Firstly, to prove the upper bound of Eq. (
3.1), we follow the approach used in the proof of Lemma 4.24 in Resnick [
19] or Theorem 2 in Bae and Ko [
1]. For any
\(0<\delta <1\) and integer
\(n_{0}\) in (
3.2), we have
$$\begin{aligned} P \Biggl(\sum_{i=1}^{\infty }\psi _{i}X_{i}>x \Biggr) \leq &P \Biggl(\sum _{i=1}^{n_{0}}\psi _{i}X_{i}^{+}>(1- \delta )x \Biggr) +P \Biggl(\sum_{i=n _{0}+1}^{\infty } \psi _{i}X_{i}^{+}>\delta x \Biggr) \\ =:&I_{1}(x)+I_{2}(x), \end{aligned}$$
(3.4)
where
\(X_{i}^{+}=\max \{X_{i},0\}\),
\(i\geq 1\). For convenience’s sake, we remark that
\(F_{i}\in \mathscr{L}\cap \mathscr{D}\),
\(i\geq 1\), can imply
\(F_{i}\in \mathscr{S}\),
\(1\leq i\leq n\), and
\(F_{i}*F_{j}\in \mathscr{S}\) for all
\(1\leq i< j\leq n\); see Jiang et al. [
13]. Therefore, the distributions
\(F_{i}\),
\(i\geq 1\), in Theorem
2.1 and Lemma
3.1, can also satisfy the conditions in Theorem
1.C. For
\(I_{1}(x)\), by Theorem
1.C or
1.D, and (
2.3), it follows that
$$\begin{aligned} I_{1}(x) \sim &\sum_{i=1}^{n_{0}}P \bigl(\psi _{i}X_{i}^{+}>(1-\delta )x\bigr) \\ \lesssim &M\sum_{i=1}^{n_{0}}\overline{F} \bigl(\psi _{i}^{-1}(1-\delta )x\bigr) \\ \leq &M\sup_{0< \psi _{i}\leq 1}\frac{\overline{F}(\psi _{i}^{-1}(1- \delta )x)}{\overline{F}(\psi _{i}^{-1}x)}\sum _{i=1}^{n_{0}} \overline{F}\bigl(\psi _{i}^{-1}x\bigr). \end{aligned}$$
(3.5)
By
\(F\in \mathscr{C}\), we get
$$ \lim_{\delta \downarrow 0}\limsup_{x\rightarrow \infty } \sup _{0< \psi _{i}\leq 1}\frac{\overline{F}(\psi _{i}^{-1}(1-\delta )x)}{ \overline{F}(\psi _{i}^{-1}x)} =\lim_{\delta \downarrow 0} \limsup _{x\rightarrow \infty }\frac{\overline{F}((1-\delta )x)}{ \overline{F}(x)}=1. $$
(3.6)
Hence, we substitute (
3.6) into (
3.5) to obtain
$$ I_{1}(x)\lesssim M\sum_{i=1}^{n_{0}} \overline{F}\bigl(\psi _{i}^{-1}x\bigr). $$
(3.7)
For
\(I_{2}(x)\), when
\(0< J_{F}^{+}<1\), we have
$$\begin{aligned} I_{2}(x) \leq &P \Biggl(\bigcup_{i=n_{0}+1}^{\infty } \bigl\{ \psi _{i}X_{i} ^{+}>\delta x\bigr\} \Biggr)+P \Biggl(\sum_{i=n_{0}+1}^{\infty }\psi _{i}X _{i}^{+}{\mathbf{1}}_{\{\psi _{i}X_{i}^{+}\leq \delta x\}}>\delta x \Biggr) \\ =:&I_{21}(x)+I_{22}(x). \end{aligned}$$
(3.8)
By (
1.2), (
2.3), (
3.3) and
\(F\in \mathscr{C} \subset \mathscr{D}\), for any
\(p_{2}>J_{F}^{+}\), there exist some large positive constants
\(C_{4}\) and
\(D_{4}\) such that, for all
\(x\geq \max \{D_{3}, D_{4}\}\),
$$\begin{aligned} I_{21}(x) \leq &\sum_{i=n_{0}+1}^{\infty }P \bigl(\psi _{i}X_{i}^{+}>\delta x\bigr) \\ \lesssim &M\sum_{i=n_{0}+1}^{\infty }\overline{F} \bigl(\psi _{i}^{-1} \delta x\bigr) \\ \leq &C_{3}C_{4}M\delta ^{-p_{2}}\varepsilon \overline{F}(x). \end{aligned}$$
(3.9)
By Markov’s inequality and the monotone convergence theorem, we see that
$$\begin{aligned} I_{22}(x) \leq &(\delta x)^{-1}E \Biggl(\sum _{i=n_{0}+1}^{\infty }\psi _{i}X_{i}^{+}{ \mathbf{1}}_{\{\psi _{i}X_{i}^{+}\leq \delta x\}} \Biggr) \\ =&(\delta x)^{-1}\sum_{i=n_{0}+1}^{\infty } \psi _{i}E\bigl(X_{i}^{+}{\mathbf{1}} _{\{X_{i}^{+}\leq \psi _{i}^{-1}\delta x\}}\bigr). \end{aligned}$$
(3.10)
By
\(F\in \mathscr{C}\subset \mathscr{D}\), (
1.2) and (
2.3), for any
\(J_{F}^{+}< p_{2}<1\), there exist some large positive constants
\(C_{5}\) and
\(D_{5}\) such that, for all
\(x\geq D_{5}\),
$$\begin{aligned} E \bigl(X_{i}^{+}{\mathbf{1}}_{\{X_{i}^{+}\leq \psi _{i}^{-1}\delta x \}} \bigr) =&- \int _{0}^{\psi _{i}^{-1}\delta x}u \, d \overline{F_{i}}(u) \\ =&-\psi _{i}^{-1}\delta x\overline{F_{i}}\bigl( \psi _{i}^{-1}\delta x\bigr)+ \int _{0}^{\psi _{i}^{-1}\delta x}\overline{F_{i}}(u)\, du \\ \leq &\psi _{i}^{-1}\delta x \int _{0}^{1}\overline{F_{i}}\bigl(t\psi _{i} ^{-1}\delta x\bigr)\, dt \\ \lesssim &M\psi _{i}^{-1}\delta x\overline{F}\bigl(\psi _{i}^{-1}\delta x\bigr) \int _{0}^{1}\frac{\overline{F}(t\psi _{i}^{-1}\delta x)}{\overline{F}( \psi _{i}^{-1}\delta x)}\, dt \\ \leq &\frac{C_{5}M}{1-p_{2}}\psi _{i}^{-1}\delta x\overline{F} \bigl(\psi _{i}^{-1}\delta x\bigr). \end{aligned}$$
(3.11)
Substituting (
3.11) into (
3.10) and using the last step of (
3.9) can lead to
$$ I_{22}(x)\lesssim \frac{C_{3}C_{4}C_{5}M}{1-p_{2}}\delta ^{-p_{2}} \varepsilon \overline{F}(x). $$
(3.12)
Therefore by (
3.4), (
3.7)–(
3.9), (
3.12) and the arbitrariness of
ε, we derive that
$$ P \Biggl(\sum_{i=1}^{\infty }\psi _{i}X_{i}>x \Biggr)\lesssim M\sum _{i=1} ^{n_{0}}\overline{F}\bigl(\psi _{i}^{-1}x\bigr)\leq M\sum_{i=1}^{\infty } \overline{F}\bigl(\psi _{i}^{-1}x\bigr). $$
(3.13)
For the case when
\(J_{F}^{+}\geq 1\), we choose some constant
\(\beta \in (J_{F}^{+},J_{F}^{-}p^{-1})\) such that
\(p<\beta ^{-1}J_{F} ^{-}\leq \beta ^{-1}J_{F}^{+}<1\). Set
\(\psi =\sum_{i=n_{0}+1}^{\infty }\psi _{i}\), which is assumed to be less than 1 without loss of generality. Then by Jensen’s inequality, it follows that
$$\begin{aligned} I_{2}(x) =&P \Biggl(\psi ^{\beta } \Biggl(\sum _{i=n_{0}+1}^{\infty }\frac{ \psi _{i}}{\psi }X_{i}^{+} \Biggr)^{\beta }>\delta ^{\beta }x^{\beta } \Biggr) \\ \leq &P \Biggl(\sum_{i=n_{0}+1}^{\infty }\psi _{i}X_{i}^{+\beta }> \psi ^{1-\beta }\delta ^{\beta } x^{\beta } \Biggr) \\ \leq &P \Biggl(\bigcup_{i=n_{0}+1}^{\infty } \bigl\{ \psi _{i}X_{i}^{+ \beta }>\psi ^{1-\beta }\delta ^{\beta } x^{\beta } \bigr\} \Biggr) \\ &{}+ P \Biggl(\sum_{i=n_{0}+1}^{\infty }\psi _{i}X_{i}^{+\beta }{\mathbf{1}} _{\{\psi _{i}X_{i}^{+\beta }\leq \psi ^{1-\beta }\delta ^{\beta } x^{ \beta }\}}>\psi ^{1-\beta }\delta ^{\beta } x^{\beta } \Biggr) \\ =:&I'_{21}(x)+I'_{22}(x). \end{aligned}$$
(3.14)
For
\(I'_{21}(x)\), by using
\(F\in \mathscr{C}\subset \mathscr{D}\) and (
1.1), and arguing as (
3.9), for any
\(p_{1}\in (\beta p,J _{F}^{-})\) and
\(p_{2}>J_{F}^{+}\), there exist some large positive constants
\(C_{6}\) and
\(D_{6}\) such that, for all
\(x\geq \max \{D_{3}, D_{4}, D_{6}\}\),
$$\begin{aligned} I'_{21}(x) \leq &\sum_{i=n_{0}+1}^{\infty }P \bigl(X_{i}^{+}>\psi _{i} ^{-\frac{1}{\beta }}\psi ^{\frac{1-\beta }{\beta }}\delta x \bigr) \\ \lesssim &M\sum_{i=n_{0}+1}^{\infty }\overline{F} \bigl(\psi _{i}^{-\frac{1}{ \beta }}\psi ^{\frac{1-\beta }{\beta }}\delta x \bigr) \\ \leq &C_{6}M\psi ^{\frac{p_{1}(\beta -1)}{\beta }}\sum_{i=n_{0}+1} ^{\infty }\overline{F} \bigl(\psi _{i}^{-\frac{1}{\beta }}\delta x \bigr) \\ \leq &C_{3}C_{4}C_{6}M\psi ^{\frac{p_{1}(\beta -1)}{\beta }} \delta ^{-p_{2}}\varepsilon \overline{F}(x). \end{aligned}$$
(3.15)
For
\(I'_{22}(x)\), by going along the same lines of the derivation of
\(I_{22}(x)\), we conclude that, for any
\(J_{F}^{+}< p_{2}<\beta \), there exist some large positive constants
\(C_{7}\) and
\(D_{7}\) such that, for all
\(x\geq \max \{D_{3}, D_{4}, D_{6}, D_{7}\}\),
$$\begin{aligned} I'_{22}(x) \lesssim &\frac{C_{7}M\beta }{\beta -p_{2}}\sum _{i=n_{0}+1} ^{\infty }\overline{F} \bigl(\psi _{i}^{-\beta } \psi ^{\frac{1-\beta }{\beta }}\delta x \bigr) \\ \leq &\frac{C_{3}C_{4}C_{6}C_{7}M\beta }{\beta -p_{2}} \psi ^{\frac{p_{1}(\beta -1)}{\beta }}\delta ^{-p_{2}}\varepsilon \overline{F}(x), \end{aligned}$$
(3.16)
where the last step is obtained similarly to (
3.15). Then by (
3.4), (
3.7), (
3.14)–(
3.16) and the arbitrariness of
ε, we prove that Eq. (
3.13) holds.
Secondly, we deal with the lower bound of Eq. (
3.1). Let
\(n_{0}\) and
p be fixed as those in (
3.2). For any
\(0<\delta <1\), we have
$$\begin{aligned} P \Biggl(\sum_{i=1}^{\infty }\psi _{i}X_{i}>x \Biggr) =&P \Biggl(\sum _{i=1}^{\infty }\psi _{i}X_{i}^{+}- \sum_{i=1}^{\infty }\psi _{i}X_{i} ^{-}>x \Biggr) \\ \geq &P \Biggl(\sum_{i=1}^{\infty }\psi _{i}X_{i}^{+}>(1+\delta )x, \sum _{i=1}^{\infty }\psi _{i}X_{i}^{-} \leq \delta x \Biggr) \\ \geq &P \Biggl(\sum_{i=1}^{\infty }\psi _{i}X_{i}^{+}>(1+\delta )x \Biggr)-P \Biggl(\sum _{i=1}^{\infty }\psi _{i}X_{i}^{-}> \delta x \Biggr) \\ =:&I_{3}(x)-I_{4}(x), \end{aligned}$$
(3.17)
where
\(X_{i}^{-}=-\min \{X_{i}, 0\}\),
\(i\geq 1\). For
\(I_{3}(x)\), by (
2.3), Theorem
1.C or
1.D, we have
$$\begin{aligned} I_{3}(x) \geq &P \Biggl(\sum_{i=1}^{n_{0}} \psi _{i}X_{i}^{+}>(1+\delta )x \Biggr) \\ \sim &\sum_{i=1}^{n_{0}}P\bigl(\psi _{i}X_{i}^{+}>(1+\delta )x\bigr) \\ \gtrsim &S\sum_{i=1}^{n_{0}}\overline{F} \bigl(\psi _{i}^{-1}(1+\delta )x\bigr) \\ \geq &S\inf_{0< \psi _{i}\leq 1}\frac{\overline{F}(\psi _{i}^{-1}(1+ \delta )x)}{\overline{F}(\psi _{i}^{-1}x)}\sum _{i=1}^{n_{0}} \overline{F}\bigl(\psi _{i}^{-1}x\bigr). \end{aligned}$$
(3.18)
By
\(F\in \mathscr{C}\), it follows that
$$ \lim_{\delta \downarrow 0}\liminf_{x\rightarrow \infty } \inf _{0< \psi _{i}\leq 1}\frac{\overline{F}(\psi _{i}^{-1}(1+\delta )x)}{ \overline{F}(\psi _{i}^{-1}x)} =\lim_{\delta \downarrow 0} \liminf _{x\rightarrow \infty }\frac{\overline{F}((1+\delta )x)}{ \overline{F}(x)}=1. $$
(3.19)
By (
3.3), (
3.18) and (
3.19), we obtain
$$\begin{aligned} I_{3}(x) \gtrsim &S\sum_{i=1}^{n_{0}} \overline{F}\bigl(\psi _{i}^{-1}x\bigr) \\ =&S\sum_{i=1}^{\infty }\overline{F}\bigl(\psi _{i}^{-1}x\bigr)-S\sum_{i=n_{0}+1} ^{\infty }\overline{F}\bigl(\psi _{i}^{-1}x\bigr) \\ \geq &S\sum_{i=1}^{\infty }\overline{F}\bigl( \psi _{i}^{-1}x\bigr)-C_{3}S\varepsilon \overline{F}(x), \end{aligned}$$
which, along with the arbitrariness of
\(0<\varepsilon <1\), implies that
$$ I_{3}(x)\gtrsim S\sum_{i=1}^{\infty } \overline{F}\bigl(\psi _{i}^{-1}x\bigr). $$
(3.20)
By (
2.2), for any
\(0<\varepsilon <1\), there exists a large positive constant
\(D'\) such that, for all
\(x\geq D'\),
$$ \sup_{i\geq 1}\frac{F_{i}(-x)}{\overline{F}(x)}< \varepsilon . $$
(3.21)
For
\(I_{4}(x)\), we only consider the case
\(0< J_{F}^{+}<1\). In fact, the case of
\(J_{F}^{+}\geq 1\) follows from similar derivations to (
3.14)–(
3.16) with slight modifications. Clearly,
$$\begin{aligned} I_{4}(x) \leq &\sum_{i=1}^{\infty }P \bigl(\psi _{i}X_{i}^{-}>\delta x \bigr)+P \Biggl( \sum_{i=1}^{\infty }\psi _{i}X_{i}^{-}{ \mathbf{1}}_{\{\psi _{i}X _{i}^{-}\leq \delta x\}}>\delta x \Biggr) \\ =&\sum_{i=1}^{\infty }P(\psi _{i}X_{i}< -\delta x)+P \Biggl(\sum _{i=1} ^{\infty }\psi _{i}X_{i}^{-}{ \mathbf{1}}_{\{\psi _{i}X_{i}^{-}\leq \delta x\}}>\delta x \Biggr) \\ =:&I_{41}(x)+I_{42}(x). \end{aligned}$$
(3.22)
For
\(I_{41}(x)\), by (
3.21) and the last step of (
3.9), for all
\(x\geq \max \{D', D_{4}\}\),
$$ I_{41}(x)< \varepsilon \sum_{i=1}^{\infty } \overline{F}\bigl(\psi _{i}^{-1} \delta x\bigr)\leq C_{4}\delta ^{-p_{2}}\varepsilon \sum _{i=1}^{\infty } \overline{F}\bigl(\psi _{i}^{-1}x \bigr). $$
(3.23)
For
\(I_{42}(x)\), similarly to (
3.10), we have
$$ I_{42}(x) \leq (\delta x)^{-1}\sum _{i=1}^{\infty }\psi _{i}E \bigl(X _{i}^{-}{\mathbf{1}}_{\{X_{i}^{-}\leq \psi _{i}^{-1}\delta x\}} \bigr). $$
(3.24)
Similarly to (
3.11), by
\(F\in \mathscr{C}\subset \mathscr{D}\), (
1.2), (
2.2) and (
2.3), for any
\(J_{F}^{+}< p _{2}<1\), there exist some large positive constants
\(C_{8}\) and
\(D_{8} \) such that, for all
\(x\geq \max \{ D', D_{8}\}\),
$$\begin{aligned}& E \bigl(X_{i}^{-}{\mathbf{1}}_{\{X_{i}^{-}\leq \psi _{i}^{-1}\delta x\}} \bigr) \\& \quad =-\psi _{i}^{-1}\delta xP \bigl(X_{i}^{-}> \psi _{i}^{-1}\delta x \bigr)+ \psi _{i}^{-1} \delta x \int _{0}^{1}P \bigl(X_{i}^{-}>t \psi _{i}^{-1} \delta x \bigr)\,dt \\& \quad \leq \psi _{i}^{-1}\delta x\overline{F} \bigl(\psi _{i}^{-1}\delta x \bigr) \int _{0}^{1}\frac{F_{i}(-t\psi _{i}^{-1}\delta x)}{\overline{F}(t\psi _{i}^{-1}\delta x)}\frac{\overline{F}(t\psi _{i}^{-1}\delta x)}{ \overline{F}(\psi _{i}^{-1}\delta x)} \,dt \\& \quad < \varepsilon \psi _{i}^{-1}\delta x\overline{F} \bigl(\psi _{i}^{-1} \delta x \bigr) \int _{0}^{1}\frac{\overline{F}(t\psi _{i}^{-1}\delta x)}{ \overline{F}(\psi _{i}^{-1}\delta x)}\,dt \\& \quad \leq \frac{C_{8}\varepsilon }{1-p_{2}}\psi _{i}^{-1}\delta x \overline{F} \bigl(\psi _{i}^{-1}\delta x \bigr). \end{aligned}$$
(3.25)
Then, by substituting (
3.25) into (
3.24) and arguing similarly to (
3.9), we prove that, for all
\(x\geq \max \{ D', D_{4}, D_{8}\}\),
$$\begin{aligned}& I_{42}(x)< \frac{C_{8}\varepsilon }{1-p_{2}}\sum_{i=1}^{\infty } \overline{F}\bigl(\psi _{i}^{-1}\delta x\bigr)\leq \frac{C_{4}C_{8}}{1-p_{2}} \delta ^{-p_{2}}\varepsilon \sum _{i=1}^{\infty }\overline{F}\bigl(\psi _{i} ^{-1}x\bigr); \end{aligned}$$
(3.26)
and further we substitute (
3.23) and (
3.26) into (
3.22) to obtain, for all
\(x\geq \max \{D', D_{4}, D_{8}\}\),
$$\begin{aligned}& I_{4}(x)< \biggl(\frac{C_{8}}{1-p_{2}}+1 \biggr)C_{4}\delta ^{-p_{2}} \varepsilon \sum_{i=1}^{\infty } \overline{F}\bigl(\psi _{i}^{-1}x\bigr), \end{aligned}$$
(3.27)
which, along with (
3.17), (
3.20) and the arbitrariness of
\(0<\varepsilon <1\), can show the lower bound of Eq. (
3.1). □