  We investigate the second order asymptotic behavior of trimmed sums
$T_n=\frac 1n \sum_{i=\kn+1}^{n-\mn}\xin$, where $\kn$, $\mn$ are sequences of
integers, $0\le \kn < n-\mn \le n$, such that $\min(\kn, \mn) \to \infty$, as
$\nty$, the $\xin$'s denote the order statistics corresponding to a sample
$X_1,...,X_n$ of $n$ i.i.d. random variables. In particular, we focus on the
case of slightly trimmed sums with vanishing trimming percentages, i.e. we
assume that $\max(\kn,\mn)/n\to 0$, as $\nty$, and heavy tailed distribution
$F$, i.e. the common distribution of the observations $F$ is supposed to have
an infinite variance.
  We derive optimal bounds of Berry -- Esseen type of the order
$O\bigl(r_n^{-1/2}\bigr)$, $r_n=\min(\kn,\mn)$, for the normal approximation to
$T_n$ and, in addition, establish one-term expansions of the Edgeworth type for
slightly trimmed sums and their studentized versions.
  Our results supplement previous work on first order approximations for
slightly trimmed sums by Csorgo, Haeusler and Mason (1988) and on second order
approximations for (Studentized) trimmed sums with fixed trimming percentages
by Gribkova and Helmers (2006, 2007).
