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Strong convergence on weakly logarithmic combinatorial assemblies

Authors
Journal
Discrete Mathematics
0012-365X
Publisher
Elsevier
Publication Date
Volume
311
Issue
6
Identifiers
DOI: 10.1016/j.disc.2010.12.014
Keywords
  • Random Combinatorial Structure
  • Total Variation Distance
  • Almost Sure Convergence
  • Iterated Logarithm Law
  • Additive Function
Disciplines
  • Mathematics

Abstract

Abstract We deal with the random combinatorial structures called assemblies. Instead of the traditional logarithmic condition which assures asymptotic regularity of the number of components of a given order, we assume only lower and upper bounds of this number. Using the author’s analytic approach, we generalize the independent process approximation in the total variation distance of the component structure of an assembly. To evaluate the influence of strongly dependent large components, we obtain estimates of the appropriate conditional probabilities by unconditioned ones. The estimates are applied to examine additive functions defined on a new class of structures, called weakly logarithmic. Some analogs of Major’s and Feller’s theorems which concern almost sure behavior of sums of independent random variables are proved.

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