  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.
