Some remarks on the accuracy of normal approximation to distributions of random sums (Q1177476)

Let \(\{\xi_ j\}^ \infty_{j=1}\) be independent r.v.s with \(E\xi_ j=0\), \(\sigma^ 2_ j=E\xi^ 2_ j<\infty\) and \(S_ n=\sum^ n_{k=1}\xi_ k\), \(B^ 2_ n=\sum^ n_{k=1}\sigma^ 2_ k\), \(F_ n(x)=P(S_ n<xB_ n)\). For integer-valued r.v.s \(\nu>0\) independent of \(\{\xi_ j\}\) denote \(B^ 2=EB^ 2_ \nu\), \(F(x)=P(S_ \nu<xB)\). Set \(\Delta_ n=\sup_ x| F_ n(x)- \Phi(x)|\), \(\Delta=\sup_ x| F(x)-\Phi(x)|\), where \(\Phi\) is the d.f. of a standard normal r.v. It is proved that \[ \Delta\leq E\Delta_ \nu+\min\{1.04 \kappa_ 1;\;1.95 \kappa_ 2\}, \leqno (*) \] where \(\kappa_ 1=E| B^{-2}B^ 2_ \nu-1|\), \(\kappa_ 2=E| B^{-2}B^ 2_ \nu-1|^ 2\). Assuming \(\mu^ 3_ j=E|\xi_ j|^ 3<\infty\), \(j\in\mathbb{N}\), the bound for \(\Delta\) is obtained in terms of \(\kappa_ 1\), \(\kappa_ 2\) and \(B^{- 3}E(\mu^ 3_ 1+\cdots+\mu^ 3_ \nu)\). Another version of (*) uses Lévy distance between distributions of \(B^{-2}B^ 2_ \nu\) and 1.