A note to the convergence rates in precise asymptotics

Abstract

Let $\{X,X_n,n\ge 1\}$ 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 ${\lambda }_{r,p}(ϵ)={\sum }_{n=1}^{\mathrm{\infty }}{n}^{r/p-2}P(|S_n|\ge n^{1/p}ϵ)$. In this paper, the author discusses the rate of approximation of $\frac{p}{r-p}E{|N|}^{2(r-p)/(2-p)}$ by ${ϵ}^{2(r-p)/(2-p)}{\lambda }_{r,p}(ϵ)$ 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 ${ϵ}^2{\lambda }_{2,1/(\beta +1)}(ϵ)$ under the condition $E{X}^{2}I(|X|>t)=O(t^{-\delta }l(t))$ for some $\delta >0$, where $l(t)$ is a slowly varying function at infinity.

MSC:60F15, 60G50.

1 Introduction

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

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

whenever $EX=0$ and $E{X}^2={\sigma }^2<\mathrm{\infty }$. Klesov [2] studied the rate of the approximation of ${\sigma }^2$ by ${ϵ}^2{\lambda }_{2,1}(ϵ)$ under the condition $E{|X|}^3<\mathrm{\infty }$. He and Xie [3] improved the results of Klesov [2]. Gut and Steinebach [4] extended the results of Klesov [2] and obtained the following Theorem A. Gut and Steinebach [5] studied the general idea of proving precise asymptotics.

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

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

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

• (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

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

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 [3] under suitable moment conditions. In addition, we shall discuss the rate at which ${ϵ}^2{\lambda }_{2,1/(\beta +1)}(ϵ)$ converges to $\frac{{\sigma }^2}{2\beta +1}$ under the condition $T(t)=O(t^{-\delta }l(t))$ for some $\delta >0$, where $T(t)=E{X}^{2}I(|X|>t)$, $l(t)$ 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 $[x]$ denotes the integer part of x. $\mathrm{\Phi }(x)$ is the standard normal distribution function, $\mathrm{\Phi }(x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^x{e}^{-t^2/2}dt$, $\phi (x)={\mathrm{\Phi }}^{\mathrm{\prime }}(x)$.

2 Main results

From Gut and Steinebach [6], it is easy to obtain the following lemma.

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

$$\begin{array}{r}{ϵ}^{2(r-p)/(2-p)}\sum _{n=1}^{\mathrm{\infty }}{n}^{(r/p)-2}P(|S_n|\ge n^{1/p}ϵ)-\frac{p}{r-p}E{|N|}^{2(r-p)/(2-p)}\\ =\{\begin{array}{cc}O({ϵ}^{2(r-p)/(2-p)}),\hfill & r<3p,\hfill \\ O({ϵ}^{4p/(2-p)}),\hfill & r\ge 3p.\hfill \end{array}\end{array}$$

(2.1)

Lemma 2.2 (Bingham et al. [7])

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

• (1) for any $\eta >0$,

  $$\underset{t\to \mathrm{\infty }}{lim}{t}^{\eta }l(t)=\mathrm{\infty },\underset{t\to \mathrm{\infty }}{lim}{t}^{-\eta }l(t)=0;$$

• (2) if $0<\delta <1$, then

  $${\int }_a^t{s}^{-\delta }l(s)ds\sim \frac{1}{1-\delta }{t}^{1-\delta }l(t),t\to \mathrm{\infty };$$

• (3) if $\delta >1$, then

  $${\int }_t^{\mathrm{\infty }}{s}^{-\delta }l(s)ds\sim -\frac{1}{1-\delta }{t}^{1-\delta }l(t),t\to \mathrm{\infty };$$

• (4) if $\delta =1$, then $L(t)={\int }_t^{\mathrm{\infty }}\frac{l(s)}{s}ds$, $m(t)={\int }_a^t\frac{l(s)}{s}ds$ are slowly varying functions; and

  $$\underset{t\to \mathrm{\infty }}{lim}\frac{l(t)}{L(t)}=0,\underset{t\to \mathrm{\infty }}{lim}\frac{l(t)}{m(t)}=0.$$

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

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

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

  (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

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

  (2.3)

• (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

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

  (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 [3] are special cases of Theorem 2.1, by taking $r=2$ and $p=1$.

Remark 2.2 If $0<p<2$, $r\ge 2$, we have $min(\frac{2(r-p)}{2-p},\frac{p\delta }{2-p})>\frac{p\delta (r-p)}{(2+\delta -p)(2-p)}$ for $r<2+\delta =q\le 3$ and $min(\frac{2(r-p)}{2-p},\frac{p}{2-p})>\frac{2(r-p)p}{(2-p)(p+2q-pq)}$ 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 $\{X,X_n;n\ge 1\}$ be a sequence of i.i.d random variables with zero mean, and let $T(t)=O(t^{-\delta }l(t))$ for some $\delta >0$, where $l(t)$ is a slowly varying function at infinity. Set $E{X}^2={\sigma }^2>0$ and $\beta >-\frac{1}{2}$.

• (1) If $\delta >1$, then

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

  (2.5)

• (2) If $0<\delta <1$, then

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

  (2.6)

• (3) If $\delta =1$, then

  $${ϵ}^2{\lambda }_{2,1/(\beta +1)}(ϵ)-\frac{{\sigma }^2}{2\beta +1}=\{\begin{array}{cc}O({ϵ}^2+{ϵ}^2{\int }_1^{{ϵ}^{-1/(2\beta +1)}}\frac{l(t)}{t}dt),\hfill & -\frac{1}{2}<\beta <-\frac{1}{4},\hfill \\ O({ϵ}^2(1+{\int }_1^{{ϵ}^{-5}}\frac{l(t)}{t}dt)log\frac{1}{ϵ}),\hfill & \beta =-\frac{1}{4},\hfill \\ O({ϵ}^{1/(2\beta +1)}(1+{\int }_1^{{ϵ}^{-(2\beta +3)/(2\beta +1)}}\frac{l(t)}{t}dt)),\hfill & \beta >-\frac{1}{4}.\hfill \end{array}$$

  (2.7)

Remark 2.3 For $r=2$, $p=\frac{1}{\beta +1}$. If $l(t)=1$, 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 $T(t)=O(t^{-\delta })$ is weaker than the condition $E{|X|}^{2+\delta }<\mathrm{\infty }$. If $l(t)\to 0$ 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 $T(t)=O(t^{-\delta }l(t))$. Here are some suitable examples.

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

Example 2 Let X be a random variable with density $f(x)=\frac{C(\delta {ln}^2|x|+|x|(ln|x|-1))}{{|x|}^{\delta +3}{ln}^2|x|e^{|x|/ln|x|}}I(|x|>e)$, where $0<\delta <1$, then $EX=0$ and $T(t)=\frac{C}{t^{\delta }{e}^{t/ln(t)}}I(t>e)$, $h(t)=\frac{1}{e^{t/lnt}}$, $E{|X|}^{2+\delta }<\mathrm{\infty }$. But $h(t)=\frac{1}{e^{t/lnt}}$ 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 $t^{\delta }T(t)\to 0$ and ${\int }_t^{\mathrm{\infty }}{s}^{\delta -1}T(s)ds\to 0$ as $t\to \mathrm{\infty }$.

Remark 2.5 If $t^{\delta }T(t)$ is bounded as $t\to \mathrm{\infty }$ for some $\delta >0$, then we have $E{|X|}^{2+\alpha }<\mathrm{\infty }$ for every $\alpha \in (0,\delta )$ 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 $C_{1}l(t)\le t^{\delta }T(t)\le C_{2}l(t)$ for sufficiently large t and some $\delta >0$, where $l(t)$ is a slowly varying function at infinity, then from Lemma 2.2(4) and Theorem 2.3, we have

$$E{|X|}^{2+\delta }<\mathrm{\infty }\iff {\int }_t^{\mathrm{\infty }}\frac{l(s)}{s}ds\to 0\text{as }t\to \mathrm{\infty }.$$

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

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

(3.1)

where

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

From (3.1), we have

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

(3.2)

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

• (1) On account of a non-uniform estimate of the central limit theorem by Nagaev [8], for every $x\in R$,

  $$|P(\frac{S_n}{\sqrt{n}}<x)-\mathrm{\Phi }(x)|\le \frac{CE{|X|}^3}{\sqrt{n}{(1+|x|)}^3}.$$

  (3.3)

By (3.3), $|R_n|\le \frac{CE{|X|}^3}{\sqrt{n}{(1+ϵn^{(2-p)/2p})}^3}$.

• (a) If $r<3p/2$, then

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

  (3.4)

• (b) If $3p/2<r<3$, then

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

  (3.5)

• (c) If $r=3p/2$, then

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

  (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).

• (2) By the inequality in Osipov and Petrov [9], there exists a bounded and decreasing function $\psi (u)$ on the interval $(0,\mathrm{\infty })$ such that ${lim}_{u\to \mathrm{\infty }}\psi (u)=0$ and

  $$|P(\frac{1}{\sqrt{n}\sigma }{S}_n<x)-\mathrm{\Phi }(x)|\le \frac{\psi (\sqrt{n}(1+|x|))}{n^{\delta /2}{(1+|x|)}^{2+\delta }}.$$

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

• (a) If $2<r<(1+\delta /2)p$, then

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

  (3.7)

• (b) If $(1+\delta /2)p<r<2+\delta$, then by noticing that ${lim}_{u\to \mathrm{\infty }}\psi (u)=0$ for any $\eta >0$, there exists a natural number $N_0$ such that $\psi (\sqrt{n})<\eta$ whenever $n>N_0$. We conclude that

  $$\begin{array}{r}{ϵ}^{2(r-p)/(2-p)}\sum _{n=1}^{\mathrm{\infty }}{n}^{r/p-2}{R}_n\\ \le C{ϵ}^{2(r-p)/(2-p)}\sum _{n=1}^{\mathrm{\infty }}\frac{2n^{r/p-2}\psi (\sqrt{n}(1+n^{(2-p)/2p}ϵ))}{n^{\delta /2}{(1+ϵn^{(2-p)/2p})}^{2+\delta }}\\ \le C{ϵ}^{2(r-p)/(2-p)}(\sum _{n=1}^{N_0}{n}^{r/p-2-\delta /2}\psi (\sqrt{n})+\eta \sum _{n=N_0+1}^{[{ϵ}^{-2p/(2-p)}]}{n}^{r/p-2-\delta /2})\\ +C{ϵ}^{2(r-p)/(2-p)-2-\delta }\psi \left({ϵ}^{-p/(2-p)}\right)\sum _{n=[{ϵ}^{-2p/(2-p)}]+1}^{\mathrm{\infty }}{n}^{r/p-2-\delta /2-(1/p-1/2)(2+\delta )}\\ \le {ϵ}^{2(r-p)/(2-p)}{N}_0^{r/p-1-\delta /2}+C\eta {ϵ}^{p\delta /(2-p)}+C\psi \left({ϵ}^{-p/(2-p)}\right){ϵ}^{p\delta /(2-p)}\\ =o\left({ϵ}^{p\delta /(2-p)}\right).\end{array}$$

  (3.8)

• (c) If $r=(1+\delta /2)p$, then

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

  (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).

• (3) We make use of the following large deviation estimate in Petrov [10]:

  $$|P(\frac{1}{\sqrt{n}\sigma }{S}_n<x)-\mathrm{\Phi }(x)|\le \frac{C}{\sqrt{n}{(1+|x|)}^q},x>0.$$

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

• (a) If $r<3p/2$, then

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

  (3.10)

• (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

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

  (3.11)

• (c) If $r=3p/2$, then

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

  (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

$$\begin{array}{r}{ϵ}^2{\lambda }_{2,1/(\beta +1)}(ϵ)-\frac{1}{2\beta +1}\\ =(\frac{2{ϵ}^2}{\sqrt{2\pi }}\sum _{n=1}^{\mathrm{\infty }}{n}^{2\beta }{\int }_{ϵn^{\beta +1/2}}^{\mathrm{\infty }}{e}^{-t^2/2}dt-\frac{1}{2\beta +1})\\ +{ϵ}^2(\sum _{n=1}^{[{ϵ}^{-4/(2\beta +1)}]}+\sum _{n=[{ϵ}^{-4/(2\beta +1)}]+1}^{\mathrm{\infty }})n^{2\beta }(P(|S_n|\ge ϵn^{\beta +1})-\frac{2}{\sqrt{2\pi }}{\int }_{ϵn^{\beta +1/2}}^{\mathrm{\infty }}{e}^{-t^2/2}dt)\\ =:I_1+I_2+I_3.\end{array}$$

(3.13)

First, according to Lemma 2.1, we have

$$I_1=\{\begin{array}{cc}O({ϵ}^2),\hfill & -\frac{1}{2}<\beta <\frac{1}{2},\hfill \\ O({ϵ}^{4/(2\beta +1)}),\hfill & \beta \ge \frac{1}{2}.\hfill \end{array}$$

(3.14)

For $I_3$, applying Lemma 2.3 of Xie and He [11], and letting $x=2y=n^{\beta +1}ϵ$, we obtain

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

(3.15)

Observing the following fact

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

(3.16)

from (3.15) and (3.16), we have

$$\begin{array}{rcl}{I}_3& \le & {ϵ}^2\sum _{n=[{ϵ}^{-4/(2\beta +1)}]+1}^{\mathrm{\infty }}{n}^{2\beta }P(|S_n|\ge ϵn^{\beta +1})+{ϵ}^2\sum _{n=[{ϵ}^{-4/(2\beta +1)}]+1}^{\mathrm{\infty }}\frac{2n^{2\beta }}{\sqrt{2\pi }}{\int }_{ϵn^{\beta +1/2}}^{\mathrm{\infty }}{e}^{-t^2/2}dt\\ \le & {ϵ}^2\sum _{n=[{ϵ}^{-4/(2\beta +1)}]+1}^{\mathrm{\infty }}{n}^{2\beta +1}P(|X|>\frac{ϵn^{\beta +1}}{2})+C{ϵ}^{-2}\sum _{n=[{ϵ}^{-4/(2\beta +1)}]+1}^{\mathrm{\infty }}{n}^{-2\beta -2}\\ +C{ϵ}^{-3}\sum _{n=[{ϵ}^{-4/(2\beta +1)}]+1}^{\mathrm{\infty }}{n}^{-3\beta -5/2}\\ \le & {ϵ}^2\sum _{n=[{ϵ}^{-4/(2\beta +1)}]+1}^{\mathrm{\infty }}{n}^{2\beta +1}{\int }_{|x|\ge \frac{1}{2}{n}^{\beta +1}ϵ}dF(x)+O\left({ϵ}^2\right)+O\left({ϵ}^3\right)\\ \le & {ϵ}^2\sum _{n=[{ϵ}^{-4/(2\beta +1)}]+1}^{\mathrm{\infty }}{n}^{2\beta +1}\sum _{k=n}^{\mathrm{\infty }}{\int }_{\frac{1}{2}{k}^{\beta +1}ϵ\le x<\frac{1}{2}{(k+1)}^{\beta +1}ϵ}dF(x)+O\left({ϵ}^2\right)\\ \le & {ϵ}^2\sum _{k=[{ϵ}^{-4/(2\beta +1)}]+1}^{\mathrm{\infty }}\sum _{n=1}^k{n}^{2\beta +1}{\int }_{\frac{1}{2}{k}^{\beta +1}ϵ\le x<\frac{1}{2}{(k+1)}^{\beta +1}ϵ}dF(x)+O\left({ϵ}^2\right)\\ \le & C{ϵ}^2\sum _{k=[{ϵ}^{-4/(2\beta +1)}]+1}^{\mathrm{\infty }}{k}^{2\beta +2}{\int }_{\frac{1}{2}{k}^{\beta +1}ϵ\le x<\frac{1}{2}{(k+1)}^{\beta +1}ϵ}dF(x)+O\left({ϵ}^2\right)\\ \le & C\sum _{k=[{ϵ}^{-4/(2\beta +1)}]+1}^{\mathrm{\infty }}{\int }_{\frac{1}{2}{k}^{\beta +1}ϵ\le x<\frac{1}{2}{(k+1)}^{\beta +1}ϵ}{x}^{2}dF(x)+O\left({ϵ}^2\right)\\ \le & C{\int }_{x\ge \frac{1}{2}{ϵ}^{-(2\beta +3)/(2\beta +1)}}{x}^{2}dF(x)+O\left({ϵ}^{2^{}}\right)\\ =& CT\left({ϵ}^{-(2\beta +3)/(2\beta +1)}\right)+O\left({ϵ}^2\right).\end{array}$$

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

$$I_3=\{\begin{array}{cc}O({ϵ}^2),\hfill & -\frac{1}{2}<\beta \le \frac{min(\delta ,1)}{4}-\frac{1}{2},\hfill \\ O({ϵ}^{1/(2\beta +1)}),\hfill & \beta \ge -\frac{1}{4},\delta \ge 1,\hfill \\ O({ϵ}^{\delta /(2\beta +1)}l({ϵ}^{-1/(2\beta +1)})),\hfill & \beta \ge -\frac{1}{2}+\frac{\delta }{4},0<\delta <1.\hfill \end{array}$$

(3.17)

For $I_2$, by Bikelis’s inequality (see [12]), we have

$$\begin{array}{rcl}{I}_2& \le & {ϵ}^2\sum _{n=1}^{[{ϵ}^{-4/(2\beta +1)}]}\frac{C{n}^{2\beta }}{{(1+ϵn^{\beta +1/2})}^3\sqrt{n}}{\int }_0^{(1+ϵn^{\beta +1/2})\sqrt{n}}T(v)dv\\ \le & {ϵ}^2\sum _{n=1}^{[{ϵ}^{-2/(2\beta +1)}]}{n}^{2\beta -1/2}{\int }_0^{(1+ϵn^{\beta +1/2})\sqrt{n}}T(v)dv\\ +{ϵ}^{-1}\sum _{n=[{ϵ}^{-2/(2\beta +1)}]+1}^{[{ϵ}^{-4/(2\beta +1)}]}{n}^{-\beta -2}{\int }_0^{(1+ϵn^{\beta +1/2})\sqrt{n}}T(v)dv.\end{array}$$

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

Case 1 of $\delta >1$.

Noting that $T(t)\le E{X}^2=1$, let ${\delta }_1$ be a real number such that $1<{\delta }_1<\delta$, by Lemma 2.2(1), ${lim}_{t\to \mathrm{\infty }}{t}^{{\delta }_1-\delta }l(t)=0$. Therefore, there is a real number $T_0>0$ such that $|\frac{l(t)}{t^{\delta -{\delta }_1}}|<1$ whenever $t>T_0$. Then

$${\int }_0^{\mathrm{\infty }}T(t)dt\le {\int }_0^{1}T(t)dt+{\int }_1^{\mathrm{\infty }}T(t)dt\le C+{\int }_{T_0}^{\mathrm{\infty }}\frac{1}{t^{{\delta }_1}}dt<\mathrm{\infty }.$$

We have

$$\begin{array}{rcl}{I}_2& \le & C{ϵ}^2\sum _{n=1}^{[{ϵ}^{-2/(2\beta +1)}]}{n}^{2\beta -1/2}+C{ϵ}^{-1}\sum _{n=[{ϵ}^{-2/(2\beta +1)}]+1}^{[{ϵ}^{-4/(2\beta +1)}]+1}{n}^{-\beta -2}\\ =& \{\begin{array}{cc}O({ϵ}^2),\hfill & -\frac{1}{2}<\beta \le -\frac{1}{4},\hfill \\ O({ϵ}^{2}log\frac{1}{ϵ}),\hfill & \beta =-\frac{1}{4},\hfill \\ O({ϵ}^{1/(2\beta +1)}),\hfill & \beta >-\frac{1}{4}.\hfill \end{array}\end{array}$$

(3.18)

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

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

• (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 ${\int }_1^{\mathrm{\infty }}{t}^{4\beta +1-\delta }l(t)dt<\mathrm{\infty }$. Making use of Lemma 2.2(2)-(3), we have

  $$\begin{array}{rcl}{I}_2& \le & C{ϵ}^2\sum _{n=1}^{[{ϵ}^{-2/(2\beta +1)}]}{n}^{2\beta -1/2}(1+{\int }_1^{2\sqrt{n}}T(t)dt)\\ +C{ϵ}^{-1}\sum _{n=[{ϵ}^{-2/(2\beta +1)}]+1}^{[{ϵ}^{-4/(2\beta +1)}]}{n}^{-\beta -2}(1+{\int }_1^{2ϵn^{\beta +1}}T(t)dt)\\ \le & C{ϵ}^2\sum _{n=1}^{[{ϵ}^{-2/(2\beta +1)}]}{n}^{2\beta -1/2}{(\sqrt{n})}^{1-\delta }l(\sqrt{n})+O\left({ϵ}^2\right)\\ +C{ϵ}^{1/(2\beta +1)}+C{ϵ}^{-1}\sum _{n=[{ϵ}^{-2/(2\beta +1)}]+1}^{[{ϵ}^{-4/(2\beta +1)}]}{n}^{-\beta -2}{\left(2n^{\beta +1}ϵ\right)}^{1-\delta }l\left(2n^{\beta +1}ϵ\right)\\ \le & C{ϵ}^2{\int }_1^{{ϵ}^{-2/(2\beta +1)}}{x}^{2\beta -1/2}{(\sqrt{x})}^{1-\delta }l(\sqrt{x})dx\\ +C{ϵ}^{-\delta }{\int }_{{ϵ}^{-2/(2\beta +1)}}^{\mathrm{\infty }}{x}^{-\beta -2}l\left(2x^{\beta +1}ϵ\right)x^{(\beta +1)(1-\delta )}dx+O\left({ϵ}^2\right)\\ \le & C{ϵ}^2{\int }_1^{{ϵ}^{-1/(2\beta +1)}}{t}^{4\beta +1-\delta }l(t)dt+C{\int }_{{ϵ}^{-1/(2\beta +1)}}^{\mathrm{\infty }}\frac{l(t)}{t^{1+\delta }}dt+O\left({ϵ}^2\right)\\ \le & C{ϵ}^2+C{ϵ}^{\delta /(2\beta +1)}l\left({ϵ}^{-1/(2\beta +1)}\right)\\ =& O\left({ϵ}^2\right).\end{array}$$

  (3.19)

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

  $$\begin{array}{rcl}{I}_2& \le & C{ϵ}^2{ϵ}^{-(4\beta +1)/(2\beta +1)}(1+{\int }_1^{2{ϵ}^{-1/(2\beta +1)}}T(t)dt)+C{ϵ}^{1/(2\beta +1)}+C{ϵ}^{\delta /(2\beta +1)}l\left({ϵ}^{-1/(2\beta +1)}\right)\\ \le & C{ϵ}^{1/(2\beta +1)}(1+\left(2{ϵ}^{-(1-\delta )/(2\beta +1)}l\left({ϵ}^{-1/(2\beta +1)}\right)\right)+C{ϵ}^{\delta /(2\beta +1)}l\left({ϵ}^{-1/(2\beta +1)}\right)\\ \le & C{ϵ}^{\delta /(2\beta +1)}l\left({ϵ}^{-1/(2\beta +1)}\right).\end{array}$$

  (3.20)

Therefore

$$I_2=\{\begin{array}{cc}O({ϵ}^2),\hfill & -\frac{1}{2}<\beta \le -\frac{1}{2}+\frac{\delta }{4},\hfill \\ O({ϵ}^{\delta /(2\beta +1)}l({ϵ}^{1/(2\beta +1)})),\hfill & \beta \ge -\frac{1}{2}+\frac{\delta }{4}.\hfill \end{array}$$

(3.21)

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

Case 3 of $\delta =1$.

• (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

  $$\begin{array}{rcl}{I}_2& \le & C{ϵ}^2(1+{\int }_1^{{ϵ}^{-1/(2\beta +1)}}T(t)dt)\sum _{n=1}^{[{ϵ}^{-2/(2\beta +1)}]}{n}^{2\beta -1/2}\\ +C{ϵ}^{-1}(1+{\int }_1^{{ϵ}^{-\frac{2\beta +3}{2\beta +1}}}T(t)dt)\sum _{n=[{ϵ}^{-2/(2\beta +1)}]+1}^{[{ϵ}^{-4/(2\beta +1)}]}{n}^{-\beta -2}\\ \le & C{ϵ}^2(1+{\int }_1^{-1/(2\beta +1)}\frac{l(t)}{t}dt)+{ϵ}^{1/(2\beta +1)}(1+{\int }_1^{{ϵ}^{-\frac{2\beta +3}{2\beta +1}}}\frac{l(t)}{t}dt)\\ \le & C{ϵ}^2(1+{\int }_1^{{ϵ}^{-1/(2\beta +1)}}\frac{l(t)}{t}dt).\end{array}$$

  (3.22)

• (b) If $\beta >-\frac{1}{4}$, then we have

  $$\begin{array}{rcl}{I}_2& \le & C{ϵ}^2{ϵ}^{\frac{-2}{2\beta +1}(2\beta +1/2)}(1+{\int }_1^{{ϵ}^{-1/(2\beta +1)}}\frac{l(t)}{t}dt)+C{ϵ}^{1/(2\beta +1)}(1+{\int }_1^{{ϵ}^{-(2\beta +3)/(2\beta +1)}}\frac{l(t)}{t}dt)\\ \le & C{ϵ}^{1/(2\beta +1)}(1+{\int }_1^{-1/(2\beta +1)}\frac{l(t)}{t}dt)+{ϵ}^{1/(2\beta +1)}(1+{\int }_1^{{ϵ}^{-(2\beta +3)/(2\beta +1)}}\frac{l(t)}{t}dt)\\ \le & C{ϵ}^{1/(2\beta +1)}(1+{\int }_1^{{ϵ}^{-(2\beta +3)/(2\beta +1)}}\frac{l(t)}{t}dt).\end{array}$$

  (3.23)

• (c) If $\beta =-\frac{1}{4}$, then we have

  $$\begin{array}{rcl}{I}_2& \le & C{ϵ}^{2}log\frac{1}{ϵ}(1+{\int }_1^{{ϵ}^{-2}}\frac{l(t)}{t}dt)+C{ϵ}^2(1+{\int }_1^{{ϵ}^{-5}}\frac{l(t)}{t}dt)\\ \le & C{ϵ}^{2}log\frac{1}{ϵ}(1+{\int }_1^{{ϵ}^{-5}}\frac{l(t)}{t}dt)\end{array}$$

so that

$$I_2=\{\begin{array}{cc}O({ϵ}^2(1+{\int }_1^{{ϵ}^{-1/(2\beta +1)}}\frac{l(t)}{t}dt)),\hfill & -\frac{1}{2}<\beta \le -\frac{1}{4},\hfill \\ O({ϵ}^2(1+{\int }_1^{{ϵ}^{-5}}\frac{l(t)}{t}dt)log\frac{1}{ϵ}),\hfill & \beta =-\frac{1}{4},\hfill \\ O({ϵ}^{1/(2\beta +1)}(1+{\int }_1^{{ϵ}^{-(2\beta +3)/(2\beta +1)}}\frac{l(t)}{t}dt)),\hfill & \beta >-\frac{1}{4}.\hfill \end{array}$$

(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 $T_1(t)=E{|X|}^{2+\delta }I(|X|>t)$. First, note that

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

We have

$$T_1(t)=\delta {\int }_t^{\mathrm{\infty }}{s}^{\delta -1}T(s)ds+t^{\delta }T(t).$$

Since ${\int }_t^{\mathrm{\infty }}{s}^{\delta -1}T(s)ds\ge 0$, $t^{\delta }T(t)\ge 0$, we have

$$T_1(t)\to 0\iff t^{\delta }T(t)\to 0\text{and}{\int }_t^{\mathrm{\infty }}{s}^{\delta -1}T(s)ds\to 0\text{as }t\to \mathrm{\infty }.$$

Next, it is easy to get

$$E{|X|}^{2+\delta }<\mathrm{\infty }\iff T_1(t)\to 0\text{as }t\to \mathrm{\infty }.$$

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