk with i.i.d. jumps. We refine Stone’s integro-local theorem by deriving the first term in the asymptotic expansion for the probability P(S_n \in [x,x+D)) as n–>oo and establishing uniform bounds for the remainder term, under the assumption that the jump distribution Cramer’s strong non-lattice condition and E|X|^r=3. (Joint work with A.A. Borovkov.)

**Andrea Collevecchio** .* Attraction properties for general urn processes.*

We study attraction properties of urns composed of balls with two distinct colours and which evolve over time. This evolution may depend on the composition of the urn as well as certain other factors, external or internal depending on the history of the urn. We prove that, under mild conditions, the model localizes on one of the two colours. We extend our discussion to a system of interacting urns, and a general class of strongly reinforced random walks. Joint Work with Jiro Akahori, Tim Garoni and Kais Hamza.

**Daniel Dufresne**. *Discounted Sums with Renewal Times.*

“Claims” (= i.i.d. random variables) {X(k)} occur at renewal times {T(k)}; each claim is discounted back to time 0, by multiplying it by exp(-r T(k)), r the rate of interest. What is the distribution of Z(t), the sum of the discounted claims occurring between time 0 and a fixed time t? This problem originates in risk theory, when trying to improve the classical ruin model by introducing a non-zero rate of return. Up to now very few results were known about Z(t), except in the case where claim arrival times are Poisson. It is well known that the distribution of Z(t=infinity) satisfies an identity of the form Z=(law) B times (Z+C), where B, C and Z are independent; there is a literature on that problem, with a number of explicit solutions. We will show that there is a similar identity for Z(S), where S is a random time, and that this allows easier calculations of the moments of Z(t<infinity) and possibly its distribution. (Joint work with Zhehao Zhang.)

**Jie Yen Fan **(University of Melbourne). *Limit theorems for the age structure of a population.*

We consider a family of general branching processes indexed by the parameter K and whose reproduction parameters may depend on the age of the individual as well as the whole age structure of the population. The parameter may, for example, represent the carrying capacity of a population that is supercritical (resp. subcritical) when the population size is below (resp. above) the carrying capacity. The law of large numbers of the population, as index K increases, was established in a previous paper of Hamza, Jagers and Klebaner. In this paper, we give the central limit theorem result. We show the convergence of the properly scaled process to a distribution-valued process. Joint work with Kais Hamza, Peter Jagers and Fima Klebaner.

**Laurence Field **(EPF Lausanne). * SLE and conformally invariant loop measures.*

We will discuss some recent developments in the theory of SLE and conformal invariance. A basic conformally invariant object is the Brownian loop measure, first considered by Symanzik in the context of Euclidean quantum field theory, and later applied to the Schramm-Loewner Evolution in work of Lawler and Werner. The same loop measure was used by Sheffield and Werner to provide the first rigorous construction of the simple Conformal Loop Ensembles (CLE). We use a renormalized version of the Brownian loop measure to construct the unique conformally covariant measure on “SLE-type” loops rooted at a point in a simply connected domain that has a natural restriction property. We then describe a procedure for “unrooting” SLE curves to be independent of a base point by using the natural parametrization for SLE, and discuss work in progress applying this procedure to SLE loops. (Joint work with Greg Lawler.

**Greg Markowsky**. *The planar Brownian Green’s function, and a probabilistic proof of the Riemann Mapping Theorem.*

It has been known for some time that the Green’s function of a planar domain can be defined in terms of Brownian motion, and that this Green’s function is conformally invariant. I will show how this conformal invariance can be generalized in order to calculate Green’s function for stopping times more general than exit times, and how these considerations can be used to give a proof that any two simply connected domains in the plane (excluding the plane itself) are conformally invariant, also known as the Riemann Mapping Theorem.

**Nathan Ross . **Limit behavior of some Polya urn models associated to preferential attachment graphs and random trees.

I will discuss a family of Pólya urn models that arise when studying degree statistics in preferential attachment graphs and lengths of spanning subtrees in some generative tree models such as Rémy’s algorithm for binary trees. The limiting distributions of the number of balls of a given color in the urn varies considerably across the family and can be difficult to describe. However, these limits are fixed points of certain probabilistic distributional transformations and this perspective provides proofs of convergence, sometimes with rates via Stein’s method, and leads to further properties of the limits. Joint work with Erol Peköz and Adrian Röllin.