An asymptotic behavior of the remainder in the central limit theorem for moments of sums of independent random variables (Q1969269)

Let \(\{X_{nj}\}_{j\geq 1}\), \(n= 1,2,\dots\), be a series of sequences of independent random variables with zero means and finite variances \(\sigma_{nj}^2\) such that \(\sum_{j\geq 1} \sigma_{nj}^2= 1\). Put \(S_n= \sum_{n\geq 1} X_{nj}\), \(F_n(x)= {\mathbf P}(S_n< x)\), \(\beta_n(k)= \sum_{j\geq 1} {\mathbf E}|X_{nj} |^k\), \(k\geq 0\). Provided that \(\beta_n(2s)+ \beta_n(\upsilon)< \infty\) for an integer \(s\geq 1\) and for \(0< \upsilon< 2s+ 2\), consider the remainder \(d_{sn} (\upsilon)= {\mathbf E}|S_n|^\upsilon- \sum_{l=0}^{s-1} \int_{-\infty}^\infty |x|^\upsilon Q_{2l,n} (dx)\), \(n\geq 1\), where \(Q_{0,n}\) is the standard normal distribution function and \(Q_{2l,n}\), \(l\geq 1\), are the even terms of the corresponding Edgeworth-Cramér expansion. The asymptotic behaviour of \(d_{sn} (\upsilon)\) as \(n\to \infty\) is derived. The results obtained generalize results of \textit{P. Hall} [Ann. Probab. 10, 1004-1018 (1982; Zbl 0516.60025)] in the i.i.d. case.