A note on the convergence rates in precise asymptotics

Abstract

Let $\{X, X_{n}, n\geq1\}$ be a sequence of i.i.d. random variables with $EX=0$, $EX^{2}=\sigma^{2}$. Set $S_{n}=\sum_{k=1}^{n}X_{k}$ and let ${\mathcal {N} }$ be the standard normal random variable. Let $g(x)$ be a positive and twice differentiable function on $[n_{0},\infty)$ such that $g(x)\nearrow\infty$, $g'(x)\searrow0$ as $x\to\infty$. In this short note, under some suitable conditions on both X and $g(x)$, we establish the following convergence rates in precise asymptotics

$$ \lim_{\varepsilon\searrow0}\Biggl[ \sum_{n=n_{0}}^{\infty} g'(n)P\biggl\{ \frac{|S_{n}|}{\sigma\sqrt{n}}\geq \varepsilon g^{s}(n) \biggr\} -{\varepsilon}^{-1/s}E|{\mathcal {N}}|^{1/s}\Biggr]=\gamma -\eta, $$

where $\gamma=\lim_{n\to\infty}(\sum_{k=n_{0}}^{n}g'(k)-g(n))$, $\eta=\sum_{n=n_{0}}^{\infty}g'(n)P\{S_{n}=0\}$. It can describe the relations among the boundary function, weighting function, convergence rate and the limit value in studies of complete convergence. The result extends and generalizes the corresponding results of Gut and Steinebach (Ann. Univ. Sci. Budapest. Sect. Comput. 39:95–110, 2013), Kong (Lith. Math. J. 56(3):318–324, 2016), Kong and Dai (Stat. Probab. Lett. 119(10):295–300, 2016).

1 Introduction and main results

Throughout this note, let $\{X, X_{n}, n\geq1\}$ be a sequence of i.i.d. random variables with $EX=0$, $EX^{2}=\sigma^{2}$ and set $S_{n}=\sum_{k=1}^{n}X_{k}$. Let ${\mathcal {N}}$ be the standard normal random variable; C denotes a positive constant, possibly varying from place to place, the notions $a_{n}\sim b_{n}$ $a_{n}=O(b_{n})$, $a_{n}\asymp b_{n}$ stand for $\lim_{n\to\infty}\frac{a_{n}}{b_{n}}=1$, $\limsup_{n\to\infty}\frac{a_{n}}{b_{n}}<\infty$ and $0<\liminf_{n\to \infty}\frac{a_{n}}{b_{n}}\leq\limsup_{n\to\infty}\frac{a_{n}}{b_{n}}<\infty$, respectively. We define $\log x=\ln{\max\{e,x\}}$ and $\log\log x=\ln\ln{\max\{e^{e},x\}}$.

The concept of complete convergence was first introduced by Hsu and Robbins [4], since then there have been extensions in several directions. One of them is to discuss the precise rate and limit value of $\sum_{n=n_{0}}^{\infty}\varphi(n)P\{|S_{n}|\geq\varepsilon g(n)\}$ as $\varepsilon\searrow a$, $a\geq0$, where the weighting function $\varphi(x)$ and boundary function $g(x)$ are positive functions defined on $[n_{0},\infty)$. A first result in this direction was given by Heyde [5], who proved that

$$ \lim_{\varepsilon\searrow0}\varepsilon^{2} \sum _{n=1}^{\infty}P\bigl\{ \vert S_{n} \vert \geq\varepsilon n\bigr\} =\sigma^{2}. $$

The research in this field is called the precise asymptotics. For analogous results in more general case, see [6,7,8,9] and the references therein.

Another interesting direction is to consider the convergence rate for the precise asymptotic problems. Klesov [10] obtained the following result:

$$ \lim_{\varepsilon\searrow0}\Biggl[ \sum_{n=1}^{\infty}P \bigl\{ \vert S_{n} \vert \geq\varepsilon n\bigr\} -\frac{\sigma ^{2}}{\varepsilon^{2}} \Biggr]=-\frac{1}{2}, $$

where $\{X, X_{n}, n\geq1\}$ is a sequence of i.i.d. normal random variables with $EX=0$, $EX^{2}=\sigma^{2}$, $S_{n}=\sum_{k=1}^{n}X_{k}$.

Recently, Gut and Steinebach [1] obtained the following results:

$$ \lim_{\varepsilon\searrow0}\Biggl[ \sum_{n=1}^{\infty} n^{\frac{r}{p}-2}P\biggl\{ \frac{|S_{n}|}{\sigma\sqrt {n}}\geq\varepsilon n^{\frac{1}{p}-\frac{1}{2}}\biggr\} -\frac {p}{r-p}{\varepsilon}^{-\frac{2(r-p)}{2-p})}E|{\mathcal {N}}|^{\frac {2(r-p)}{2-p}} \Biggr] =\gamma_{\frac{r}{p}-2}-\eta_{r,p}, $$

where $1\leq p<2$, $p< r<3p/2$, $E|X|^{\frac{2r}{p}}<\infty$, $\gamma _{\theta}=\lim_{n\to\infty}(\sum_{j=1}^{n} j^{\theta}-\frac{n^{\theta +1}}{\theta+1})$, $\eta_{r,p}=\sum_{n=1}^{\infty} n^{\frac{r}{p}-2} P\{S_{n}=0\}$.

Later, Kong [2] proved the convergence rate in precise asymptotics for the law of iterated logarithm as follows:

$$ \lim_{\varepsilon\searrow0}\Biggl[ \sum_{n=3}^{\infty} \frac{1}{n\log n}P\biggl\{ \frac{|S_{n}|}{\sigma\sqrt {n}}\geq\varepsilon\sqrt{\log\log n}\biggr\} -{\varepsilon}^{-2}\Biggr]=\gamma-\eta, $$

where $E|X|^{q}<\infty$, $2< q\leq3$, $\gamma=\lim_{n\to\infty}(\sum_{j=3}^{n}\frac{1}{j\log j}-\log\log n)$, and $\eta=\sum_{n=3}^{\infty }\frac{1}{n\log n}P\{S_{n}=0\}$.

Also Kong and Dai [3] established the following convergence rate in precise asymptotics for the Davis law of large numbers with $EX^{2}(\log(1+|X|))^{1+\delta}<\infty$:

$$ \lim_{\varepsilon\searrow0}\Biggl[ \sum_{n=1}^{\infty} \frac{(\log n)^{\delta}}{n}P\biggl\{ \frac{|S_{n}|}{\sigma \sqrt{n}}\geq\varepsilon\sqrt{\log n}\biggr\} -{\varepsilon}^{-2(\delta +1)}\frac{E|{\mathcal {N}}|^{2(\delta+1)}}{\delta+1}\Biggr]=\gamma_{\delta}- \eta _{\delta}, $$

where $\gamma_{\delta}=\lim_{n\to\infty}(\sum_{j=1}^{n}\frac{(\log j)^{\delta}}{j}-\frac{(\log n)^{\delta+1}}{\delta+1})$, $\eta_{\delta}=\sum_{n=1}^{\infty}\frac{(\log n)^{\delta}}{n}P\{S_{n}=0\}$, and $\delta\geq0$.

In this note we will extend the scope of the weighting and boundary functions, and give more general convergence rates in precise asymptotics of i.i.d. random variables, which extends and generalizes the above results. The main result of this note is the following.

Theorem 1.1

Let $g(x)$ be a positive and twice differentiable function defined on $[n_{0},\infty)$, which is strictly increasing to ∞. Assume $g'(x)$ is strictly decreasing to 0 and $g'(n)\asymp\frac {1}{n^{\alpha_{1}}(\log n)^{\alpha_{2}}(\log\log n)^{\alpha_{3}}}$ where $\alpha_{i}$, $i=1,2,3$ are defined later. Assume that

• (1) $EX^{2}(\log(1+|X|))<\infty$, for $\alpha_{1}=1$, $\alpha_{2}>0$, $\alpha _{3}\in{\mathbb {R}}$;

• (2) $EX^{2}(\log(1+|X|))^{1-\alpha_{2}}<\infty$, for $\alpha_{1}=1$, $\alpha _{2}\leq0$, $\alpha_{3}\geq0$;

• (3) $E|X|^{4-2\alpha_{1}}<\infty$, for $1/2<\alpha_{1}<1$, $\alpha_{2}\geq0$, $\alpha_{3}\geq0$.

Then for any $s>0$, we have

$$ \lim_{\varepsilon\searrow0}\Biggl[ \sum _{n=n_{0}}^{\infty} g'(n)P\biggl\{ \frac{|S_{n}|}{\sigma\sqrt{n}}\geq \varepsilon g^{s}(n)\biggr\} -{ \varepsilon}^{-1/s}E|{\mathcal {N}}|^{1/s}\Biggr]=\gamma _{g}-\eta_{g}, $$

(1.1)

where $\gamma_{g}=\lim_{n\to\infty}(\sum_{k=n_{0}}^{n}g'(k)-g(n))$, $\eta_{g}=\sum_{n=n_{0}}^{\infty}g'(n)P\{S_{n}=0\}$.

Remark 1.2

There are many functions satisfying the assumptions of $g(x)$, such as $g(x)=x^{\beta_{1}}(\log x)^{\beta_{2}}(\log\log x)^{\beta_{3}}$ with some suitable conditions of $\beta_{i}$, $i=1,2,3$. The following corollaries are some typical examples.

Corollary 1.3

Let $g(x)=(\log x)^{\delta+1}$, $\delta>-1$, and $s>0$ in Theorem 1.1. If $EX^{2}(\log(1+|X|))^{\max\{1,1+\delta\}}<\infty$, then

$$ \lim_{\varepsilon\searrow0}\Biggl[ \sum_{n=1}^{\infty} \frac{(\log n)^{\delta}}{n}P\biggl\{ \frac{|S_{n}|}{\sigma \sqrt{n}}\geq\varepsilon({\log n)^{(\delta+1)s}}\biggr\} -{\varepsilon }^{-1/s}\frac{E|{\mathcal {N}}|^{1/s}}{\delta+1} \Biggr]=\gamma_{\delta}-\eta _{\delta}, $$

where $\gamma_{\delta}=\lim_{n\to\infty}(\sum_{j=1}^{n}\frac{(\log j)^{\delta}}{j}-\frac{(\log n)^{\delta+1}}{\delta+1})$, $\eta_{\delta}=\sum_{n=1}^{\infty}\frac{(\log n)^{\delta}}{n}P\{S_{n}=0\}$.

Corollary 1.4

Let $g(x)=(\log\log x)^{b+1}$, $b>-1$, and $s>0$ in Theorem 1.1. If $EX^{2}(\log(1+|X|))<\infty$, then

$$ \lim_{\varepsilon\searrow0}\Biggl[ \sum_{n=1}^{\infty} \frac{(\log n)^{b}}{n\log n}P\biggl\{ \frac{|S_{n}|}{\sigma \sqrt{n}}\geq\varepsilon({\log\log n})^{(b+1)s}\biggr\} -{\varepsilon }^{-1/s}\frac{E|{\mathcal {N}}|^{1/s}}{b+1} \Biggr]=\gamma_{b}-\eta_{b}, $$

where $\gamma_{b}=\lim_{n\to\infty}(\sum_{j=1}^{n}\frac{(\log j)^{b}}{j\log j}-\frac{(\log\log n)^{b+1}}{b+1})$, $\eta_{b}=\sum_{n=1}^{\infty}\frac{(\log n)^{b}}{n\log n}P\{S_{n}=0\}$.

Corollary 1.5

Let $g(x)=x^{\frac{r}{p}-1}$, $0< p<2$, $p< r<3p/2$, and $s>0$ in Theorem 1.1. If $E|X|^{\frac{2r}{p}}<\infty$, then

$$ \lim_{\varepsilon\searrow0}\Biggl[ \sum_{n=1}^{\infty} n^{\frac{r}{p}-2}P\biggl\{ \frac{|S_{n}|}{\sigma\sqrt {n}}\geq\varepsilon n^{\frac{(r-p)s}{p}}\biggr\} -\frac{p}{r-p}{\varepsilon }^{-1/s}E|{\mathcal {N}}|^{1/s} \Biggr] =\gamma_{\frac{r}{p}-2}-\varrho_{r,p}, $$

where $\gamma_{\theta}=\lim_{n\to\infty}(\sum_{j=1}^{n} j^{\theta}-\frac {n^{\theta+1}}{\theta+1})$, $\varrho_{r,p}=\sum_{n=1}^{\infty} n^{\frac {r}{p}-2}P\{S_{n}=0\}$.

Corollary 1.6

Let $g(x)=\frac{x^{a}}{(\log x)^{b}}$, $0< a<1/2$, $b>0$, and $s>0$ in Theorem 1.1. If $E|X|^{2(1+a)}<\infty$, then

$$ \lim_{\varepsilon\searrow0}\Biggl[ \sum_{n=1}^{\infty} \biggl[a-\frac{b}{\log n}\biggr]\frac{1}{n^{1-a}(\log n)^{b}}P\biggl\{ \frac{|S_{n}|}{\sigma\sqrt{n}} \geq\varepsilon\frac{n^{as}}{(\log n)^{bs}}\biggr\} -{\varepsilon}^{-1/s}E|{\mathcal {N}}|^{1/s}\Biggr] =\gamma_{a,b}-\varrho_{a,b}, $$

where $\gamma_{a,b}=\lim_{n\to\infty}(\sum_{j=1}^{n} [a-\frac{b}{\log j}]\frac{1}{j^{1-a}(\log j)^{b}}-\frac{n^{a}}{(\log n)^{b}})$ and $\varrho_{a,b}=\sum_{n=1}^{\infty} [a-\frac{b}{\log n}]\frac {1}{n^{1-a}(\log n)^{b}} P\{S_{n}=0\}$.

Remark 1.7

Obviously, Corollary 1.3 with $s=\frac{1}{2(\delta+1)}$ extends Theorem 1.1 in Kong and Dai [3] with the scope of δ from $\delta\geq0$ to $\delta>-1$; Corollary 1.4 with $s=\frac{1}{2(b+1)}$ extends Theorem 1 from Kong [2] with the scope of b from $b=0$ to $b>-1$ and the moment condition from $E|X|^{q}<\infty$ ($2< q\leq3$) to $EX^{2}(\log(1+|X|))<\infty$; Corollary 1.5 with $s=\frac{2-p}{2(r-p)}$ extends Theorem 2.2(a) from Gut and Steinebach [1] with the scope of r, p from $1\leq p<2$, $p< r<3p/2$ to $0< p<2$, $p< r<3p/2$. Therefore our results extend the known results.

2 Proof of Theorem 1.1

The following lemmas are useful for the proof of Theorem 1.1.

Lemma 2.1

Let $\{X, X_{n}, n\geq1\}$ be a sequence of i.i.d. random variables with $EX=0$, $EX^{2}=\sigma^{2}$ and set $S_{n}=\sum_{k=1}^{n}X_{k}$. Then

$$ \sum_{n=1}^{\infty} \frac{1}{n^{1-\delta/2}}\sup _{x} \biggl\vert P\biggl\{ \frac {S_{n}}{\sigma\sqrt{n}}\leq x\biggr\} -P\{{\mathcal {N}}\leq x\} \biggr\vert < \infty, $$

if and only if $E|X|^{2+\delta}<\infty$ for $0<\delta<1$. Also

$$ \sum_{n=1}^{\infty} \frac{(\log n)^{\delta}}{n}\sup _{x} \biggl\vert P\biggl\{ \frac {S_{n}}{\sigma\sqrt{n}}\leq x\biggr\} -P\{{\mathcal {N}}\leq x\} \biggr\vert < \infty, $$

if and only if $E|X|^{2}(\log(1+|X|))^{1+\delta}<\infty$ for $\delta\geq0$.

Proof

The first part can be found in a theorem from Heyde [11] or Theorem 1 from Heyde and Leslie [12] (with $k=0$), the second part can be found in Proposition 3.2 from Kong and Dai [3]. □

Lemma 2.2

Under the conditions of Theorem 1.1, we have

$$ \sum_{n=n_{0}}^{\infty} g'(n)\sup _{x} \biggl\vert P\biggl\{ \frac{S_{n}}{\sigma\sqrt{n}}\leq x\biggr\} -P\{{\mathcal {N}}\leq x\} \biggr\vert < \infty. $$

Proof

If $\alpha_{1}=1$, $\alpha_{2}>0$, $\alpha_{3}\in{\mathbb {R}}$, we know that $EX^{2}\log(1+|X|)<\infty$, and then, by the second part of Lemma 2.1 (with $\delta=0$),

$$\begin{aligned}& \sum_{n=n_{0}}^{\infty} g'(n)\sup _{x} \biggl\vert P\biggl\{ \frac{S_{n}}{\sigma\sqrt {n}}\leq x\biggr\} -P\{{\mathcal {N}}\leq x\} \biggr\vert \\& \quad \leq \sum_{n=n_{0}}^{\infty} \frac{C}{n(\log n)^{\alpha_{2}}(\log\log n)^{\alpha_{3}}}\sup_{x} \biggl\vert P\biggl\{ \frac{S_{n}}{\sigma\sqrt{n}}\leq x\biggr\} -P\{ {\mathcal {N}}\leq x\} \biggr\vert \\& \quad \leq \sum_{n=n_{0}}^{\infty} \frac{C}{n}\sup_{x} \biggl\vert P\biggl\{ \frac{S_{n}}{\sigma \sqrt{n}}\leq x\biggr\} -P\{{\mathcal {N}}\leq x\} \biggr\vert < \infty. \end{aligned}$$

If $\alpha_{1}=1$, $\alpha_{2}\leq0$, $\alpha_{3}\geq0$, we know that $EX^{2}(\log(1+|X|))^{1-\alpha_{2}}<\infty$, and then, by the second part of Lemma 2.1 (with $\delta=-\alpha_{2}\geq0$),

$$\begin{aligned}& \sum_{n=n_{0}}^{\infty} g'(n)\sup _{x} \biggl\vert P\biggl\{ \frac{S_{n}}{\sigma\sqrt {n}}\leq x\biggr\} -P\{{\mathcal {N}}\leq x\} \biggr\vert \\& \quad \leq \sum_{n=n_{0}}^{\infty} \frac{C}{n(\log n)^{\alpha_{2}}(\log\log n)^{\alpha_{3}}}\sup_{x} \biggl\vert P\biggl\{ \frac{S_{n}}{\sigma\sqrt{n}}\leq x\biggr\} -P\{ {\mathcal {N}}\leq x\} \biggr\vert \\& \quad \leq \sum_{n=n_{0}}^{\infty} \frac{C(\log n)^{-\alpha_{2}}}{n}\sup_{x} \biggl\vert P\biggl\{ \frac{S_{n}}{\sigma\sqrt{n}}\leq x\biggr\} -P\{{\mathcal {N}}\leq x\} \biggr\vert < \infty. \end{aligned}$$

If $1/2<\alpha_{1}<1$, $\alpha_{2}\geq0$, $\alpha_{3}\geq0$, we know that $E|X|^{4-2\alpha_{1}}=E|X|^{2+\delta}<\infty$ (with $\delta =2(1-\alpha_{1})\in(0,1)$), and then, by the first part of Lemma 2.1,

$$\begin{aligned}& \sum_{n=n_{0}}^{\infty} g'(n)\sup _{x} \biggl\vert P\biggl\{ \frac{S_{n}}{\sigma\sqrt {n}}\leq x\biggr\} -P\{{\mathcal {N}}\leq x\} \biggr\vert \\& \quad \leq \sum_{n=n_{0}}^{\infty} \frac{1}{n^{\alpha_{1}}(\log n)^{\alpha _{2}}(\log\log n)^{\alpha_{3}}}\sup_{x} \biggl\vert P\biggl\{ \frac{S_{n}}{\sigma\sqrt{n}}\leq x\biggr\} -P\{{\mathcal {N}}\leq x\} \biggr\vert \\& \quad \leq \sum_{n=n_{0}}^{\infty} \frac{C}{n^{\alpha_{1}}}\sup_{x} \biggl\vert P\biggl\{ \frac {S_{n}}{\sigma\sqrt{n}}\leq x\biggr\} -P\{{\mathcal {N}}\leq x\} \biggr\vert \\& \quad \leq \sum_{n=n_{0}}^{\infty} \frac{C}{n^{1-\delta/2}}\sup_{x} \biggl\vert P\biggl\{ \frac {S_{n}}{\sigma\sqrt{n}}\leq x\biggr\} -P\{{\mathcal {N}}\leq x\} \biggr\vert < \infty. \end{aligned}$$

The proof of Lemma 2.2 is completed. □

The existence and finiteness of $\gamma_{g}$ and $\eta_{g}$ can be obtained by the following two lemmas.

Lemma 2.3

Under the conditions of Theorem 1.1, we have

$$ \gamma_{n,g}=\gamma_{g}+O\bigl(g'(n)\bigr), $$

where $\gamma_{n,g}=\sum_{k=n_{0}}^{n}g'(k)-g(n)$ and $\gamma_{g}$ is a constant depending only on function g satisfying

$$-g(n_{0})\leq\gamma_{g}\leq g'(n_{0})-g(n_{0}). $$

Proof

Note that $g(x)$ is a positive and twice differentiable function defined on $[n_{0},\infty)$, which is strictly increasing to ∞; $g'(x)$ is strictly decreasing to 0. Then, by the mean value theorem for $g(x)$, we know

$$ \gamma_{n+1,g}-\gamma_{n,g}=g'(n+1)- \bigl(g(n+1)-g(n)\bigr)=g'(n+1)-g'(\xi_{n}) \leq0, $$

where $n<\xi_{n}<n+1$, then we obtain that $\{\gamma_{n,g},n\geq n_{0}\}$ is a decreasing sequence. Note that $g'(x)$ is strictly decreasing to 0, and then we have

$$ \gamma_{n,g}=\sum_{k=n_{0}+1}^{n} \int _{k-1}^{k}\bigl[g'(k)-g'(x) \bigr]\,dx+g'(n_{0})-g(n_{0})\leq g'(n_{0})-g(n_{0}), $$

and

$$\begin{aligned} \gamma_{n,g} =&\sum_{k=n_{0}+1}^{n} \int _{k-1}^{k}\bigl[g'(k)-g'(x) \bigr]\,dx+g'(n_{0})-g(n_{0}) \\ \geq& \sum_{k=n_{0}+1}^{n} \bigl[g'(k)-g'(k-1)\bigr]+g'(n_{0})-g(n_{0}) \\ =&g'(n)-g'(n_{0})+g'(n_{0})-g(n_{0}) \geq-g(n_{0}), \quad \text{as } n\to\infty. \end{aligned}$$

The above two inequalities mean that $\{\gamma_{n,g},n\geq n_{0}\}$ is a bounded sequence, so, by the monotone bounded sequence theorem, also that $\{\gamma_{n,g},n\geq n_{0}\}$ is a convergent sequence, and therefore $-g(n_{0})\leq\gamma_{g}\leq g'(n_{0})-g(n_{0})$.

Finally, for any $m>n$, by the monotonicity of $g'(x)$, we have

$$\begin{aligned} \gamma_{g}-\gamma_{n,g} =&\lim_{m\to\infty}[ \gamma_{m,g}-\gamma _{n,g}] =\lim_{m\to\infty} \Biggl[\sum_{k=n+1}^{m}g'(k)- \int_{n}^{m} g'(x)\,dx\Biggr] \\ =&\lim_{m\to\infty}\Biggl[\sum_{k=n+1}^{m} \int_{k-1}^{k}\bigl[g'(k)-g'(x) \bigr]\,dx\Biggr] \\ \geq&\lim_{m\to\infty}\Biggl[\sum_{k=n+1}^{m} \bigl[g'(k)-g'(k-1)\bigr]\Biggr] \\ =&\lim_{m\to \infty}\bigl[g'(m)-g'(n) \bigr]=-g'(n), \end{aligned}$$

which means that Lemma 2.3 holds since the sequence $\{\gamma _{n,g},n\geq n_{0}\}$ is decreasing. □

Lemma 2.4

Under the conditions of Theorem 1.1, we have

$$ \eta_{g}=\sum_{n=n_{0}}^{\infty}g'(n)P \{S_{n}=0\}< \infty. $$

Proof

Note that

$$P\biggl\{ |{\mathcal {N}}|< \frac{1}{n^{2}}\biggr\} =\sqrt{\frac{2}{\pi}} \int_{0}^{\frac {1}{n^{2}}}e^{-t^{2}/2}\,dt\leq C \frac{1}{n^{2}}, $$

so, by Lemma 2.2 and the monotonicity of $g'(x)$, we know

$$\begin{aligned} \eta_{g} =&\sum_{n=n_{0}}^{\infty}g'(n)P \{S_{n}=0\}\leq\sum_{n=n_{0}}^{\infty }g'(n)P \biggl\{ \frac{|S_{n}|}{\sigma\sqrt{n}}< \frac{1}{n^{2}}\biggr\} \\ \leq&\sum_{n=n_{0}}^{\infty}g'(n) \biggl\vert P\biggl\{ \frac{|S_{n}|}{\sigma\sqrt{n}}< \frac {1}{n^{2}}\biggr\} -P\biggl\{ |{ \mathcal {N}}|< \frac{1}{n^{2}}\biggr\} \biggr\vert +\sum _{n=n_{0}}^{\infty }g'(n)P\biggl\{ |{\mathcal {N}}|< \frac{1}{n^{2}}\biggr\} \\ \leq&\sum_{n=n_{0}}^{\infty}g'(n) \sup_{x} \biggl\vert P\biggl\{ \frac{|S_{n}|}{\sigma\sqrt {n}}\leq x\biggr\} -P\bigl\{ |{\mathcal {N}}|\leq x\bigr\} \biggr\vert +\sum_{n=n_{0}}^{\infty }Cg'(n) \frac{1}{n^{2}}< \infty. \end{aligned}$$

 □

Remark 2.5

Obviously, if X is a continuous random variable, then $\eta_{g}=0$. For the simplest discrete case, if we take $P(X=1)=P(X=-1)=\frac{1}{2}$, it is easy to check that $P\{S_{2n+1}=0\}=0$ and $P\{S_{2n}=0\}=C_{2n}^{n}\frac{1}{2^{2n}}\sim \frac{1}{\sqrt{\pi}}\frac{1}{n^{1/2}}$; therefore, $0<\eta_{g}<\infty$.

Lemma 2.6

Under the conditions of Theorem 1.1, we have

$$\begin{aligned}& \lim_{\varepsilon\searrow0}\Biggl[\frac{2}{\sqrt{2\pi}}\sum _{k=n_{0}}^{\infty }g(k) \int_{\varepsilon g^{s}(k)}^{\varepsilon g^{s}(k+1)}e^{-t^{2}/2}\,dt- \varepsilon^{-1/s} E|{\mathcal {N}}|^{1/s}\Biggr]=0, \end{aligned}$$

(2.1)

$$\begin{aligned}& \lim_{\varepsilon\searrow0}\frac{2\gamma_{g}}{\sqrt{2\pi}}\sum _{k=n_{0}}^{\infty} \int_{\varepsilon g^{s}(k)}^{\varepsilon g^{s}(k+1)}e^{-t^{2}/2}\,dt= \gamma_{g}, \end{aligned}$$

(2.2)

$$\begin{aligned}& \lim_{\varepsilon\searrow0}\frac{2}{\sqrt{2\pi}}\sum _{k=n_{0}}^{\infty }g'(k) \int_{\varepsilon g^{s}(k)}^{\varepsilon g^{s}(k+1)}e^{-t^{2}/2}\,dt=0. \end{aligned}$$

(2.3)

Proof

By the mean value theorem for integrals, there exists a constant $\xi _{k}\in(k,k+1)$ such that

$$ \int_{\varepsilon g^{s}(k)}^{\varepsilon g^{s}(k+1)}e^{-t^{2}/2}\,dt=\varepsilon \bigl[g^{s}(k+1)-g^{s}(k)\bigr]e^{-\varepsilon^{2}g^{2s}(\xi_{k})/2}. $$

(2.4)

Using Taylor expansion, we know

$$\begin{aligned}& e^{-\varepsilon^{2}g^{2s}(\xi_{k})/2}=e^{-\varepsilon ^{2}g^{2s}(k)/2}+\varepsilon^{2}O \bigl(g^{2s-1}(k)g'(k)e^{-\varepsilon^{2}g^{2s}(k)/2}\bigr), \\& g^{s}(k+1)-g^{s}(k)=sg^{s-1}(k)g'(k)+O \bigl(g^{s-2}(k) \bigl(g'(k)\bigr)^{2}-g^{s-1}(k)g''(k) \bigr). \end{aligned}$$

Therefore we have

$$\begin{aligned} \int_{\varepsilon g^{s}(k)}^{\varepsilon g^{s}(k+1)}e^{-t^{2}/2}\,dt =& s\varepsilon g^{s-1}(k)g'(k)e^{-\varepsilon ^{2}g^{2s}(k)/2} \\ &{}+\varepsilon O\bigl(\bigl(g^{s-2}(k) \bigl(g'(k) \bigr)^{2}-g^{s-1}(k)g''(k) \bigr)e^{-\varepsilon^{2}g^{2s}(k)/2}\bigr) \\ &{} +\varepsilon^{2}O\bigl(g^{3s-2}(k)\bigl[g'(k) \bigr]^{2}e^{-\varepsilon^{2}g^{2s}(k)/2}\bigr). \end{aligned}$$

(2.5)

By the monotonicity of $g(x)$ and $g'(x)$, we have

$$\begin{aligned}& \frac{2s\varepsilon}{\sqrt{2\pi}}\sum_{k=n_{0}}^{\infty }g(k)g^{s-1}(k)g'(k)e^{-\varepsilon^{2}g^{2s}(k)/2} \\& \quad = \frac{2s\varepsilon}{\sqrt{2\pi}} \int_{n_{0}}^{\infty }g^{s}(x)g'(x)e^{-\varepsilon^{2}g^{2s}(x)/2} \,dx+O(\varepsilon) \\& \quad = \frac{2s\varepsilon}{\sqrt{2\pi}} \int_{g(n_{0})}^{\infty }y^{s}(x)e^{-\varepsilon^{2}y^{2s}/2} \,dy+O(\varepsilon) \\& \quad = \varepsilon^{-1/s}\frac{2^{\frac{1}{2s}}}{\sqrt{\pi}} \int _{\varepsilon^{2}g^{2s}(n_{0})/2}^{\infty}t^{\frac{1}{2s}-\frac {1}{2}}e^{-t} \,dt+O(\varepsilon) \\& \quad = \varepsilon^{-1/s}\frac{2^{\frac{1}{2s}}}{\sqrt{\pi}} \int _{0}^{\infty}t^{\frac{1}{2s}-\frac{1}{2}}e^{-t}\,dt - \varepsilon^{-1/s}\frac{2^{\frac{1}{2s}}}{\sqrt{\pi}} \int _{0}^{\varepsilon^{2}g^{2s}(n_{0})/2}t^{\frac{1}{2s}-\frac {1}{2}}e^{-t} \,dt+O(\varepsilon) \\& \quad = \varepsilon^{-1/s}\frac{2^{\frac{1}{2s}}}{\sqrt{\pi}}\varGamma \biggl( \frac{1}{2s}+\frac{1}{2}\biggr)+O(\varepsilon)= \varepsilon^{-1/s}E|{ \mathcal {N}}|^{1/s}+O(\varepsilon). \end{aligned}$$

(2.6)

Since $g'(x)$ is strictly decreasing to 0, there exists a constant $k_{0}$ such that $g'(k)<\delta$, for any $k>k_{0}$. Then by monotonicity of $g(x)$ and $g'(x)$, we obtain

$$\begin{aligned}& \lim_{\varepsilon\searrow0}\Biggl[\frac{2\varepsilon}{\sqrt{2\pi}}\sum _{k=n_{0}}^{\infty}g(k)g^{s-2}(k) \bigl[g'(k)\bigr]^{2}e^{-\varepsilon^{2}g^{2s}(k)/2}\Biggr] \\& \quad \leq \lim_{\varepsilon\searrow0}\Biggl[\frac{2\varepsilon}{\sqrt{2\pi }}\sum _{k=n_{0}}^{k_{0}}g^{s-1}(k)\bigl[g'(k) \bigr]^{2}e^{-\varepsilon^{2}g^{2s}(k)/2}\Biggr] \\& \qquad {} +\lim_{\varepsilon\searrow0}\Biggl[\frac{2\varepsilon\delta}{\sqrt{2\pi }}\sum _{k=k_{0}+1}^{\infty}g^{s-1}(k)\bigl[g'(k) \bigr]e^{-\varepsilon ^{2}g^{2s}(k)/2}\Biggr] \\& \quad \leq \lim_{\varepsilon\searrow0}C\varepsilon\delta \int_{k_{0}}^{\infty }g^{s-1}(x)g'(x)e^{-\varepsilon^{2}g^{2s}(x)/2} \,dx \\& \quad = \lim_{\varepsilon\searrow0}C\varepsilon\delta \int_{g(k_{0})}^{\infty }y^{s-1}e^{-\varepsilon^{2}y^{2s}/2}\,dy \\& \quad \leq \lim_{\varepsilon\searrow0}C\delta \int_{\varepsilon ^{2}g^{2s}(k_{0})/2}^{\infty}t^{-1/2}e^{-t}\,dy \\& \quad = C\delta \int_{0}^{\infty}t^{-1/2}e^{-t}\,dy =C\delta\varGamma\biggl(\frac{1}{2}\biggr), \end{aligned}$$

and

$$\begin{aligned}& \lim_{\varepsilon\searrow0}\Biggl[\frac{2\varepsilon^{3}}{\sqrt{2\pi}}\sum _{k=n_{0}}^{\infty}g(k)g^{3s-2}(k) \bigl[g'(k)\bigr]^{2}e^{-\varepsilon ^{2}g^{2s}(k)/2}\Biggr] \\& \quad \leq \lim_{\varepsilon\searrow0}C\varepsilon^{3}\delta \int _{k_{0}}^{\infty}g^{3s-1}(x)g'(x)e^{-\varepsilon^{2}g^{2s}(x)/2} \,dx \\& \quad = \lim_{\varepsilon\searrow0}C\varepsilon^{3}\delta \int _{g(k_{0})}^{\infty}y^{3s-1}e^{-\varepsilon^{2}y^{2s}/2}\,dy \\& \quad \leq \lim_{\varepsilon\searrow0}C\delta \int_{\varepsilon ^{2}g^{2s}(k_{0})/2}^{\infty}t^{1/2}e^{-t}\,dy \\& \quad = C\delta \int_{0}^{\infty}t^{1/2}e^{-t}\,dy =C\delta\varGamma\biggl(\frac{3}{2}\biggr). \end{aligned}$$

Then, by the arbitrariness of δ and letting $\delta\to0$, we derive

$$\begin{aligned}& \lim_{\varepsilon\searrow0}\Biggl[\frac{2\varepsilon}{\sqrt{2\pi}}\sum _{k=n_{0}}^{\infty}g(k)g^{s-2}(k) \bigl[g'(k)\bigr]^{2}e^{-\varepsilon^{2}g^{2s}(k)/2}\Biggr]=0, \end{aligned}$$

(2.7)

$$\begin{aligned}& \lim_{\varepsilon\searrow0}\Biggl[\frac{2\varepsilon^{3}}{\sqrt{2\pi}}\sum _{k=n_{0}}^{\infty}g(k)g^{3s-2}(k) \bigl[g'(k)\bigr]^{2}e^{-\varepsilon^{2}g^{2s}(k)/2}\Biggr]=0. \end{aligned}$$

(2.8)

From the fact that $g'(x)$ is strictly decreasing to 0, we know $g''(x)\leq0$, and then, by using integration by parts, similar to the above discussion, one can get

$$\begin{aligned}& \lim_{\varepsilon\searrow0} -\varepsilon \int_{n_{0}}^{\infty }g^{s}(x)g''(x)e^{-\varepsilon^{2}g^{2s}(x)/2} \,dx \\& \quad = \lim_{\varepsilon\searrow0} -\varepsilon \int_{n_{0}}^{\infty }g^{s}(x)e^{-\varepsilon^{2}g^{2s}(x)/2} \,dg'(x) \\& \quad = \lim_{\varepsilon\searrow0} -\varepsilon\bigl[g^{s}(x)e^{-\varepsilon ^{2}g^{2s}(x)/2}g'(x) \bigr]\big|_{n_{0}}^{\infty} \\& \qquad {} + \lim_{\varepsilon\searrow0} \varepsilon \int_{n_{0}}^{\infty }g'(x) \bigl[sg^{s-1}(x)g'(x)-s\varepsilon^{2}g^{2s-1}(x)g'(x) \bigr]e^{-\varepsilon ^{2}g^{2s}(x)/2}\,dx \\& \quad = s \lim_{\varepsilon\searrow0} \varepsilon \int_{n_{0}}^{\infty }g^{s-1}(x) \bigl[g'(x)\bigr]^{2}e^{-\varepsilon^{2}g^{2s}(x)/2}\,dx -s\lim _{\varepsilon\searrow0} \varepsilon^{3} \int_{n_{0}}^{\infty }g^{2s-1}(x) \bigl[g'(x)\bigr]^{2}e^{-\varepsilon^{2}g^{2s}(x)/2}\,dx \\& \quad \leq s \delta\lim_{\varepsilon\searrow0} \varepsilon \int _{k_{0}}^{\infty}g^{s-1}(x)g'(x)e^{-\varepsilon^{2}g^{2s}(x)/2} \,dx +s\delta\lim_{\varepsilon\searrow0} \varepsilon^{3} \int_{k_{0}}^{\infty }g^{2s-1}(x)g'(x)e^{-\varepsilon^{2}g^{2s}(x)/2} \,dx \\& \quad \leq C\delta. \end{aligned}$$

Then, by letting $\delta\to0$, we can derive

$$\begin{aligned}& \lim_{\varepsilon\searrow0}\Biggl[\frac{-2\varepsilon}{\sqrt{2\pi}}\sum _{k=n_{0}}^{\infty}g(k)g^{s-1}(k)g''(k)e^{-\varepsilon^{2}g^{2s}(k)/2} \Biggr] \\& \quad \leq\lim_{\varepsilon\searrow0} \biggl[-C\varepsilon \int_{n_{0}}^{\infty }g^{s}(x)g''(x)e^{-\varepsilon^{2}g^{2s}(x)/2} \,dx\biggr]=0. \end{aligned}$$

(2.9)

Finally, (2.1) can be obtained by combining (2.4)–(2.9).

It is obvious that

$$ \lim_{\varepsilon\searrow0} \frac{2\gamma_{g}}{\sqrt{2\pi}}\sum _{k=n_{0}}^{\infty} \int_{\varepsilon g^{s}(k)}^{\varepsilon g^{s}(k+1)}e^{-t^{2}/2}\,dt = \lim _{\varepsilon\searrow0} \frac{2\gamma_{g}}{\sqrt{2\pi}} \int _{\varepsilon g^{s}(n_{0})}^{\infty}e^{-t^{2}/2}\,dt = \frac{2\gamma_{g}}{\sqrt{2\pi}} \int_{0}^{\infty}e^{-t^{2}/2}\,dt = \gamma_{g}, $$

thus (2.2) is proved.

Since $g'(x)$ is strictly decreasing to 0, there exists a constant $k_{0}$ such that $g'(k)<\delta$, for any $k>k_{0}$, and

$$\begin{aligned}& \lim_{\varepsilon\searrow0}\frac{2}{\sqrt{2\pi}}\sum_{k=n_{0}}^{\infty }g'(k) \int_{\varepsilon g^{s}(k)}^{\varepsilon g^{s}(k+1)}e^{-t^{2}/2}\,dt \\& \quad \leq \lim_{\varepsilon\searrow0} C P\bigl\{ |{\mathcal {N}}|\leq\varepsilon g^{s}(k_{0}+1)\bigr\} +\delta P\bigl\{ |{\mathcal {N}}|\geq \varepsilon g^{s}(k_{0}+1)\bigr\} =\delta. \end{aligned}$$

Thus (2.3) holds in view of the arbitrariness of δ. This completes the proof. □

Lemma 2.7

Under the conditions of Theorem 1.1, we have

$$ \lim_{\varepsilon\searrow0}\Biggl[\sum_{k=n_{0}}^{\infty}g'(k)P \bigl\{ |{\mathcal {N}}|\geq\varepsilon g^{s}(k)\bigr\} - \varepsilon^{-1/s} E|{\mathcal {N}}|^{1/s}\Biggr]= \gamma_{g}. $$

Proof

By Fubini’s theorem, Lemmas 2.3 and 2.6, we derive that

$$\begin{aligned}& \lim_{\varepsilon\searrow0}\Biggl[\sum_{k=n_{0}}^{\infty}g'(k)P \bigl\{ |{\mathcal {N}}|\geq\varepsilon g^{s}(k)\bigr\} - \varepsilon^{-1/s} E|{\mathcal {N}}|^{1/s}\Biggr] \\& \quad = \lim_{\varepsilon\searrow0}\Biggl[\frac{2}{\sqrt{2\pi}}\sum _{k=n_{0}}^{\infty}g'(k) \sum _{j=k}^{\infty} \int_{\varepsilon g^{s}(j)}^{\varepsilon g^{s}(j+1)}e^{-t^{2}/2}\,dt - \varepsilon^{-1/s} E|{\mathcal {N}}|^{1/s}\Biggr] \\& \quad = \lim_{\varepsilon\searrow0}\Biggl[\frac{2}{\sqrt{2\pi}}\sum _{j=n_{0}}^{\infty} \sum_{k=n_{0}}^{j}g'(k) \int_{\varepsilon g^{s}(j)}^{\varepsilon g^{s}(j+1)}e^{-t^{2}/2}\,dt - \varepsilon^{-1/s} E|{\mathcal {N}}|^{1/s}\Biggr] \\& \quad = \lim_{\varepsilon\searrow0}\Biggl[\frac{2}{\sqrt{2\pi}}\sum _{j=n_{0}}^{\infty} g(j) \int_{\varepsilon g^{s}(j)}^{\varepsilon g^{s}(j+1)}e^{-t^{2}/2}\,dt - \varepsilon^{-1/s} E|{\mathcal {N}}|^{1/s}\Biggr] \\& \qquad {} +\lim_{\varepsilon\searrow0}\frac{2}{\sqrt{2\pi}}\sum _{j=n_{0}}^{\infty} \gamma_{g} \int_{\varepsilon g^{s}(j)}^{\varepsilon g^{s}(j+1)}e^{-t^{2}/2}\,dt \\& \qquad {} +\lim_{\varepsilon\searrow0}\frac{2}{\sqrt{2\pi}}\sum _{j=n_{0}}^{\infty} O\bigl(g'(j)\bigr) \int_{\varepsilon g^{s}(j)}^{\varepsilon g^{s}(j+1)}e^{-t^{2}/2}\,dt \\& \quad = \gamma_{g}. \end{aligned}$$

 □

Lemma 2.8

Under the conditions of Theorem 1.1, we have

$$ \lim_{\varepsilon\searrow0} \sum_{k=n_{0}}^{\infty}g'(k) \biggl[P\biggl\{ \frac {|S_{k}|}{\sigma\sqrt{k}}\geq\varepsilon g^{s}(k)\biggr\} -P \bigl\{ |{\mathcal {N}}|\geq \varepsilon g^{s}(k)\bigr\} \biggr] =- \eta_{g}. $$

Proof

By Lemma 2.2, we have

$$\begin{aligned}& \sum_{k=n_{0}}^{\infty}g'(k) \biggl\vert P\biggl\{ \frac{|S_{k}|}{\sigma\sqrt{k}}\geq \varepsilon g^{s}(k)\biggr\} -P \bigl\{ |{\mathcal {N}}|\geq\varepsilon g^{s}(k)\bigr\} \biggr\vert \\& \quad \leq \sum_{k=n_{0}}^{\infty}g'(k) \sup_{x} \biggl\vert P\biggl\{ \frac{|S_{k}|}{\sigma\sqrt {k}}\geq x \biggr\} -P\bigl\{ \vert {\mathcal {N}} \vert \geq x\bigr\} \biggr\vert < \infty. \end{aligned}$$

Then, by the dominated convergence theorem and continuity of ${\mathcal {N}}$, we have

$$\begin{aligned}& \lim_{\varepsilon\searrow0} \sum_{k=n_{0}}^{\infty}g'(k) \biggl[P\biggl\{ \frac {|S_{k}|}{\sigma\sqrt{k}}\geq\varepsilon g^{s}(k)\biggr\} -P \bigl\{ |{\mathcal {N}}|\geq \varepsilon g^{s}(k)\bigr\} \biggr] \\& \quad = \lim_{\varepsilon\searrow0} \sum_{k=n_{0}}^{\infty}g'(k) \biggl[P\bigl\{ |{\mathcal {N}}|< \varepsilon g^{s}(k)\bigr\} -P\biggl\{ \frac{|S_{k}|}{\sigma\sqrt{k}}< \varepsilon g^{s}(k)\biggr\} \biggr] \\& \quad = \sum_{k=n_{0}}^{\infty}g'(k) \lim_{\varepsilon\searrow0}\biggl[P\bigl\{ |{\mathcal {N}}|< \varepsilon g^{s}(k)\bigr\} -P\biggl\{ \frac{|S_{k}|}{\sigma\sqrt{k}}< \varepsilon g^{s}(k)\biggr\} \biggr] \\& \quad = -\sum_{k=n_{0}}^{\infty}g'(k)P \{S_{k}=0] =-\eta_{g}. \end{aligned}$$

 □

Proof of Theorem 1.1

By combining Lemmas 2.7 and 2.8, we obtain Theorem 1.1. □

3 Conclusions

In this paper, using the rate of convergence to the normal distribution and Fubini theorem, under some suitable conditions, the convergence rates in precise asymptotics for the complete convergence have been discussed with more general boundary functions. The result extends and generalizes the corresponding results of Gut and Steinebach [1], Kong [2], and Kong and Dai [3]. However, this paper has only studied the convergence rates for complete convergence. In the future research, we will discuss the convergence rates in precise asymptotics for complete moment convergence, which was first studied by Liu and Lin [13], as it is more difficult to handle the moment terms.