Strong Convergence on Weakly Logarithmic Combinatorial Assemblies

TL;DR

This paper extends the theory of assemblies by weakening the regularity conditions, providing new approximation methods and analyzing the behavior of large components and additive functions within these structures.

Contribution

It generalizes the Fundamental Lemma for assemblies under weaker conditions and develops estimates for dependent large components and additive functions.

Findings

Generalized the Fundamental Lemma for weakly logarithmic assemblies.

Provided estimates for dependent large components in assemblies.

Proved analogs of Major's and Feller's theorems for these structures.

Abstract

We deal with the random combinatorial structures called assemblies. By weakening the logarithmic condition which assures regularity of the number of components of a given order, we extend the notion of logarithmic assemblies. Using the author's analytic approach, we generalize the so-called Fundamental Lemma giving 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. These estimates are applied to examine additive functions defined on such a class of structures. Some analogs of Major's and Feller's theorems which concern almost sure behavior of sums of independent random variables are proved.