- 8 April 2011, The University of Melbourne
- Supported by MASCOS
- Organiser: K. Borovkov

**Vladimir A Vatutin**(Steklov Matematical Institute, Moscow).*Multitype branching processes in random environment and polling systems.***Aihua Xia**(Melbourne Uni)**.***Clubbed Binomial Approximation for the Lightbulb Process.*.**Daniel Dufresne**(Melbourne Uni). Changes of measure for the square-root process and their consequences**Kostya Borovkov**(Melbourne Uni).*Ornstein-Uhlenbeck type processes with heavy distribution tails*

**Abstracts**

**Vladimir A Vatutin**. *Evaluation of ruin probability by simulations. Multitype branching processes in random environment and polling systems.*

We consider a polling system consisting of a single server and m stations with infinite-buffer queues indexed by i=0,1,…,m-1. Initially there are no customers in the system. When customers arrive to the system the server starts immediately the service by visiting the stations in cyclic order starting at station 0 according to a selected service policy (to be described). Two basic models are considered: polling systems with zero switchover times from station to station and polling systems with positive random switchover times from station to station. Assuming that the arrival and service policies meet the so-called branching and immigration conditions and allowing the arrival and service policies to change in a random manner we describe the tail distributions of various characteristics of the polling systems including busy periods and the total number of customers served within a busy period. The results obtained can be applied to a wide class of polling systems including systems subject to exhaustive, gated, exhaustive-gated and binomial gated disciplines. Our proofs are based on a connection between the polling systems under consideration and the multitype branching processes in random environment without immigration (for the case of zero switchover times) and with immigration (for the case of positive random switchover times).

**Aihua Xia ****. ***Clubbed Binomial Approximation for the Lightbulb Process*

The lightbulb process was motivated by a pharmaceutical study of the effect of dermal patches designed to activate/deactivate targeted receptors [see Rao, Rao and Zhang~(2007)] and it evolves as follows. On days r=1,…,n, out of n lightbulbs, all initially off, exactly r bulbs selected uniformly and independent of the past have their status changed from off to on, or vice versa and our interest is on the distribution of the number W_n of bulbs on at the terminal time n. We demonstrate that, with C_n a suitable clubbed binomial distribution, the total variation distance between the distributions of C_n and W_n is bounded by 2.7314 \sqrt{n} e^{-(n+1)/3}. The talk is based on a joint work with L. Goldstein.

* Daniel Dufresne . *Changes of measure for the square-root process and their consequences.

The square-root process includes the squared Bessel process as a special case. Its study goes back several decades. Properties of the square-root process have been found using various methods, often not so simple. In particular, (1) the absolute continuity relationships that hold between the processes with different parameters are obtained using some specific properties of the process themselves (coupled with Kazamaki’s criterion), and (2) the joint Laplace transform of the variable and its integral has been derived by solving ordinary or partial differential equations. Using some recent results on changes of measures (Mijatovic and Urusov, 2010), we show how to simplify the derivation of these properties by turning them on their heads, finding the changes of measures first and the transforms second. The Laplace transform of the variable is obtained from its moments; next, it is shown that the Laplace transform of the variable and the Radon-Nikodym derivatives suffice to obtain the triple transform of the variable, the integral of the variable and the integral of one over the variable. Several of the formulas appear to be new.

**Kostya Borovkov** . *Ornstein-Uhlenbeck type processes with heavy distribution tails.*

We consider a transformed Ornstein-Uhlenbeck process model that can be a good candidate for modelling real-life processes characterized by acombination of time-reverting behaviour with heavy distribution tails. We begin with presenting the results of an exploratory statistical analysis of the log prices of a major Australian public company, demonstrating several key features typical of such time series. Motivated by these findings, we suggest a simple transformed Ornstein-Uhlenbeck process model and analyze its properties showing that the model is capable of replicating our empirical findings. We also discuss three different estimators for the drift coefficient in the underlying (unobservable) Ornstein-Uhlenbeck process which is the key descriptor of dependence in the process. [Joint work with G. Decrouez]