Past Seminars

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Random thinnings and point processes under cover
Dominic Schumacher
The University of Western Australia and Swiss National Science Foundation
Start time: 3:15 pm
Date: Friday 9 November 2007
Location: Room 213, Richard Berry Building, Department of Mathematics and Statistics, The University of Melbourne

A thinning of a point process (a "random point pattern") on R^D is obtained by deleting points independently according to probabilities supplied by a random function R^D -> [0,1]. In this talk I present a general estimate of the total variation distance between the distribution of a thinned point process and a Poisson process distribution and apply the result to a point process that is covered by i.i.d. balls. No previous knowledge about point processes or distances between probability distributions is required.

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Pair-copula constructions of multiple dependence
Claudia Czado
Technische Universität, München, Germany
Start time: 3:15 pm
Date: Monday 1 October 2007
Location: Theatre 2, Ground Floor, 111 Barry Street, Carlton

Building on the work of Bedford, Cooke and Joe, we show how multivariate data, which exhibit complex patterns of dependence in the tails, can be modelled using a cascade of pair-copulae, acting on two variables at a time. We use the pair-copula decomposition of a general multivariate distribution and propose a method to perform inference. The model construction is hierarchical in nature, the various levels corresponding to the incorporation of more variables in the conditioning sets, using pair-copulae as simple building blocks. Pair-copula decomposed models also represent a very flexible way to construct higher-dimensional copulae. We apply the methodology to a financial data set. This is joint work with K. Aas, A. Frigessi, H. Bakken and A. Min.

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Student Processes
Nikolai Leonenko
Cardiff University, UK
Start time: 3:15 pm
Date: Friday 14 September 2007
Location: Theatre 1, Ground Floor, 111 Barry Street, Carlton

With the aim of modelling key stylized features of observational series from economic and finance a number of stochastic processes with Student marginals and various types of dependence structures are discussed. In the field of finance, distributions of logarithmic asset returns can often be fitted extremely well by Student t-distribution with "?" degrees of freedom, typically 3 = ? = 5. This implies infinite k-th moments for k =?. Another issue in modelling economic and financial time series is that their sample autocorrelation functions (acf) may decay quickly, but their absolute or squared increments may have acfs with non-negligible values for large lags. These ubiquitous phenomena call for on effort to develop reasonable models which can be integrated into the economic and financial theory as well as theories of turbulence. This approach has a long history since Mandebrot, who advocated to use heavy tails distributions (stable or Poreto type) and multifractals.

Alternatively, Barndorff-Nielsen has proposed to use different type of hyperbolic distributions in financial econometrics. In fact, the symmetric scaled t-distribution can be considered as limiting case of generalized hyperbolic distribution However, some properties of t-distribution can not be obtained from the corresponding limiting procedure. For instance, the characteristic function of the symmetric scaled t-distribution can not be obtained from the expression for haracteristic function of generalized hyperbolic distribution, based on the Laplace transform technique, which does not exist for the symmetric scaled t-distribution. Moreover, most of hyperbolic distributions are semiheavy tailed, while the symmetric scaled t-distributions heavy tailed. However, the basic ideas and constructions similar to symmetric scaled t-distribution is and more large class of the generalized hyperbolic distributions.

In this talk we propose a number of stochastic processes with Student marginal and various types of dependence structures, which are, in our opinion, interesting for economic and finance. This talk is based on joint work with Chris Heyde (Australian National University and Columbia University)

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Path to extinction
Fima Klebaner
Monash University, Melbourne
Start time: 1:15 pm
Date: Tuesday 11 September 2007
Location: Russell Love Theatre, Richard Berry Building, Department of Mathematics and Statistics, The University of Melbourne

We consider the size of a branching population at the time uT, where 0<u<1 and T is the time of extinction. For example, for u=0.5, it gives us size half-way to extinction, and when u varies between 0 and 1 we recover the path to extinction. We give an approximation when the population starts with a large number of individuals x, and show that the population is of order x^{1-u}. This approximation is not so precise just before extinction when u is close 1. For this case we consider random time T-a for small a, and give an approximation for this case. It is joint work with P. Jagers and S. Sagitov, Chalmers, Sweden.

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Maxmoments and mixing in stochastic systems
Daniel Tokarev
MASCOS
Start time: 1:15 pm
Date: Thursday 6 September 2007
Location: Russell Love Theatre, Richard Berry Building, Department of Mathematics and Statistics, The University of Melbourne

We consider the expectations of the maxima of independent non-negativerandom variables. In the case of identically distributed random variables, when such expectations are referred to as maxmoments, we solve an analogue of the moment problem and present the asymptotics of the maxmoments as the number of random variables increases to infinity, and give an application to simple branching processes. In the case of non-identically distributed independent random variables, we derive tight upper and lower bounds for these expectations in terms of the maxmoments (which, incidentally, is equivalent to deriving tight bounds for the L_1-distance between the arithmetic and geometric means of the respective distribution functions). We also investigate a related optimization problem in the case when the random variables can follow one of two fixed different distributions. [This talk is based on joint research with K. Hamza, A. Sudbury, F. Klebaner and P. Jagers.]

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