- 20 March 2013, The University of Melbourne
- Proudly supported by MASCOS
- Organiser: K. Borovkov

**Speakers and talks**

(Lehigh University).**Joe Yukich***Variance asymptotics for random polytopes in smooth convex bodies.*(The University of Melbourne).**Aihua Xia***A revisit of approximate Melamed’s theorem*.( Osaka City University).*Atsushi Takeuchi**Large deviations for stochastic functional differential equations.*(Monash University).**Andrea Collevecchio***On Large deviations and interacting particles systems.*

* Joe Yukich*.

*Variance asymptotics for random polytopes in smooth convex bodies.*

Let $K \subset \R^d$ be a smooth convex set and let $\P_\lambda$ be a Poisson point process on $\R^d$ of intensity $\lambda$. The convex hull of $\P_\lambda \cap K$ is a random convex polytope $K_\lambda$. As $\lambda \to \infty$, we show that the variance of the number of $k$-dimensional faces of $K_\lambda$, when properly scaled, converges to a scalar multiple of the affine surface area of $K$. Similar asymptotics hold for the variance of the number of $k$-dimensional faces for the convex hull of a binomial process in $K$. This is joint work with Pierre Calka.

* Aihua Xia*. A

*revisit of approximate Melamed’s theorem*.

We demonstrate that the equilibrium customer flow process (ECFP) along the links in a Jackson queuing network is asymptotically a negative binomial process. Our result provides a better alternative than the approximate Melamed’s theorem proposed by Barbour and Brown (1996) that the ECFP process is approximately Poisson if the probability of customers travelling along the links more than once is very small.

** Atsushi Takeuchi**.

*Large deviations for stochastic functional differential equations.*

Consider stochastic functional differential equations depending on past histories. We shall study the large deviations for the family of the solution process, and apply it to the asymptotic behavior of the density. The Malliavin calculus plays a crucial role in our argument.

* Andrea Collevecchio* .

*On Large deviations and interacting particles systems.*

We talk about general large deviations principles and analyze their application to few interacting particles systems. In particular we study a system of Bosons and express important quantities related to this system in terms of variational formulas.