The order of approximation in the central limit theorem for random summation (Q1102027)

Let \(\{X_ n,n\geq 1\}\) be a sequence of independent random variables such that E \(X_ n=a_ n\), \(\sigma^ 2X_ n=\sigma^ 2_ n<\infty\), \(n\geq 1\), and let \(\{N_ n,n\geq 1\}\) be a sequence of positive integer- valued random variables. Let us put \[ S_ n=\sum^{n}_{k=1}X_ k,\quad A_ n=\sum^{n}_{k=1}a_ k,\quad s^ 2_ n=\sum^{n}_{k=1}\sigma^ 2_ k. \] It is known that if there exists a sequence \(\{k_ n,n\geq 1\}\) of positive integers such that \(k_ n\to \infty\) as \(n\to \infty\), and \(s^ 2_{N_ n}/s^ 2_{k_ n}\to^{p}\lambda\) as \(n\to \infty\), for some positive random variable \(\lambda\), then \((S_ n-A_ n)/s_ n\to^{{\mathcal D}}N(0,1)\) implies \[ (1)\quad (S_{N_ n}-A_{N_ n})/s_{N_ n}\to^{{\mathcal D}}N(0,1)\quad and \] \[ (2)\quad (S_{N_ n}-A_{N_ n})/\lambda^{1/2}s_{k_ n}\to^{{\mathcal D}}N(0,1), \] where N(0,1) denotes a standard normal random variable. In this paper the rate of convergence in (1) and (2) is presented under the assumption that \(\lambda\) is independent of \(\{X_ n,n\geq 1\}\).