  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.
