Asymptotic expansions for distributions of compound sums of light   subexponential random variables
  We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same light-tailed subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, 7 terms expansions. 
Ph .Barbe (CNRS), W.P. McCormick (UGA), C. Zhang (UGA)