Moments of Sums of Independent and Identically Distributed Random   Variables
  We present an analytic method for computing the moments of a sum of independent and identically distributed random variables. The limiting behavior of these sums is very important to statistical theory, and the moment expressions that we derive allow for it to be studied relatively easily. We show this by presenting a new proof of the central limit theorem and several other convergence results. 
Daniel M. Packwood
math.ST stat.TH