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Approximation by the Dickman distribution and quasi-logarithmic combinatorial structures


Barbour, A D; Nietlispach, B (2011). Approximation by the Dickman distribution and quasi-logarithmic combinatorial structures. Electronic Journal of Probability, 16:880-902.

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). In order to obtain asymptotic approximations to their component spectrum, it is necessary first to establish an approximation to the sum of an associated sequence of independent random variables in terms of the Dickman distribution. 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.

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). In order to obtain asymptotic approximations to their component spectrum, it is necessary first to establish an approximation to the sum of an associated sequence of independent random variables in terms of the Dickman distribution. 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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:May 2011
Deposited On:17 Feb 2012 19:25
Last Modified:05 Apr 2016 15:33
Publisher:Institute of Mathematical Statistics
ISSN:1083-6489
Publisher DOI:https://doi.org/10.1214/EJP.v16-881
Related URLs:http://arxiv.org/abs/1007.5269

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