# Probability Mini-Conference @ Melbourne Uni, 21 February 2013

• 21 February 2013, The University of Melbourne
• Organiser: Aihua Xia

Speakers and talks

Abstract

Louis H. Y. Chen. On the error bound in a combinatorial central limit theorem.

The notion of exchangeable pair is central to Stein’s method. The use of concentration inequalities is an effective means for bounding the Kolmogorov distance in normal approximation. Let $\{X_{ij}: i, j = 1, …, n\}$ be independent random variables with finite 3rd moments and let $\pi$ be a random permutation of $(1, …, n)$, independent of the $X_{ij}$. Let $U = \sum X_i \pi(i)$ and let $W = (U – EU)/(Var(U))^{1/2}$. In this talk we will use exchangeable pairs and the concentration inequality approach to obtain a 3rd-moment error bound on $|P(W \le x) – \Phi(x)|$, where $\Phi$ is the standard normal distribution function. This result includes the case where the $X_{ij}$ are constants and the case of sampling without replacement from independent random variables. This talk is based on joint work with Xiao Fang.

Søren Asmussen (Aarhus University). L\’evy processes, phase-type distributions, and martingales.

L\’evy processes are defined as processes with stationary independent increments and have become increasingly popular as models in queueing, finance etc.; apart from Brownian motion and compound Poisson processes, some popular examples are stable processes, variance Gamma processes, CGMY L\’evy processes (tempered stable processes), NIG (normal inverse Gaussian) L\’evy processes, hyperbolic L\’evy processes. We consider here a dense class of L\’evy processes, compound Poisson processes with phase-type jumps in both directions and an added Brownian component. Within this class, we survey how to explicitly compute a number of quantities that are traditionally studied in the area of L\’evy processes, in particular two-sided exit probabilities and associated Laplace transforms, the closely related scale function, one-sided exit probabilities and associated Laplace transforms coming up in queueing problems, and similar quantities for a L\’evy process with reflection in 0. The solutions are in terms of roots to polynomials, and the basic equations are derived by purely probabilistic arguments using martingale optional stopping; a particularly useful martingale is the so-called Kella-Whitt martingale. Also the relation to fluid models with a Brownian component is discussed.

Just as ARMA processes play a central role in the representation of stationary time series with discrete time parameter, $(Y_n)_{n\in Z}$, CARMA processes play an analogous role in the representation of stationary time series with continuous time parameter, $(Y(t))_{t\in R}$. L\’evy-driven CARMA processes permit the modelling of heavy-tailed and asymmetric time series and incorporate both distributional and sample-path information. In this talk we review recent results in the theory and application of these processes, including existence and uniqueness, causality and invertibility, sequences derived by sampling and integration, prediction of CARMA processes and inference based on high-frequency observations.