On the Convergence of Moments in the Central Limit Theorem

Abstract

Let $X_1, X_2, \cdots, X_n$ be a sequence of independent random variables (r.v.'s) with zero mean and finite standard deviation $\sigma_i, 1 \leqq i \leqq n$. According to the central limit theorem, the normed sum $Y_n = (1/s_n) \sum^n_{i=1} X_i,$ where $s_n = \sum^n_{i=1} \sigma^2_i$, is under certain additional conditions approximatively normally distributed. We will here examine the convergence of the moments and the absolute moments of $Y_n$ towards the corresponding moments of the normal distribution. The results in this general case are stated in Theorem 3 and Theorem 4, but, in order to avoid repetition and unnecessary complication, explicit proofs will only be given in the case of equally distributed random variables. (Theorem 1 and Theorem 2).