On the Convergence of Moments in the Central Limit Theorem

Let X₁, X₂, ⋯, Xₙ be a sequence of independent random variables (r.v.'s) with zero mean and finite standard deviation σᵢ, 1 i n. According to the central limit theorem, the normed sum Yₙ = (1/sₙ) ∑ⁿᵢ₌₁ Xᵢ, where sₙ = ∑ⁿᵢ₌₁ σ²ᵢ, is under certain additional conditions approximatively normally distributed. We will here examine the convergence of the moments and the absolute moments of Yₙ 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).