# Quasi-logarithmic combinatorial structures, and approximation by the Dickman distribution

*Stochastic Processes and Financial Mathematics*

*by Professor Andrew Barbour*

*Institution:*Universitaet Zuerich, Switzerland

*Date: Wed 28th July 2010*

*Time: 3:15 PM*

*Location: Old Geology Theatre 2, The University of Melbourne*

*Abstract*: Quasi-logarithmic combinatorial structures are a class of decomposable combinatorial structures which extend the logarithmic class considered by Arratia, Barbour and Tavar\'{e} (2003), examples of which may arise as additive algebraic semigroups in the sense of Knopfmacher (1979). In order to obtain asymptotic approximations to their component spectra, it is first necessary to find good approximations to the distribution of the sum of an associated sequence of independent random variables, whose limit is not normal, but has density related to the Dickman function. This in turn requires an argument that refines the Mineka coupling by incorporating a blocking construction, leading to exponentially sharper coupling rates for the sums in question. Applications include distributional limit theorems for the size of the largest component and for the vector of counts of the small components in a quasi-logarithmic combinatorial structure. (This is joint work with Bruno Nietlispach)

*For More Information:* contact: Prof Daniel Dufresne - dufresne@unimelb.edu.au OR Dr Aihua Xia - xia@ms.unimelb.edu.au