Convergence rates in precise asymptotics (Q408244)

Let \(X,X_1,X_2,\ldots\) be i.i.d. random variables with partial sums \(S_n=\sum_{k=1}^nX_k\). Suppose that \(\operatorname{E}[X]=0\) and \(\operatorname{E}[X^2]=\sigma^2\in(0,\infty)\), then \textit{C. C. Heyde}'s result [J. Appl. Probab. 12, 173--175 (1975; Zbl 0305.60008)] on precise asymptotics states that NEWLINE\[NEWLINE\lim_{\varepsilon\downarrow0}\varepsilon^2\sum_{n=1}^\infty \operatorname{P}(|S_n|\geq\varepsilon n)=\sigma^2,NEWLINE\]NEWLINE and a corresponding rate of convergence result NEWLINE\[NEWLINE\lim_{\varepsilon\downarrow0}\varepsilon^{3/2}\left(\sum_{n=1}^\infty \operatorname{P}(|S_n|\geq\varepsilon n)-\frac{\sigma^2}{\varepsilon^2}\right)=0NEWLINE\]NEWLINE was shown by \textit{O. I. Klesov} [Theory Probab. Math. Stat. 49, 83--87 (1994); translation from Teor. Jmovirn. Mat. Stat. 49, 119--125 (1993; Zbl 0861.60054)] under the additional assumption of finite third moments.NEWLINENEWLINEA very careful and technical analysis leads the authors to similar rate of convergence results (extending Klesov's) corresponding to \textit{R. Chen}'s [J. Multivariate Anal. 8, 328--333 (1978; Zbl 0376.60033)] precise asymptotics NEWLINE\[NEWLINE\lim_{\varepsilon\downarrow0}\varepsilon^{2\frac{r-p}{2-p}}\sum_{n=1}^\infty n^{\frac{r}{p}-2}\operatorname{P}(|S_n|\geq\varepsilon n)=\frac{p}{r-p}\,\operatorname{E}\left[|Z|^{2\frac{r-p}{2-p}}\right]NEWLINE\]NEWLINE for \(0<p<2\) and some \(r\geq2\) with \(\operatorname{E}[|X|^r]<\infty\), where \(Z\) is nomal with mean zero and variance \(\sigma^2\). The authors' only additional assumption for their main result is that \(\operatorname{E}[|Z|^q]<\infty\) for some \(q>r\).