Strong laws of large numbers for intermediately trimmed sums of i.i.d.   random variables with infinite mean
  We consider moderately trimmed sums of non-negative i.i.d. random variables. We show that for every distribution function there exists a proper moderate trimming such that for the trimmed sum a non-trivial strong law of large numbers holds. In case that the distribution function has regularly varying tails we give necessary and sufficient conditions on the trimming for a strong law of large numbers to hold. 
Marc Kesseb\"ohmer and Tanja Schindler
math.PR