# UofM & Monash micro-conference on Probability Theory & Its Applications, 20 March 2013

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

Speakers and talks

Abstracts

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.