# A note to the convergence rates in precise asymptotics

## Abstract

Let $\left\{X,{X}_{n},n\ge 1\right\}$ be a sequence of i.i.d. random variables with zero mean. Set ${S}_{n}={\sum }_{k=1}^{n}{X}_{k}$, $E{X}^{2}={\sigma }^{2}>0$, and ${\lambda }_{r,p}\left(ϵ\right)={\sum }_{n=1}^{\mathrm{\infty }}{n}^{r/p-2}P\left(|{S}_{n}|\ge {n}^{1/p}ϵ\right)$. In this paper, the author discusses the rate of approximation of $\frac{p}{r-p}E{|N|}^{2\left(r-p\right)/\left(2-p\right)}$ by ${ϵ}^{2\left(r-p\right)/\left(2-p\right)}{\lambda }_{r,p}\left(ϵ\right)$ under suitable moment conditions, where N is normal with zero mean and variance ${\sigma }^{2}>0$, which improves the results of Gut and Steinebach (J. Math. Anal. Appl. 390:1-14, 2012) and extends the work He and Xie (Acta Math. Appl. Sin. 29:179-186, 2013). Specially, for the case $r=2$ and $p=\frac{1}{\beta +1}$, $\beta >-\frac{1}{2}$, the author discusses the rate of approximation of $\frac{{\sigma }^{2}}{2\beta +1}$ by ${ϵ}^{2}{\lambda }_{2,1/\left(\beta +1\right)}\left(ϵ\right)$ under the condition $E{X}^{2}I\left(|X|>t\right)=O\left({t}^{-\delta }l\left(t\right)\right)$ for some $\delta >0$, where $l\left(t\right)$ is a slowly varying function at infinity.

MSC:60F15, 60G50.

## 1 Introduction

Let $\left\{X,{X}_{n},n\ge 1\right\}$ be a sequence of i.i.d. random variables. Set ${S}_{n}={\sum }_{k=1}^{n}{X}_{k}$ and ${\lambda }_{r,p}\left(ϵ\right)={\sum }_{n=1}^{\mathrm{\infty }}{n}^{r/p-2}P\left(|{S}_{n}|\ge {n}^{1/p}ϵ\right)$. Heyde  proved that

$\underset{ϵ\to 0}{lim}{ϵ}^{2}{\lambda }_{2,1}\left(ϵ\right)={\sigma }^{2},$

whenever $EX=0$ and $E{X}^{2}={\sigma }^{2}<\mathrm{\infty }$. Klesov  studied the rate of the approximation of ${\sigma }^{2}$ by ${ϵ}^{2}{\lambda }_{2,1}\left(ϵ\right)$ under the condition $E{|X|}^{3}<\mathrm{\infty }$. He and Xie  improved the results of Klesov . Gut and Steinebach  extended the results of Klesov  and obtained the following Theorem A. Gut and Steinebach  studied the general idea of proving precise asymptotics.

Theorem A Let $\left\{X,{X}_{n},n\ge 1\right\}$ be a sequence of i.i.d.random variables with zero mean and $0, $r\ge 2$.

1. (1)

If $E{X}^{2}={\sigma }^{2}>0$ and $E{|X|}^{q}<\mathrm{\infty }$ for some $r, then

${ϵ}^{2\left(r-p\right)/\left(2-p\right)}{\lambda }_{r,p}\left(ϵ\right)-\frac{p}{r-p}E{|N|}^{2\left(r-p\right)/\left(2-p\right)}=o\left({ϵ}^{\frac{p\left(q-2\right)\left(r-p\right)}{\left(q-p\right)\left(2-p\right)}}\right).$
2. (2)

If $E{X}^{2}={\sigma }^{2}>0$ and $E{|X|}^{q}<\mathrm{\infty }$ for some $q\ge 3$ with $q>\frac{2r-3p}{2-p}$, then

${ϵ}^{2\left(r-p\right)/\left(2-p\right)}{\lambda }_{r,p}\left(ϵ\right)-\frac{p}{r-p}E{|N|}^{2\left(r-p\right)/\left(2-p\right)}=o\left({ϵ}^{\frac{2p\left(r-p\right)}{\left(2-p\right)\left(p+2q-pq\right)}}\right),$

where N is normal with mean 0 and variance ${\sigma }^{2}>0$.

The purpose of this paper is to strengthen Theorem A and extend the theorem of He and Xie  under suitable moment conditions. In addition, we shall discuss the rate at which ${ϵ}^{2}{\lambda }_{2,1/\left(\beta +1\right)}\left(ϵ\right)$ converges to $\frac{{\sigma }^{2}}{2\beta +1}$ under the condition $T\left(t\right)=O\left({t}^{-\delta }l\left(t\right)\right)$ for some $\delta >0$, where $T\left(t\right)=E{X}^{2}I\left(|X|>t\right)$, $l\left(t\right)$ is a slowly varying function at infinity. Throughout this paper, C represents a positive constant, though its value may change from one appearance to the next, and $\left[x\right]$ denotes the integer part of x. $\mathrm{\Phi }\left(x\right)$ is the standard normal distribution function, $\mathrm{\Phi }\left(x\right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{x}{e}^{-{t}^{2}/2}\phantom{\rule{0.2em}{0ex}}dt$, $\phi \left(x\right)={\mathrm{\Phi }}^{\mathrm{\prime }}\left(x\right)$.

## 2 Main results

From Gut and Steinebach , it is easy to obtain the following lemma.

Lemma 2.1 Let $\left\{X,{X}_{n},n\ge 1\right\}$ be a sequence of i.i.d. normal distribution random variables with zero mean and variance ${\sigma }^{2}>0$. Set $0 and $r\ge 2$, then

$\begin{array}{r}{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{\left(r/p\right)-2}P\left(|{S}_{n}|\ge {n}^{1/p}ϵ\right)-\frac{p}{r-p}E{|N|}^{2\left(r-p\right)/\left(2-p\right)}\\ \phantom{\rule{1em}{0ex}}=\left\{\begin{array}{cc}O\left({ϵ}^{2\left(r-p\right)/\left(2-p\right)}\right),\hfill & r<3p,\hfill \\ O\left({ϵ}^{4p/\left(2-p\right)}\right),\hfill & r\ge 3p.\hfill \end{array}\end{array}$
(2.1)

Lemma 2.2 (Bingham et al. )

Let $l\left(t\right)$ be a slowly varying function. We have

1. (1)

for any $\eta >0$,

$\underset{t\to \mathrm{\infty }}{lim}{t}^{\eta }l\left(t\right)=\mathrm{\infty },\phantom{\rule{2em}{0ex}}\underset{t\to \mathrm{\infty }}{lim}{t}^{-\eta }l\left(t\right)=0;$
2. (2)

if $0<\delta <1$, then

${\int }_{a}^{t}{s}^{-\delta }l\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\sim \frac{1}{1-\delta }{t}^{1-\delta }l\left(t\right),\phantom{\rule{1em}{0ex}}t\to \mathrm{\infty };$
3. (3)

if $\delta >1$, then

${\int }_{t}^{\mathrm{\infty }}{s}^{-\delta }l\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\sim -\frac{1}{1-\delta }{t}^{1-\delta }l\left(t\right),\phantom{\rule{1em}{0ex}}t\to \mathrm{\infty };$
4. (4)

if $\delta =1$, then $L\left(t\right)={\int }_{t}^{\mathrm{\infty }}\frac{l\left(s\right)}{s}\phantom{\rule{0.2em}{0ex}}ds$, $m\left(t\right)={\int }_{a}^{t}\frac{l\left(s\right)}{s}\phantom{\rule{0.2em}{0ex}}ds$ are slowly varying functions; and

$\underset{t\to \mathrm{\infty }}{lim}\frac{l\left(t\right)}{L\left(t\right)}=0,\phantom{\rule{2em}{0ex}}\underset{t\to \mathrm{\infty }}{lim}\frac{l\left(t\right)}{m\left(t\right)}=0.$

Theorem 2.1 Let $\left\{X,{X}_{n},n\ge 1\right\}$ be a sequence of i.i.d.random variables with zero mean and $0, $r\ge 2$.

1. (1)

If $E{X}^{2}={\sigma }^{2}>0$ and $E{|X|}^{3}<\mathrm{\infty }$ for some $r<3$, then

${ϵ}^{2\left(r-p\right)/\left(2-p\right)}{\lambda }_{r,p}\left(ϵ\right)-\frac{p}{r-p}E{|N|}^{2\left(r-p\right)/\left(2-p\right)}=\left\{\begin{array}{cc}O\left({ϵ}^{2\left(r-p\right)/\left(2-p\right)}\right),\hfill & 2\le r<\frac{3p}{2},\hfill \\ O\left({ϵ}^{p/\left(2-p\right)}log\frac{1}{ϵ}\right),\hfill & r=\frac{3p}{2},\hfill \\ O\left({ϵ}^{p/\left(2-p\right)}\right),\hfill & \frac{3p}{2}
(2.2)
2. (2)

If $E{X}^{2}={\sigma }^{2}>0$ and $E{|X|}^{2+\delta }<\mathrm{\infty }$ for some $0<\delta <1$, $r<2+\delta$, then

$\begin{array}{r}{ϵ}^{2\left(r-p\right)/\left(2-p\right)}{\lambda }_{r,p}\left(ϵ\right)-\frac{p}{r-p}E{|N|}^{2\left(r-p\right)/\left(2-p\right)}\\ \phantom{\rule{1em}{0ex}}=\left\{\begin{array}{cc}O\left({ϵ}^{2\left(r-p\right)/\left(2-p\right)}\right),\hfill & 2\le r<\left(1+\delta /2\right)p,\hfill \\ O\left({ϵ}^{p\delta /\left(2-p\right)}log\frac{1}{ϵ}\right),\hfill & r=\left(1+\delta /2\right)p,\hfill \\ o\left({ϵ}^{p\delta /\left(2-p\right)}\right),\hfill & \left(1+\delta /2\right)p
(2.3)
3. (3)

If $E{X}^{2}={\sigma }^{2}>0$ and $E{|X|}^{q}<\mathrm{\infty }$ for some $q\ge 3$ with $q>\frac{2r-3p}{2-p}$, then

${ϵ}^{2\left(r-p\right)/\left(2-p\right)}{\lambda }_{r,p}\left(ϵ\right)-\frac{p}{r-p}E{|N|}^{2\left(r-p\right)/\left(2-p\right)}=\left\{\begin{array}{cc}O\left({ϵ}^{2\left(r-p\right)/\left(2-p\right)}\right),\hfill & 2\le r<\frac{3p}{2},\hfill \\ O\left({ϵ}^{p/\left(2-p\right)}log\frac{1}{ϵ}\right),\hfill & r=3p/2,\hfill \\ O\left({ϵ}^{p/\left(2-p\right)}\right),\hfill & r>3p/2,\hfill \end{array}$
(2.4)

where N is normal with mean 0 and variance ${\sigma }^{2}>0$.

Remark 2.1 Clearly, Theorem 1 and Theorem 2 in He and Xie  are special cases of Theorem 2.1, by taking $r=2$ and $p=1$.

Remark 2.2 If $0, $r\ge 2$, we have $min\left(\frac{2\left(r-p\right)}{2-p},\frac{p\delta }{2-p}\right)>\frac{p\delta \left(r-p\right)}{\left(2+\delta -p\right)\left(2-p\right)}$ for $r<2+\delta =q\le 3$ and $min\left(\frac{2\left(r-p\right)}{2-p},\frac{p}{2-p}\right)>\frac{2\left(r-p\right)p}{\left(2-p\right)\left(p+2q-pq\right)}$ for some $q\ge 3$ with $q>\frac{2r-3p}{2-p}$. So, the results of Theorem 2.1 are stronger than those of Theorem A.

Theorem 2.2 Let $\left\{X,{X}_{n};n\ge 1\right\}$ be a sequence of i.i.d random variables with zero mean, and let $T\left(t\right)=O\left({t}^{-\delta }l\left(t\right)\right)$ for some $\delta >0$, where $l\left(t\right)$ is a slowly varying function at infinity. Set $E{X}^{2}={\sigma }^{2}>0$ and $\beta >-\frac{1}{2}$.

1. (1)

If $\delta >1$, then

${ϵ}^{2}{\lambda }_{2,1/\left(\beta +1\right)}\left(ϵ\right)-\frac{{\sigma }^{2}}{2\beta +1}=\left\{\begin{array}{cc}O\left({ϵ}^{2}\right),\hfill & -\frac{1}{2}<\beta <-\frac{1}{4},\hfill \\ O\left({ϵ}^{2}log\frac{1}{ϵ}\right),\hfill & \beta =-\frac{1}{4},\hfill \\ O\left({ϵ}^{1/\left(2\beta +1\right)}\right),\hfill & \beta >-\frac{1}{4}.\hfill \end{array}$
(2.5)
2. (2)

If $0<\delta <1$, then

${ϵ}^{2}{\lambda }_{2,1/\left(\beta +1\right)}\left(ϵ\right)-\frac{{\sigma }^{2}}{2\beta +1}=\left\{\begin{array}{cc}O\left({ϵ}^{2}\right),\hfill & -\frac{1}{2}<\beta <-\frac{1}{2}+\frac{\delta }{4},\hfill \\ O\left({ϵ}^{\delta /\left(2\beta +1\right)}l\left({ϵ}^{-1/\left(2\beta +1\right)}\right)\right),\hfill & \beta \ge -\frac{1}{2}+\frac{\delta }{4}.\hfill \end{array}$
(2.6)
3. (3)

If $\delta =1$, then

${ϵ}^{2}{\lambda }_{2,1/\left(\beta +1\right)}\left(ϵ\right)-\frac{{\sigma }^{2}}{2\beta +1}=\left\{\begin{array}{cc}O\left({ϵ}^{2}+{ϵ}^{2}{\int }_{1}^{{ϵ}^{-1/\left(2\beta +1\right)}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right),\hfill & -\frac{1}{2}<\beta <-\frac{1}{4},\hfill \\ O\left({ϵ}^{2}\left(1+{\int }_{1}^{{ϵ}^{-5}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right)log\frac{1}{ϵ}\right),\hfill & \beta =-\frac{1}{4},\hfill \\ O\left({ϵ}^{1/\left(2\beta +1\right)}\left(1+{\int }_{1}^{{ϵ}^{-\left(2\beta +3\right)/\left(2\beta +1\right)}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right)\right),\hfill & \beta >-\frac{1}{4}.\hfill \end{array}$
(2.7)

Remark 2.3 For $r=2$, $p=\frac{1}{\beta +1}$. If $l\left(t\right)=1$, then the result of Theorem 2.2 is weaker than that of Theorem 2.1 for $0<\delta <1$, $\beta \ge -\frac{1}{2}+\frac{\delta }{4}$, and weaker than that of Theorem 2.1 for $\delta =1$. But the condition $T\left(t\right)=O\left({t}^{-\delta }\right)$ is weaker than the condition $E{|X|}^{2+\delta }<\mathrm{\infty }$. If $l\left(t\right)\to 0$ as $t\to \mathrm{\infty }$, then the result of Theorem 2.2 is the same as that of Theorem 2.1 for $0<\delta <1$.

Remark 2.4 For $\delta >0$, the condition $E{|X|}^{2+\delta }<\mathrm{\infty }$ is neither sufficient nor necessary for the condition $T\left(t\right)=O\left({t}^{-\delta }l\left(t\right)\right)$. Here are some suitable examples.

Example 1 Let X be a random variable with density $f\left(x\right)=\frac{C\left(1+\delta ln|x|\right)}{{|x|}^{3+\delta }{ln}^{2}|x|}I\left(|x|>e\right)$, where C is a normalizing constant, and $0<\delta <1$, then $EX=0$ and $T\left(t\right)=\frac{C}{{t}^{\delta }lnt}I\left(t>e\right)$, $l\left(t\right)=\frac{1}{lnt}$ is a slowly varying function at infinity. But $E{|X|}^{2+\delta }=C{\int }_{|x|>e}\frac{1+\delta ln|x|}{|x|{ln}^{2}|x|}\phantom{\rule{0.2em}{0ex}}dx=\mathrm{\infty }$.

Example 2 Let X be a random variable with density $f\left(x\right)=\frac{C\left(\delta {ln}^{2}|x|+|x|\left(ln|x|-1\right)\right)}{{|x|}^{\delta +3}{ln}^{2}|x|{e}^{|x|/ln|x|}}I\left(|x|>e\right)$, where $0<\delta <1$, then $EX=0$ and $T\left(t\right)=\frac{C}{{t}^{\delta }{e}^{t/ln\left(t\right)}}I\left(t>e\right)$, $h\left(t\right)=\frac{1}{{e}^{t/lnt}}$, $E{|X|}^{2+\delta }<\mathrm{\infty }$. But $h\left(t\right)=\frac{1}{{e}^{t/lnt}}$ is not a slowly varying function at infinity.

In fact, we have the following result.

Theorem 2.3 Suppose X is a real random variable and $\delta >0$. Then $E{|X|}^{2+\delta }<\mathrm{\infty }$ if and only if ${t}^{\delta }T\left(t\right)\to 0$ and ${\int }_{t}^{\mathrm{\infty }}{s}^{\delta -1}T\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\to 0$ as $t\to \mathrm{\infty }$.

Remark 2.5 If ${t}^{\delta }T\left(t\right)$ is bounded as $t\to \mathrm{\infty }$ for some $\delta >0$, then we have $E{|X|}^{2+\alpha }<\mathrm{\infty }$ for every $\alpha \in \left(0,\delta \right)$ from Theorem 2.3.

Remark 2.6 Let X be a random variable with zero mean. If there exist positive constants ${C}_{1}$ and ${C}_{2}$ such that ${C}_{1}l\left(t\right)\le {t}^{\delta }T\left(t\right)\le {C}_{2}l\left(t\right)$ for sufficiently large t and some $\delta >0$, where $l\left(t\right)$ is a slowly varying function at infinity, then from Lemma 2.2(4) and Theorem 2.3, we have

## 3 Proofs of the main results

Proof of Theorem 2.1 Without loss of generality, we suppose that ${\sigma }^{2}=1$, $0<ϵ<1$. Since

$P\left(|{S}_{n}|\ge {n}^{1/p}ϵ\right)=2\left(1-\mathrm{\Phi }\left({n}^{\left(2-p\right)/2p}ϵ\right)\right)+{R}_{n},$
(3.1)

where

${R}_{n}=P\left({S}_{n}\le -{n}^{1/p}ϵ\right)-\mathrm{\Phi }\left(-{n}^{1/p-1/2}ϵ\right)+\mathrm{\Phi }\left({n}^{1/p-1/2}ϵ\right)-P\left({S}_{n}\le {n}^{1/p}ϵ\right).$

From (3.1), we have

$\begin{array}{r}{ϵ}^{2\left(r-p\right)/\left(2-p\right)}{\lambda }_{r,p}\left(ϵ\right)-\frac{p}{r-p}E{|N|}^{2\left(r-p\right)/\left(2-p\right)}\\ \phantom{\rule{1em}{0ex}}=2{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-2}\left(1-\mathrm{\Phi }\left({n}^{\left(2-p\right)/2p}ϵ\right)\right)-\frac{p}{r-p}E{|N|}^{2\left(r-p\right)/\left(2-p\right)}\\ \phantom{\rule{2em}{0ex}}+{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-2}{R}_{n}.\end{array}$
(3.2)

By Lemma 2.1, in order to prove Theorem 2.1, we only need to estimate ${ϵ}^{2\left(r-p\right)/\left(2-p\right)}{\sum }_{n=1}^{\mathrm{\infty }}{n}^{r/p-2}{R}_{n}$.

1. (1)

On account of a non-uniform estimate of the central limit theorem by Nagaev , for every $x\in R$,

$|P\left(\frac{{S}_{n}}{\sqrt{n}}
(3.3)

By (3.3), $|{R}_{n}|\le \frac{CE{|X|}^{3}}{\sqrt{n}{\left(1+ϵ{n}^{\left(2-p\right)/2p}\right)}^{3}}$.

1. (a)

If $r<3p/2$, then

${ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-2}{R}_{n}\le C{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-5/2}=O\left({ϵ}^{2\left(r-p\right)/\left(2-p\right)}\right).$
(3.4)
2. (b)

If $3p/2, then

$\begin{array}{r}{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-2}{R}_{n}\\ \phantom{\rule{1em}{0ex}}\le C{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}\frac{{n}^{r/p-2}}{\sqrt{n}{\left(1+ϵ{n}^{\left(2-p\right)/2p}\right)}^{3}}\\ \phantom{\rule{1em}{0ex}}\le C{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\left(\sum _{n=1}^{\left[{ϵ}^{-2p/\left(2-p\right)}\right]}\frac{{n}^{r/p-2}}{\sqrt{n}}+{ϵ}^{-3}\sum _{n=\left[{ϵ}^{-2p/\left(2-p\right)}\right]+1}^{\mathrm{\infty }}{n}^{r/p-5/2-\left(6-3p\right)/2p}\right)\\ \phantom{\rule{1em}{0ex}}=O\left({ϵ}^{p/\left(2-p\right)}\right).\end{array}$
(3.5)
3. (c)

If $r=3p/2$, then

$\begin{array}{rcl}{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-2}{R}_{n}& \le & C{ϵ}^{p/\left(2-p\right)}\left(\sum _{n=1}^{\left[{ϵ}^{-2p/\left(2-p\right)}\right]}\frac{1}{n}+{ϵ}^{-3}\sum _{n=\left[{ϵ}^{-2p/\left(2-p\right)}\right]+1}^{\mathrm{\infty }}{n}^{-1-\left(6-3p\right)/2p}\right)\\ =& O\left({ϵ}^{p/\left(2-p\right)}log\frac{1}{ϵ}\right).\end{array}$
(3.6)

From (2.1), (3.2), (3.4), (3.5) and (3.6), we obtain (2.2). This completes the proof of part (1).

1. (2)

By the inequality in Osipov and Petrov , there exists a bounded and decreasing function $\psi \left(u\right)$ on the interval $\left(0,\mathrm{\infty }\right)$ such that ${lim}_{u\to \mathrm{\infty }}\psi \left(u\right)=0$ and

$|P\left(\frac{1}{\sqrt{n}\sigma }{S}_{n}

Let $x={n}^{\left(2-p\right)/2p}ϵ$, we have $|{R}_{n}|\le \frac{2\psi \left(\sqrt{n}\left(1+{n}^{\left(2-p\right)/2p}ϵ\right)\right)}{{n}^{\delta /2}{\left(1+{n}^{\left(2-p\right)/2p}ϵ\right)}^{2+\delta }}$, so that:

1. (a)

If $2, then

${ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-2}{R}_{n}\le {ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-2-\delta /2}=O\left({ϵ}^{2\left(r-p\right)/\left(2-p\right)}\right).$
(3.7)
2. (b)

If $\left(1+\delta /2\right)p, then by noticing that ${lim}_{u\to \mathrm{\infty }}\psi \left(u\right)=0$ for any $\eta >0$, there exists a natural number ${N}_{0}$ such that $\psi \left(\sqrt{n}\right)<\eta$ whenever $n>{N}_{0}$. We conclude that

$\begin{array}{r}{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-2}{R}_{n}\\ \phantom{\rule{1em}{0ex}}\le C{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}\frac{2{n}^{r/p-2}\psi \left(\sqrt{n}\left(1+{n}^{\left(2-p\right)/2p}ϵ\right)\right)}{{n}^{\delta /2}{\left(1+ϵ{n}^{\left(2-p\right)/2p}\right)}^{2+\delta }}\\ \phantom{\rule{1em}{0ex}}\le C{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\left(\sum _{n=1}^{{N}_{0}}{n}^{r/p-2-\delta /2}\psi \left(\sqrt{n}\right)+\eta \sum _{n={N}_{0}+1}^{\left[{ϵ}^{-2p/\left(2-p\right)}\right]}{n}^{r/p-2-\delta /2}\right)\\ \phantom{\rule{2em}{0ex}}+C{ϵ}^{2\left(r-p\right)/\left(2-p\right)-2-\delta }\psi \left({ϵ}^{-p/\left(2-p\right)}\right)\sum _{n=\left[{ϵ}^{-2p/\left(2-p\right)}\right]+1}^{\mathrm{\infty }}{n}^{r/p-2-\delta /2-\left(1/p-1/2\right)\left(2+\delta \right)}\\ \phantom{\rule{1em}{0ex}}\le {ϵ}^{2\left(r-p\right)/\left(2-p\right)}{N}_{0}^{r/p-1-\delta /2}+C\eta {ϵ}^{p\delta /\left(2-p\right)}+C\psi \left({ϵ}^{-p/\left(2-p\right)}\right){ϵ}^{p\delta /\left(2-p\right)}\\ \phantom{\rule{1em}{0ex}}=o\left({ϵ}^{p\delta /\left(2-p\right)}\right).\end{array}$
(3.8)
3. (c)

If $r=\left(1+\delta /2\right)p$, then

${ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-2}{R}_{n}=O\left({ϵ}^{p\delta /\left(2-p\right)}log\frac{1}{ϵ}\right).$
(3.9)

By (2.1) and combining with (3.2), (3.7), (3.8) and (3.9), we obtain (2.3), which completes the proof of part (2).

1. (3)

We make use of the following large deviation estimate in Petrov :

$|P\left(\frac{1}{\sqrt{n}\sigma }{S}_{n}0.$

So, $|{R}_{n}|\le \frac{C}{\sqrt{n}{\left(1+ϵ{n}^{\left(2-p\right)/2p}\right)}^{q}}$. Hence we have the following.

1. (a)

If $r<3p/2$, then

${ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-2}{R}_{n}\le {ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-5/2}=O\left({ϵ}^{2\left(r-p\right)/\left(2-p\right)}\right).$
(3.10)
2. (b)

If $r>3p/2$, then $\frac{r}{p}-\frac{5}{2}-\frac{2q-pq}{2p}<-1$. By noting that $q>\frac{2r-3p}{2-p}$, we obtain

$\begin{array}{rcl}{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-2}{R}_{n}& \le & C{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}\frac{{n}^{r/p-2}}{{\left(1+ϵ{n}^{\left(2-p\right)/2p}\right)}^{q}\sqrt{n}}\\ \le & C{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\left[{ϵ}^{-2p/\left(2-p\right)}\right]}{n}^{r/p-2-1/2}\\ +C{ϵ}^{2\left(r-p\right)/\left(2-p\right)-q}\sum _{n=\left[{ϵ}^{-2p/\left(2-p\right)}\right]+1}^{\mathrm{\infty }}{n}^{r/p-2-1/2-\left(2-p\right)q/2p}\\ =& O\left({ϵ}^{p/\left(2-p\right)}\right).\end{array}$
(3.11)
3. (c)

If $r=3p/2$, then

$\begin{array}{rcl}{ϵ}^{2\left(r-p\right)/\left(2-p\right)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-2}{R}_{n}& \le & C{ϵ}^{p/\left(2-p\right)}\sum _{n=1}^{\left[{ϵ}^{-2p/\left(2-p\right)}\right]}\frac{1}{n}+C{ϵ}^{p/\left(2-p\right)-q}\sum _{n=\left[{ϵ}^{-2p/\left(2-p\right)}\right]+1}^{\mathrm{\infty }}{n}^{-1-\left(2-p\right)q/2p}\\ =& O\left({ϵ}^{p/\left(2-p\right)}log\frac{1}{ϵ}\right).\end{array}$
(3.12)

By (2.1), from (3.2), (3.10), (3.11) and (3.12), we have (2.4), which completes the proof of part (3). □

Proof of Theorem 2.2 We write

$\begin{array}{r}{ϵ}^{2}{\lambda }_{2,1/\left(\beta +1\right)}\left(ϵ\right)-\frac{1}{2\beta +1}\\ \phantom{\rule{1em}{0ex}}=\left(\frac{2{ϵ}^{2}}{\sqrt{2\pi }}\sum _{n=1}^{\mathrm{\infty }}{n}^{2\beta }{\int }_{ϵ{n}^{\beta +1/2}}^{\mathrm{\infty }}{e}^{-{t}^{2}/2}\phantom{\rule{0.2em}{0ex}}dt-\frac{1}{2\beta +1}\right)\\ \phantom{\rule{2em}{0ex}}+{ϵ}^{2}\left(\sum _{n=1}^{\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]}+\sum _{n=\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]+1}^{\mathrm{\infty }}\right){n}^{2\beta }\left(P\left(|{S}_{n}|\ge ϵ{n}^{\beta +1}\right)-\frac{2}{\sqrt{2\pi }}{\int }_{ϵ{n}^{\beta +1/2}}^{\mathrm{\infty }}{e}^{-{t}^{2}/2}\phantom{\rule{0.2em}{0ex}}dt\right)\\ \phantom{\rule{1em}{0ex}}=:{I}_{1}+{I}_{2}+{I}_{3}.\end{array}$
(3.13)

First, according to Lemma 2.1, we have

${I}_{1}=\left\{\begin{array}{cc}O\left({ϵ}^{2}\right),\hfill & -\frac{1}{2}<\beta <\frac{1}{2},\hfill \\ O\left({ϵ}^{4/\left(2\beta +1\right)}\right),\hfill & \beta \ge \frac{1}{2}.\hfill \end{array}$
(3.14)

For ${I}_{3}$, applying Lemma 2.3 of Xie and He , and letting $x=2y={n}^{\beta +1}ϵ$, we obtain

$P\left(|{S}_{n}|\ge {n}^{\beta +1}ϵ\right)\le nP\left(|X|\ge \frac{1}{2}{n}^{\beta +1}ϵ\right)+8{e}^{2}{ϵ}^{-4}{n}^{-4\beta -2}.$
(3.15)

Observing the following fact

$\frac{2}{\sqrt{2\pi }}{\int }_{ϵ{n}^{\beta +1/2}}^{\mathrm{\infty }}{e}^{-{t}^{2}/2}\phantom{\rule{0.2em}{0ex}}dt=2\left(1-\mathrm{\Phi }\left({n}^{\beta +\frac{1}{2}}ϵ\right)\right)\le \frac{2\phi \left({n}^{\beta +1/2}ϵ\right)}{{n}^{\beta +1/2}ϵ}=O\left({ϵ}^{-5}{n}^{-5\beta -5/2}\right),$
(3.16)

from (3.15) and (3.16), we have

$\begin{array}{rcl}{I}_{3}& \le & {ϵ}^{2}\sum _{n=\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]+1}^{\mathrm{\infty }}{n}^{2\beta }P\left(|{S}_{n}|\ge ϵ{n}^{\beta +1}\right)+{ϵ}^{2}\sum _{n=\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]+1}^{\mathrm{\infty }}\frac{2{n}^{2\beta }}{\sqrt{2\pi }}{\int }_{ϵ{n}^{\beta +1/2}}^{\mathrm{\infty }}{e}^{-{t}^{2}/2}\phantom{\rule{0.2em}{0ex}}dt\\ \le & {ϵ}^{2}\sum _{n=\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]+1}^{\mathrm{\infty }}{n}^{2\beta +1}P\left(|X|>\frac{ϵ{n}^{\beta +1}}{2}\right)+C{ϵ}^{-2}\sum _{n=\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]+1}^{\mathrm{\infty }}{n}^{-2\beta -2}\\ +C{ϵ}^{-3}\sum _{n=\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]+1}^{\mathrm{\infty }}{n}^{-3\beta -5/2}\\ \le & {ϵ}^{2}\sum _{n=\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]+1}^{\mathrm{\infty }}{n}^{2\beta +1}{\int }_{|x|\ge \frac{1}{2}{n}^{\beta +1}ϵ}\phantom{\rule{0.2em}{0ex}}dF\left(x\right)+O\left({ϵ}^{2}\right)+O\left({ϵ}^{3}\right)\\ \le & {ϵ}^{2}\sum _{n=\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]+1}^{\mathrm{\infty }}{n}^{2\beta +1}\sum _{k=n}^{\mathrm{\infty }}{\int }_{\frac{1}{2}{k}^{\beta +1}ϵ\le x<\frac{1}{2}{\left(k+1\right)}^{\beta +1}ϵ}\phantom{\rule{0.2em}{0ex}}dF\left(x\right)+O\left({ϵ}^{2}\right)\\ \le & {ϵ}^{2}\sum _{k=\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]+1}^{\mathrm{\infty }}\sum _{n=1}^{k}{n}^{2\beta +1}{\int }_{\frac{1}{2}{k}^{\beta +1}ϵ\le x<\frac{1}{2}{\left(k+1\right)}^{\beta +1}ϵ}\phantom{\rule{0.2em}{0ex}}dF\left(x\right)+O\left({ϵ}^{2}\right)\\ \le & C{ϵ}^{2}\sum _{k=\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]+1}^{\mathrm{\infty }}{k}^{2\beta +2}{\int }_{\frac{1}{2}{k}^{\beta +1}ϵ\le x<\frac{1}{2}{\left(k+1\right)}^{\beta +1}ϵ}\phantom{\rule{0.2em}{0ex}}dF\left(x\right)+O\left({ϵ}^{2}\right)\\ \le & C\sum _{k=\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]+1}^{\mathrm{\infty }}{\int }_{\frac{1}{2}{k}^{\beta +1}ϵ\le x<\frac{1}{2}{\left(k+1\right)}^{\beta +1}ϵ}{x}^{2}\phantom{\rule{0.2em}{0ex}}dF\left(x\right)+O\left({ϵ}^{2}\right)\\ \le & C{\int }_{x\ge \frac{1}{2}{ϵ}^{-\left(2\beta +3\right)/\left(2\beta +1\right)}}{x}^{2}\phantom{\rule{0.2em}{0ex}}dF\left(x\right)+O\left({ϵ}^{{2}^{}}\right)\\ =& CT\left({ϵ}^{-\left(2\beta +3\right)/\left(2\beta +1\right)}\right)+O\left({ϵ}^{2}\right).\end{array}$

Using the assumption on $T\left(t\right)$ and Lemma 2.2(1), we can obtain

${I}_{3}=\left\{\begin{array}{cc}O\left({ϵ}^{2}\right),\hfill & -\frac{1}{2}<\beta \le \frac{min\left(\delta ,1\right)}{4}-\frac{1}{2},\hfill \\ O\left({ϵ}^{1/\left(2\beta +1\right)}\right),\hfill & \beta \ge -\frac{1}{4},\delta \ge 1,\hfill \\ O\left({ϵ}^{\delta /\left(2\beta +1\right)}l\left({ϵ}^{-1/\left(2\beta +1\right)}\right)\right),\hfill & \beta \ge -\frac{1}{2}+\frac{\delta }{4},0<\delta <1.\hfill \end{array}$
(3.17)

For ${I}_{2}$, by Bikelis’s inequality (see ), we have

$\begin{array}{rcl}{I}_{2}& \le & {ϵ}^{2}\sum _{n=1}^{\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]}\frac{C{n}^{2\beta }}{{\left(1+ϵ{n}^{\beta +1/2}\right)}^{3}\sqrt{n}}{\int }_{0}^{\left(1+ϵ{n}^{\beta +1/2}\right)\sqrt{n}}T\left(v\right)\phantom{\rule{0.2em}{0ex}}dv\\ \le & {ϵ}^{2}\sum _{n=1}^{\left[{ϵ}^{-2/\left(2\beta +1\right)}\right]}{n}^{2\beta -1/2}{\int }_{0}^{\left(1+ϵ{n}^{\beta +1/2}\right)\sqrt{n}}T\left(v\right)\phantom{\rule{0.2em}{0ex}}dv\\ +{ϵ}^{-1}\sum _{n=\left[{ϵ}^{-2/\left(2\beta +1\right)}\right]+1}^{\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]}{n}^{-\beta -2}{\int }_{0}^{\left(1+ϵ{n}^{\beta +1/2}\right)\sqrt{n}}T\left(v\right)\phantom{\rule{0.2em}{0ex}}dv.\end{array}$

Now, the proof of Theorem 2.2 will be divided into the following cases.

Case 1 of $\delta >1$.

Noting that $T\left(t\right)\le E{X}^{2}=1$, let ${\delta }_{1}$ be a real number such that $1<{\delta }_{1}<\delta$, by Lemma 2.2(1), ${lim}_{t\to \mathrm{\infty }}{t}^{{\delta }_{1}-\delta }l\left(t\right)=0$. Therefore, there is a real number ${T}_{0}>0$ such that $|\frac{l\left(t\right)}{{t}^{\delta -{\delta }_{1}}}|<1$ whenever $t>{T}_{0}$. Then

${\int }_{0}^{\mathrm{\infty }}T\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\le {\int }_{0}^{1}T\left(t\right)\phantom{\rule{0.2em}{0ex}}dt+{\int }_{1}^{\mathrm{\infty }}T\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\le C+{\int }_{{T}_{0}}^{\mathrm{\infty }}\frac{1}{{t}^{{\delta }_{1}}}\phantom{\rule{0.2em}{0ex}}dt<\mathrm{\infty }.$

We have

$\begin{array}{rcl}{I}_{2}& \le & C{ϵ}^{2}\sum _{n=1}^{\left[{ϵ}^{-2/\left(2\beta +1\right)}\right]}{n}^{2\beta -1/2}+C{ϵ}^{-1}\sum _{n=\left[{ϵ}^{-2/\left(2\beta +1\right)}\right]+1}^{\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]+1}{n}^{-\beta -2}\\ =& \left\{\begin{array}{cc}O\left({ϵ}^{2}\right),\hfill & -\frac{1}{2}<\beta \le -\frac{1}{4},\hfill \\ O\left({ϵ}^{2}log\frac{1}{ϵ}\right),\hfill & \beta =-\frac{1}{4},\hfill \\ O\left({ϵ}^{1/\left(2\beta +1\right)}\right),\hfill & \beta >-\frac{1}{4}.\hfill \end{array}\end{array}$
(3.18)

From (3.13), (3.14), (3.17) and (3.18), we obtain (2.5).

Case 2 of $0<\delta <1$.

1. (a)

If $-\frac{1}{2}<\beta <-\frac{1}{2}+\frac{\delta }{4}$, then ${\sum }_{n=1}^{\mathrm{\infty }}{n}^{2\beta -1/2}<\mathrm{\infty }$ and ${\int }_{1}^{\mathrm{\infty }}{t}^{4\beta +1-\delta }l\left(t\right)\phantom{\rule{0.2em}{0ex}}dt<\mathrm{\infty }$. Making use of Lemma 2.2(2)-(3), we have

$\begin{array}{rcl}{I}_{2}& \le & C{ϵ}^{2}\sum _{n=1}^{\left[{ϵ}^{-2/\left(2\beta +1\right)}\right]}{n}^{2\beta -1/2}\left(1+{\int }_{1}^{2\sqrt{n}}T\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\right)\\ +C{ϵ}^{-1}\sum _{n=\left[{ϵ}^{-2/\left(2\beta +1\right)}\right]+1}^{\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]}{n}^{-\beta -2}\left(1+{\int }_{1}^{2ϵ{n}^{\beta +1}}T\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\right)\\ \le & C{ϵ}^{2}\sum _{n=1}^{\left[{ϵ}^{-2/\left(2\beta +1\right)}\right]}{n}^{2\beta -1/2}{\left(\sqrt{n}\right)}^{1-\delta }l\left(\sqrt{n}\right)+O\left({ϵ}^{2}\right)\\ +C{ϵ}^{1/\left(2\beta +1\right)}+C{ϵ}^{-1}\sum _{n=\left[{ϵ}^{-2/\left(2\beta +1\right)}\right]+1}^{\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]}{n}^{-\beta -2}{\left(2{n}^{\beta +1}ϵ\right)}^{1-\delta }l\left(2{n}^{\beta +1}ϵ\right)\\ \le & C{ϵ}^{2}{\int }_{1}^{{ϵ}^{-2/\left(2\beta +1\right)}}{x}^{2\beta -1/2}{\left(\sqrt{x}\right)}^{1-\delta }l\left(\sqrt{x}\right)\phantom{\rule{0.2em}{0ex}}dx\\ +C{ϵ}^{-\delta }{\int }_{{ϵ}^{-2/\left(2\beta +1\right)}}^{\mathrm{\infty }}{x}^{-\beta -2}l\left(2{x}^{\beta +1}ϵ\right){x}^{\left(\beta +1\right)\left(1-\delta \right)}\phantom{\rule{0.2em}{0ex}}dx+O\left({ϵ}^{2}\right)\\ \le & C{ϵ}^{2}{\int }_{1}^{{ϵ}^{-1/\left(2\beta +1\right)}}{t}^{4\beta +1-\delta }l\left(t\right)\phantom{\rule{0.2em}{0ex}}dt+C{\int }_{{ϵ}^{-1/\left(2\beta +1\right)}}^{\mathrm{\infty }}\frac{l\left(t\right)}{{t}^{1+\delta }}\phantom{\rule{0.2em}{0ex}}dt+O\left({ϵ}^{2}\right)\\ \le & C{ϵ}^{2}+C{ϵ}^{\delta /\left(2\beta +1\right)}l\left({ϵ}^{-1/\left(2\beta +1\right)}\right)\\ =& O\left({ϵ}^{2}\right).\end{array}$
(3.19)
2. (b)

If $\beta \ge -\frac{1}{2}+\frac{\delta }{4}$, then we have

$\begin{array}{rcl}{I}_{2}& \le & C{ϵ}^{2}{ϵ}^{-\left(4\beta +1\right)/\left(2\beta +1\right)}\left(1+{\int }_{1}^{2{ϵ}^{-1/\left(2\beta +1\right)}}T\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\right)+C{ϵ}^{1/\left(2\beta +1\right)}+C{ϵ}^{\delta /\left(2\beta +1\right)}l\left({ϵ}^{-1/\left(2\beta +1\right)}\right)\\ \le & C{ϵ}^{1/\left(2\beta +1\right)}\left(1+\left(2{ϵ}^{-\left(1-\delta \right)/\left(2\beta +1\right)}l\left({ϵ}^{-1/\left(2\beta +1\right)}\right)\right)+C{ϵ}^{\delta /\left(2\beta +1\right)}l\left({ϵ}^{-1/\left(2\beta +1\right)}\right)\\ \le & C{ϵ}^{\delta /\left(2\beta +1\right)}l\left({ϵ}^{-1/\left(2\beta +1\right)}\right).\end{array}$
(3.20)

Therefore

${I}_{2}=\left\{\begin{array}{cc}O\left({ϵ}^{2}\right),\hfill & -\frac{1}{2}<\beta \le -\frac{1}{2}+\frac{\delta }{4},\hfill \\ O\left({ϵ}^{\delta /\left(2\beta +1\right)}l\left({ϵ}^{1/\left(2\beta +1\right)}\right)\right),\hfill & \beta \ge -\frac{1}{2}+\frac{\delta }{4}.\hfill \end{array}$
(3.21)

Combining the estimate with (3.11) and (3.14), by (3.10), this implies that (2.6) follows.

Case 3 of $\delta =1$.

1. (a)

If $-\frac{1}{2}<\beta <-\frac{1}{4}$, then ${\sum }_{n=1}^{\mathrm{\infty }}{n}^{2\beta -\frac{1}{2}}<\mathrm{\infty }$. We have

$\begin{array}{rcl}{I}_{2}& \le & C{ϵ}^{2}\left(1+{\int }_{1}^{{ϵ}^{-1/\left(2\beta +1\right)}}T\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\right)\sum _{n=1}^{\left[{ϵ}^{-2/\left(2\beta +1\right)}\right]}{n}^{2\beta -1/2}\\ +C{ϵ}^{-1}\left(1+{\int }_{1}^{{ϵ}^{-\frac{2\beta +3}{2\beta +1}}}T\left(t\right)\phantom{\rule{0.2em}{0ex}}dt\right)\sum _{n=\left[{ϵ}^{-2/\left(2\beta +1\right)}\right]+1}^{\left[{ϵ}^{-4/\left(2\beta +1\right)}\right]}{n}^{-\beta -2}\\ \le & C{ϵ}^{2}\left(1+{\int }_{1}^{-1/\left(2\beta +1\right)}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right)+{ϵ}^{1/\left(2\beta +1\right)}\left(1+{\int }_{1}^{{ϵ}^{-\frac{2\beta +3}{2\beta +1}}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right)\\ \le & C{ϵ}^{2}\left(1+{\int }_{1}^{{ϵ}^{-1/\left(2\beta +1\right)}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right).\end{array}$
(3.22)
2. (b)

If $\beta >-\frac{1}{4}$, then we have

$\begin{array}{rcl}{I}_{2}& \le & C{ϵ}^{2}{ϵ}^{\frac{-2}{2\beta +1}\left(2\beta +1/2\right)}\left(1+{\int }_{1}^{{ϵ}^{-1/\left(2\beta +1\right)}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right)+C{ϵ}^{1/\left(2\beta +1\right)}\left(1+{\int }_{1}^{{ϵ}^{-\left(2\beta +3\right)/\left(2\beta +1\right)}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right)\\ \le & C{ϵ}^{1/\left(2\beta +1\right)}\left(1+{\int }_{1}^{-1/\left(2\beta +1\right)}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right)+{ϵ}^{1/\left(2\beta +1\right)}\left(1+{\int }_{1}^{{ϵ}^{-\left(2\beta +3\right)/\left(2\beta +1\right)}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right)\\ \le & C{ϵ}^{1/\left(2\beta +1\right)}\left(1+{\int }_{1}^{{ϵ}^{-\left(2\beta +3\right)/\left(2\beta +1\right)}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right).\end{array}$
(3.23)
3. (c)

If $\beta =-\frac{1}{4}$, then we have

$\begin{array}{rcl}{I}_{2}& \le & C{ϵ}^{2}log\frac{1}{ϵ}\left(1+{\int }_{1}^{{ϵ}^{-2}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right)+C{ϵ}^{2}\left(1+{\int }_{1}^{{ϵ}^{-5}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right)\\ \le & C{ϵ}^{2}log\frac{1}{ϵ}\left(1+{\int }_{1}^{{ϵ}^{-5}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right)\end{array}$

so that

${I}_{2}=\left\{\begin{array}{cc}O\left({ϵ}^{2}\left(1+{\int }_{1}^{{ϵ}^{-1/\left(2\beta +1\right)}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right)\right),\hfill & -\frac{1}{2}<\beta \le -\frac{1}{4},\hfill \\ O\left({ϵ}^{2}\left(1+{\int }_{1}^{{ϵ}^{-5}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right)log\frac{1}{ϵ}\right),\hfill & \beta =-\frac{1}{4},\hfill \\ O\left({ϵ}^{1/\left(2\beta +1\right)}\left(1+{\int }_{1}^{{ϵ}^{-\left(2\beta +3\right)/\left(2\beta +1\right)}}\frac{l\left(t\right)}{t}\phantom{\rule{0.2em}{0ex}}dt\right)\right),\hfill & \beta >-\frac{1}{4}.\hfill \end{array}$
(3.24)

Combining the estimate with (3.14) and (3.17), by (3.13), this implies that (2.7) follows, and hence Theorem 2.2 is proved. □

Proof of Theorem 2.3 Set ${T}_{1}\left(t\right)=E{|X|}^{2+\delta }I\left(|X|>t\right)$. First, note that

$\begin{array}{rcl}E{|X|}^{2+\delta }I\left(|X|>t\right)& =& {\int }_{|x|>t}{|x|}^{2+\delta }\phantom{\rule{0.2em}{0ex}}dF\left(x\right)\\ =& {\int }_{|x|>t}{x}^{2}\left({\int }_{t}^{|x|}\delta {y}^{\delta -1}\phantom{\rule{0.2em}{0ex}}dy\right)\phantom{\rule{0.2em}{0ex}}dF\left(x\right)+{t}^{\delta }{\int }_{|x|>t}{x}^{2}\phantom{\rule{0.2em}{0ex}}dF\left(x\right)\\ =& {\int }_{t}^{\mathrm{\infty }}\delta {y}^{\delta -1}\left({\int }_{|x|>y}{x}^{2}\phantom{\rule{0.2em}{0ex}}dF\left(x\right)\right)\phantom{\rule{0.2em}{0ex}}dy+{t}^{\delta }T\left(t\right)\\ =& \delta {\int }_{t}^{\mathrm{\infty }}{s}^{\delta -1}T\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{t}^{\delta }T\left(t\right).\end{array}$

We have

${T}_{1}\left(t\right)=\delta {\int }_{t}^{\mathrm{\infty }}{s}^{\delta -1}T\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{t}^{\delta }T\left(t\right).$

Since ${\int }_{t}^{\mathrm{\infty }}{s}^{\delta -1}T\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\ge 0$, ${t}^{\delta }T\left(t\right)\ge 0$, we have

Next, it is easy to get

From the above facts, the proof of Theorem 2.3 is complete. □

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He, J. A note to the convergence rates in precise asymptotics. J Inequal Appl 2013, 378 (2013). https://doi.org/10.1186/1029-242X-2013-378 