An Inequality for the Asymmetry of Distributions and a Berry-Esseen Theorem for Random Summation  
  Authors: Hendrik Kläver, Norbert Schmitz,  
  Keywords: Random number of i.i.d. random variables, Central limit theorem for random sums, Asymmetry of distributions, Berry-Esseen theorem for random sums.  
  Date Received: 16/03/04  
  Date Accepted: 15/12/05  
  Subject Codes:

60E15, 60F05, 60G40.

  Editors: Sever S. Dragomir,  

We consider random numbers $ N_n$ of independent, identically distributed (i.i.d.) random variables $ X_i$ and their sums $ sum_{i=1}^{N_n} X_i$. Whereas Blum, Hanson and Rosenblatt [3] proved a central limit theorem for such sums and Landers and Rogge [8] derived the corresponding approximation order, a Berry-Esseen type result seems to be missing. Using an inequality for the asymmetry of distributions, which seems to be of its own interest, we prove, under the assumption $ N_n/ntotau$ (in an appropriate sense), a Berry-Esseen theorem for random summation. ;

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