Normal approximation for calculating the rate of convergence in the weak law of large numbers (Q1822407)

Let \(\{X_ k\), \(k\in {\mathbb{N}}\}\) be a sequence of real independent r.v., \(S_ n=\sum^{n}_{k=1}X_ k\). Furthermore, let \(\{A_ n\), \(n\in {\mathbb{N}}\}\) and \(\{b_ n\), \(n\in {\mathbb{N}}\}\) be sequences of real numbers, \(b_ n\) nondecreasing, \(b_ n\to \infty\) as \(n\to \infty\) and \(\limsup_{n\to \infty}b_{n+1}/b_ n<\infty\). The author gives various estimates for the limit behaviour of \(P\{| S_ n-A_ n| >b_ n\}\). For instance, let g be a positive real function such that for some \(\delta >0\), \(g(u)/u^{\delta}\to 0\) as \(u\to \infty\). Let \(\bar S{}_ n=\sum^{n}_{k=1}X_ k\chi [| X_ k| \leq b_ n]\) (\(\chi\) is the indicator function), \(\bar A{}_ n=A_ n-E(\bar S_ n)\), and for some \(\alpha >0\) let \(\limsup_{n\to \infty}| \bar A_ n| /b_ n<\alpha.\) If \(\sup_{n\geq 1}[g(u)/n]\sum^{n}_{k=1}P\{X_ k| >u\}<\infty\), then the relations \[ \sup_{n\geq 1}\frac{g(b_ n)}{n}P\{| S_ n- A_ n| >\alpha b_ n\}<\infty \] and \[ \sup_{n\geq 1}\frac{g(b_ n)}{n}\Phi (-\frac{\alpha b_ n-| \bar A_ n|}{\sqrt{Var \bar S_ n}})<\infty \] are equivalent, where \(\Phi\) is the standard d.f.