Local Closure under Infinitely Divisible Distribution Roots and Esscher Transform

Abstract

In this paper, we show that the local distribution class ${\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ is not closed under infinitely divisible distribution roots, i.e., there is an infinitely divisible distribution which belongs to the class, while the corresponding Lévy distribution does not. Conversely, we give a condition, under which, if an infinitely divisible distribution belongs to the class ${\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, then so does the Lévy distribution. Furthermore, we find some sufficient conditions that are more concise and intuitive. Using different methods, we also give a corresponding result for another local distribution class, which is larger than the above class. To prove the above results, we study the local closure under random convolution roots. In particular, we obtain a result on the local closure under the convolution root. In these studies, the Esscher transform of distribution plays a key role, which clarifies the relationship between these local distribution classes and related global distribution classes.

1. Preliminary

In this paper, we study the closure under infinitely divisible distribution (I.I.D.) roots for some local distribution classes, also known simply as the local closure under I.I.D. roots. In other words, we discuss the following problem, if an I.I.D. belongs to a local distribution class, does its corresponding Lévy distribution also belong to this class? These results are closely related to some local distribution classes and Esscher transform of distributions. Thus, in order to better illustrate the main results of this paper, we first introduce the above concepts and their basic properties in this section.

Throughout the paper, unless stated otherwise, all limits are taken as x tends to infinity; for two positive functions f and g, $f\left(x\right)\sim g\left(x\right)$ means $lim\; supf\left(x\right)/g\left(x\right)=1$, $f\left(x\right)\asymp g\left(x\right)$ means $0<lim\; inff\left(x\right)/g\left(x\right)\le lim\; supg\left(x\right)/f\left(x\right)<\infty$, $f\left(x\right)=o\left(g\left(x\right)\right)$ means $limf\left(x\right)/g\left(x\right)=0$; for a distribution V, let $\overline{V}=1-V$ be the tail distribution of V, $V^{*k}$ be the k-fold convolution of V with itself for all integers $k\ge 2$, $V^{*1}=V$ and $V^{*0}$ be the distribution degenerate at zero; and all distributions are supported on $[0,\infty )$.

1.1. Infinitely Divisible Distribution

Let H be an I.D.D. with the Laplace transform

$$\begin{array}{c}\hfill {\int }_0^{\infty }{e}^{-\lambda y}H\left(dy\right)=exp\left\{-a\lambda -{\int }_0^{\infty }(1-e^{\lambda y})\upsilon \left(dy\right)\right\},\end{array}$$

(1)

where $a\ge 0$ is a constant, and $\upsilon$ is a Borel measure on $(0,\infty )$ with the properties $\mu =\upsilon (1,\infty )<\infty$ and ${\int }_0^{\infty }min\{1,y^2\}\nu \left(dy\right)<\infty$. Let

$$F\left(x\right)=\upsilon (0,x]{\mathbf{1}}_{\{x>1\}}/\mu ,x\in (-\infty ,\infty )$$

be the Lévy distribution generated by the measure $\upsilon$. The distribution H admits the representation $H=H_1\ast H_2$, which is reserved for the convolution of two distributions $H_1$ and $H_2$ satisfying

$$\begin{array}{c}\hfill {\overline{H}}_1\left(x\right)=O\left(e^{-\beta x}\right)\mathrm{for}\mathrm{each}\beta >0\end{array}$$

(2)

and

$$\begin{array}{c}\hfill H_2\left(x\right)=e^{-\mu }\sum _{k=0}^{\infty }{F}^{*k}\left(x\right){\mu }^k/k!,x\in (-\infty ,\infty ).\end{array}$$

(3)

See, for example, pages 450 and 571 of Feller [1], Embrechts et al. [2] and Chapter 4 of Sato [3].

One of the research topics of I.D.D is the closure under I.D.D. roots for all types of distribution classes. More precisely, we say that a certain distribution class is closed under I.D.D. roots, if an I.D.D. belongs to the class, then its Lévy distribution also belongs to the same one; otherwise, we say that the class is not closed under the I.D.D. roots.

This paper mainly studies the closure of some local distribution classes under the I.D.D. roots, known simply as the local closure under the I.D.D. roots.

1.2. Related Distribution Classes

In this paper, for each $0<T\le \infty$, we denote

$$V(x+{\Delta }_T)=V(x,x+T]=\overline{V}\left(x\right)-\overline{V}(x+T)\mathrm{and}V(x+{\Delta }_{\infty })=\overline{V}\left(x\right),x\ge 0.$$

For each distribution V and $0<T<\infty$, we set that there is a $x_0=x_0(V,T)\ge 0$ such that $V(x+{\Delta }_T)>0,x\ge x_0$.

We say that a distribution V belongs to the distribution class ${\mathcal{L}}_{loc}$, if for each $0<T\le \infty$,

$$V(x-t+{\Delta }_T)\sim V(x+{\Delta }_T)\mathrm{for}\mathrm{each}t>0.$$

We say that a distribution V belongs to the distribution class ${\mathcal{S}}_{loc}$, if V belongs to the class ${\mathcal{L}}_{loc}$ and for each $0<T\le \infty$,

$$V^{*2}(x+{\Delta }_T)\sim 2V(x+{\Delta }_T).$$

See, for example, Borokov and Borokov [4].

The classes ${\mathcal{L}}_{loc}$ and ${\mathcal{S}}_{loc}$ are included in two new distribution classes ${\mathcal{OS}}_{loc}$ and ${\mathcal{OL}}_{loc}$ defined by the following conditions that, for each $0<T\le \infty$,

$$C_{{\Delta }_T}^{*}(V,t)=lim\; supV(x-t+{\Delta }_T)/V(x+{\Delta }_T)<\infty \mathrm{for}\mathrm{each}t>0;$$

and for each $0<T\le \infty$,

$$C_{{\Delta }_T}^{*}\left(V\right)=lim\; sup{V}^{*2}(x+{\Delta }_T)/V(x+{\Delta }_T)<\infty ,$$

respectively.

In the definitions of the above-mentioned local distribution classes, if “for each $0<T\le \infty$” is replaced by “for some $0<T\le \infty$”, then these classes are successively called local long-tailed distribution class, local subexponential distribution class, O-local long-tailed distribution class and O-local subexponential distribution class, denoted by ${\mathcal{L}}_{{\Delta }_T}$, ${\mathcal{S}}_{{\Delta }_T}$, ${\mathcal{OL}}_{{\Delta }_T}$ with indicator $C_{{\Delta }_T}^{*}(V,t)$ for each $0<t<\infty$ and ${\mathcal{OS}}_{{\Delta }_T}$ with indicator $C_{{\Delta }_T}^{*}\left(V\right)$, respectively. The classes ${\mathcal{L}}_{{\Delta }_T}$ and ${\mathcal{S}}_{{\Delta }_T}$ for some $0<T\le \infty$ were introduced by Asmussen et al. [5]. The class ${\mathcal{OS}}_{{\Delta }_T}$ for some $0<T\le \infty$ originates from the work of Wang et al. [6]. Clearly, the inclusion relations ${\mathcal{L}}_{loc}\subset {\mathcal{L}}_{{\Delta }_T}$ and ${\mathcal{S}}_{loc}\subset {\mathcal{S}}_{{\Delta }_T}$ for each $0<T\le \infty$ are proper.

For research on the local distribution classes, in addition to the above-mentioned references, please refer to Wang et al. [7], Wang et al. [8], Denisov et al. [9], Yang et al. [10], Watanabe [11], etc.

In particular, when $T=\infty$, we get the corresponding global distribution classes $\mathcal{L}$, $\mathcal{S}$, $\mathcal{OL}$ with indicator

$$C^{*}(V,t)=C_{{\Delta }_{\infty }}^{*}(V,t)=lim\; sup\overline{V}(x-t)/\overline{V}\left(x\right)<\infty \mathrm{for}\mathrm{each}t>0$$

and $\mathcal{OS}$ with indicator

$$C^{*}\left(V\right)=C_{{\Delta }_{\infty }}^{*}\left(V\right)=lim\; sup\overline{V^{*2}}\left(x\right)/\overline{V}\left(x\right)<\infty ,$$

respectively. The classes $\mathcal{L}$ and $\mathcal{S}$ were introduced by Chistyakov [12], and the classes $\mathcal{OL}$ and $\mathcal{OS}$ come from Shimula and Watanabe [13] and Klüppelberg [14], respectively.

Further, Lemma 2 of Chistyakov [12] shows $\mathcal{S}\subset \mathcal{L}$. This inclusion relation is proper, see Section 3 of Embrechts and Goldie [15], and so on. However, with respect to the prerequisite that $V\in {\mathcal{L}}_{{\Delta }_T}$ is necessary in the definition of the class ${\mathcal{S}}_{{\Delta }_T}$ for some $0<T<\infty$, see Propositions 3.1 and 3.2 of Chen et al. [16]. Another proper inclusion relation $\mathcal{OS}\subset \mathcal{OL}$ is given by Proposition 2.1 of Shimura and Watanabe [13].

Clearly, the class $\mathcal{OL}$ contains the heavy-tailed distribution classes ${\cup }_{0<T\le \infty }{\mathcal{L}}_{{\Delta }_T}$ and the class $\mathcal{OS}$ contains the heavy-tailed distribution classes ${\cup }_{0<T\le \infty }{\mathcal{S}}_{{\Delta }_T}$. Here, a distribution V is called the heavy-tailed distribution, if $M(V,\alpha )={\int }_0^{\infty }{e}^{\alpha y}V\left(dy\right)=\infty$ for each $\alpha >0$; otherwise, it is called the light-tailed distribution. Furthermore, for some $\gamma >0$, the following light-tailed distribution class $\mathcal{L}\left(\gamma \right)$ is a subclass of $\mathcal{OL}$, another light-tailed distribution class $\mathcal{S}\left(\gamma \right)$ is a subclass of $\mathcal{OS}$. Both classes were introduced by Chover et al. [17,18].

A distribution V belongs to the distribution class $\mathcal{L}\left(\gamma \right)$ for some $\gamma >0$, if

$$\overline{V}(x-t)\sim \overline{V}\left(x\right)e^{\gamma t}\mathrm{for}\mathrm{each}t>0.$$

A distribution V belongs to the distribution class $\mathcal{S}\left(\gamma \right)$ for some $\gamma >0$, if $V\in \mathcal{L}\left(\gamma \right)$, $M(V,\gamma )={\int }_0^{\infty }{e}^{\gamma y}V\left(dy\right)<\infty$ and

$$\overline{V^{*2}}\left(x\right)\sim 2M(V,\gamma )\overline{V}\left(x\right).$$

Clearly, here $M(V,\gamma )\ge 1$. In addition, the prerequisite that $V\in \mathcal{L}\left(\gamma \right)$ for some $\gamma >0$ also is necessary in the definition of the class $\mathcal{S}\left(\gamma \right)$, because the distribution here is closely related to its local distribution. In fact, if we define two distribution classes ${\mathcal{L}}_{{\Delta }_T}\left(\gamma \right)$ and ${\mathcal{S}}_{{\Delta }_T}\left(\gamma \right)$ for some $0<\gamma ,T<\infty$, then we can easily find that ${\mathcal{L}}_{{\Delta }_T}\left(\gamma \right)=\mathcal{L}\left(\gamma \right)$ and ${\mathcal{S}}_{{\Delta }_T}\left(\gamma \right)=\mathcal{S}\left(\gamma \right)$.

In the definition of the class $\mathcal{L}\left(\gamma \right)$, if V is a lattice, then x and t should be restricted to values of the lattice span of V, see Bertoin and Doney [19].

In addition, we might also set $\mathcal{L}=\mathcal{L}\left(0\right)$ and $\mathcal{S}=\mathcal{S}\left(0\right)$.

There are many research results on the distribution classes mentioned above, see Foss et al. [20], Wang [21] and the references therein.

1.3. Esscher Transform

Now, we use the Esscher transform to show the relationship between some heavy-tailed local distribution and the corresponding light-tailed global distribution.

For any distribution V and $\gamma \ne 0$, by $M(V,\gamma )\ge min\{1,e^{\gamma x}V\left(x\right)\},x\ge 0$, we know that $M(V,\gamma )>0$. Further, if $M(V,\gamma )<\infty$, then we define a distribution $V_{\gamma }$ such that

$$\begin{array}{c}\hfill V_{\gamma }\left(x\right)={\int }_{0-}^x{e}^{\gamma y}V\left(dy\right){\mathbf{1}}_{[0,\infty )}\left(x\right)/M(V,\gamma ),x\in (-\infty ,\infty ),\end{array}$$

(4)

which is called the Esscher transform (or the exponential tilting) of V. Clearly, for $\gamma >0$, we have

$$\begin{array}{c}\hfill 0<M(V,-\gamma )<1,V={\left(V_{-\gamma }\right)}_{\gamma }\mathrm{and}M(V,-\gamma )M(V_{-\gamma },\gamma )=1,\end{array}$$

(5)

and for all $k\ge 1$,

$$\begin{array}{c}\hfill {\left(V^{*k}\right)}_{\gamma }={\left(V_{\gamma }\right)}^{*k}=V_{\gamma }^{*k},{\left(V^{*k}\right)}_{-\gamma }={\left(V_{-\gamma }\right)}^{*k}=V_{-\gamma }^{*k}\mathrm{and}M(V^{*k},-\gamma )=M^k(V,-\gamma ),\end{array}$$

(6)

see Teugels [22], Veraverbeke [23] and Embrechts and Goldie [24] for technical details.

Further, for some $0<T<\infty$ and $\gamma >0$, Definitions 1.1 and 1.2 of Wang and Wang [25] define four global distribution classes as follows:

$${\mathcal{TL}}_{{\Delta }_T}\left(\gamma \right)=\left\{V:M(V,\gamma )<\infty \mathrm{and}{V}_{\gamma }\in {\mathcal{L}}_{{\Delta }_T}\right\},$$

$${\mathcal{TS}}_{{\Delta }_T}\left(\gamma \right)=\left\{V:M(V,\gamma )<\infty \mathrm{and}{V}_{\gamma }\in {\mathcal{S}}_{{\Delta }_T}\right\};$$

$${\mathcal{TL}}_{loc}\left(\gamma \right)=\left\{V:M(V,\gamma )<\infty \mathrm{and}{V}_{\gamma }\in {\mathcal{L}}_{loc}\right\}$$

and

$${\mathcal{TS}}_{loc}\left(\gamma \right)=\left\{V:M(V,\gamma )<\infty \mathrm{and}{V}_{\gamma }\in {\mathcal{S}}_{loc}\right\}.$$

The following proposition reveals the important role of the Esscher transform for the study of local distribution classes, see Propositions 2.1 and 2.2 of Wang and Wang [25]. On the contrary, this result also shows that some local distribution classes give new vitality to the Esscher transform.

Proposition 1.

$\left(i\right)$ For some $0<T<\infty$ and $\gamma >0$, a distribution $V\in {\mathcal{L}}_{{\Delta }_T}$ (or ${\mathcal{S}}_{{\Delta }_T})⟺V_{-\gamma }\in {\mathcal{TL}}_{{\Delta }_T}\left(\gamma \right)$ (or ${\mathcal{TS}}_{{\Delta }_T}\left(\gamma \right))$.

$\left(ii\right)$ A distribution $V\in {\mathcal{L}}_{loc}$ (or ${\mathcal{S}}_{loc})⟺V_{-\gamma }\in \mathcal{L}\left(\gamma \right)$ (or $\mathcal{S}\left(\gamma \right))$, that is ${\mathcal{TL}}_{loc}\left(\gamma \right)=\mathcal{L}\left(\gamma \right)\hspace{1em}(\mathrm{or}\hspace{1em}{\mathcal{TS}}_{loc}\left(\gamma \right)=\mathcal{S}\left(\gamma \right)).$ Furthermore, each of them implies that, for each $0<T<\infty$

$$V(x+{\Delta }_T)\sim \gamma T{e}^{\gamma x}\overline{V_{-\gamma }}\left(x\right)/M(V_{-\gamma },\gamma )=M(V,-\gamma )\gamma T{\mathit{e}}^{\gamma x}\overline{V_{-\gamma }}\left(x\right).$$

(7)

More results of the Esscher transform can be found in the above references and the others therein.

The paper is organized as follows. In Section 2, we present the main results for Theorems 1–3 related to local closure under I.I.D. roots. In Section 3, we prove the above results. To this end, we study the local closure under random convolution roots. Then in Section 4, we show that the condition (10) of Theorem 3 can be replaced by a more concise and intuitive condition (11). Finally, in Section 5, we briefly introduce some applications of the obtained results and further research problems. As an application of Theorem 2, we give a positive result on the local closure under the convolution root, which represents the local version of common Embrechts and Goldie conjecture.

2. Main Results

Before giving the main results of this paper, we recall some existing results on closure under I.I.D. roots.

For the global distribution classes, the class $\mathcal{S}\left(\gamma \right)$ is closed under I.D.D. roots, see Embrechts et al. [2] for the case $\gamma =0$, Sgibnev [26], Pakes [27] and Watanabe [28] for the case $\gamma >0$. Recently, Cui et al. [29] proved that the class $\mathcal{L}\left(\gamma \right)\cap \mathcal{OS}$ for some $\gamma \ge 0$ is closed under the roots with some restrictive condition.

However, for some global distribution classes without special restrictions, there were some negative results, i.e., there exists an I.D.D. H belonging to some class, while its Lévy distribution F does not belong to the same class; see Theorem 1.1 (iii) of Shimura and Watanabe [13] for the class $\mathcal{OS}$, Theorem 1.2 (3) of Xu et al. [30] for the class $\mathcal{L}\cap \mathcal{OS}$ and $\mathcal{L}\setminus \mathcal{OS}$ and Theorem 1.1 of Xu et al. [31] for the class $\mathcal{L}\left(\gamma \right)\cap \mathcal{OS}$ with some $\gamma >0$.

As previously mentioned, this paper mainly studies the closure of some local distribution classes under the I.D.D. roots. Clearly, if a distribution $V\in {\mathcal{L}}_{{\Delta }_T}$ for some $0<T<\infty$, then $V(x+{\Delta }_T)=o\left(V\left(x\right)\right)$. Therefore, the study of local distribution cannot be replaced by that of global distribution.

One of the difficulties in the study of local distributions is the loss of their almost monotonic decreasing property. Corollary 3.1 of Jiang et al. [32] shows that some local distributions in the class ${\mathcal{S}}_{loc}$ and the class ${\mathcal{L}}_{loc}\setminus {\mathcal{S}}_{loc}$ are not even close to decreasing. Therefore, the study of local distribution is definitely more challenging than that of global distribution. Furthermore, we find hardly any existing results regarding local closure under I.I.D. roots.

Now, we first give a negative conclusion for the class ${\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$.

Theorem 1.

The class ${\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ is not closed under I.D.D. roots.

Next, we give two positive conclusions for the class ${\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ and ${\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$ with some $0<\gamma ,T_0<\infty$, respectively.

Theorem 2.

Let H be an I.D.D. with the Lévy distribution F. Assume that $H\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, and for all $k\ge 1$,

$$\begin{array}{c}\hfill lim\; inf\overline{F_{-\gamma }^{*k}}(x-t)/\overline{F_{-\gamma }^{*k}}\left(x\right)\ge e^{\gamma t}foreacht>0.\end{array}$$

(8)

Then the following two conclusions hold.

$\left(i\right)$$H_2\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ and $H_2(x+{\Delta }_T)\sim H(x+{\Delta }_T)$ for each $0<T\le \infty$.

$\left(ii\right)$ There exists an integer $l_0\ge 1$ such that $F^{*n}\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ for all $n\ge l_0$ and $F^{*n}\notin {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ for all $1\le n\le l_0-1$. In particular, if $F\in {\mathcal{OS}}_{loc}$, then $F^{*n}\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ for all $n\ge 1$.

Remark 1.$\left(i\right)$ According to Corollary 1.1 of Cui et al. [29], the condition (8) can be implied by some more concise and convenient conditions that

$$\begin{array}{c}\hfill F_{-\gamma }\in \mathcal{OL},lim\overline{F_{-\gamma }}\left(x\right)C^{*}(F_{-\gamma },x)=0and\left(8\right)holdsfork=1.\end{array}$$

(9)

Therefore, all conclusions of Theorem 2 hold under the conditions (9) and $H\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$. Some related examples can be found in Corollary 1.2 and Example 4.1 of Cui et al. [29].

$\left(ii\right)$ In the proof of Theorem 1, we can find that there exists an I.D.D. H with Lévy distribution F such that $l_0=2$. This fact shows that there are many distributions F that satisfy condition (8), but which do not belong to the class $\mathcal{L}\left(\gamma \right)$.

Clearly, the local distribution class ${\mathcal{L}}_{{\Delta }_{T_0}}\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$ for some $0<T_0\le \infty$ is larger than the class ${\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$. Therefore, it is natural to investigate the corresponding result for the former. To this end, we first consider its corresponding light-tailed global distribution class ${\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$ for some $0<\gamma ,T_0<\infty$, which is larger than the class $\mathcal{L}\left(\gamma \right)\cap \mathcal{OS}$. We will find that the research method of the following result is different from that of Theorem 2.

Theorem 3.

Let H be an I.D.D. with the Lévy distribution F. For some $0<\gamma ,T_0<\infty$, assume that $H\in {\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$ and for all $k\ge 1$,

$$\begin{array}{c}\hfill lim\; inf{F}_{\gamma }^{*k}(x-t+{\Delta }_{T_0})/F_{\gamma }^{*k}(x+{\Delta }_{T_0})\ge 1foreacht>0.\end{array}$$

(10)

Then the following two conclusions hold.

$\left(i\right)$$H_2\in {\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$ and $H_2(x+{\Delta }_{T_0})\asymp H(x+{\Delta }_{T_0})$.

$\left(ii\right)$ There is an integer $l_0\ge 1$ such that $F^{*n}\in {\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$ for all $n\ge l_0$ and $F^{*n}\notin {\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$ for all $1\le n\le l_0-1$. In particular, if $F\in {\mathcal{OS}}_{{\Delta }_{T_0}}$, then $F^{*n}\in {\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$ for all $n\ge 1$.

Remark 2.

The condition (10) can also be replaced by the following more concise and convenient conditions:

$$\begin{array}{c}\hfill F_{\gamma }\in {\mathcal{OL}}_{{\Delta }_{T_0}},lim{F}_{\gamma }(x+{\Delta }_{T_0})C_{{\Delta }_{T_0}}^{*}(F_{\gamma },x)=0and\left(10\right)holdsfork=1.\end{array}$$

(11)

See Theorem 6 with $V=F_{\gamma }$ below.

3. The Proofs of Theorems 1–3

3.1. Proof of Theorem 1

Let $F_{\left(0\right)}$ be a heavy-tailed distribution such that

$$\begin{array}{c}\hfill \overline{F_{\left(0\right)}}\left(x\right)={\mathbf{1}}_{(-\infty ,a_0)}\left(x\right)+C\sum _{n=0}^{\infty }\left(\left(\sum _{i=n}^{\infty }\frac{1}{a_i^{\alpha }}-\frac{x-a_n}{a_n^{\alpha +1}}\right){\mathbf{1}}_{[a_n,2a_n)}\left(x\right)+\left(\sum _{i=n+1}^{\infty }\frac{1}{a_i^{\alpha }}\right){\mathbf{1}}_{[2a_n,a_{n+1})}\left(x\right)\right)\end{array}$$

(12)

with the density $f_{\left(0\right)}\left(x\right)=C{\sum }_{n=0}^{\infty }{a}_n^{-\alpha -1}{\mathbf{1}}_{[a_n,2a_n)}\left(x\right)$ for all x, where

$$\alpha \in \left(\frac{3}{2},\frac{\sqrt{5}+1}{2}\right),a_n=a^{r^n}\mathrm{for}r=1+\frac{1}{\alpha },\mathrm{some}a>8^{\alpha }\mathrm{and}\mathrm{all}n\ge 1,\mathrm{and}C={\left(\sum _{n=0}^{\infty }{a}_n^{-\alpha }\right)}^{-1}.$$

Let ${\mathcal{F}}_1\left(0\right)$ be the class comprising the above distributions $F_{\left(0\right)}$ defined by (12). Further, for some $\gamma >0$ and distribution $F_{\left(0\right)}\in {\mathcal{F}}_1\left(0\right)$, define the light-tailed distribution $F_{\left(\gamma \right)}$ in the form

$$\begin{array}{c}\hfill \overline{F_{\left(\gamma \right)}}\left(x\right)={\mathbf{1}}_{(-\infty ,0)}\left(x\right)+e^{-\gamma x}\overline{F_{\left(0\right)}}\left(x\right){\mathbf{1}}_{[0,\infty )}\left(x\right)\end{array}$$

(13)

with its density $f_{\left(\gamma \right)}$ for all x. Then we can construct a new distribution class

$${\mathcal{F}}_1\left(\gamma \right)=\{F_{\left(\gamma \right)}\mathrm{defined}\mathrm{by}\left(13\right):F_{\left(0\right)}\in {\mathcal{F}}_1\left(0\right)\}.$$

See the proof of Theorem 1 of Xu et al. [31].

Let $H=H_1\ast H_2$ is an I.D.D. with Lévy distribution $F_{\left(\gamma \right)}\in {\mathcal{F}}_1\left(\gamma \right)$ for some $\gamma >0$. Then Proposition 1 and Theorem 1 of Xu et al. [31] show that, $H,H_2$ and $F_{\left(\gamma \right)}^{*k}$ for all $k\ge 2$ belong to the class $\left(\mathcal{L}\left(\gamma \right)\cap \mathcal{OS}\right)\setminus \mathcal{S}\left(\gamma \right)$, while $F_{\left(\gamma \right)}$ with $M(F_{\left(\gamma \right)},\gamma )<\infty$ belongs to the class $\mathcal{OL}\setminus \left(\mathcal{L}\left(\gamma \right)\cup \mathcal{OS}\right)$.

Because $M(F_{\left(\gamma \right)},\gamma )<\infty$, $M(H,\gamma )<\infty$, then $H_{\gamma }=H_{1,\gamma }\ast H_{2,\gamma }$, as the Esscher transform of H, is defined and is I.D.D. with Lévy distribution $F_{\left(\gamma \right),\gamma }$. To reveal the properties of $H_{\gamma }$ and $F_{\left(\gamma \right),\gamma }$, we need the following result.

Lemma 1.

For some $0<\gamma ,T<\infty$, $V_{-\gamma }\in {\mathcal{OS}}_{{\Delta }_T}⟺V\in {\mathcal{OS}}_{{\Delta }_T}$. Thus, $V_{-\gamma }\in {\mathcal{OS}}_{loc}⟺V\in {\mathcal{OS}}_{loc}$. Further, if $V_{-\gamma }\in \mathcal{L}\left(\gamma \right)$, then $V_{-\gamma }\in \mathcal{OS}⟺V_{-\gamma }\in {\mathcal{OS}}_{loc}$. Therefore,

$$V_{-\gamma }\in \mathcal{L}\left(\gamma \right)\cap \mathcal{OS}⟺V\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}.$$

Proof. 

We now prove the first conclusion. From (2.4) of Wang and Wang [25], we have

$$\begin{array}{c}\hfill V_{-\gamma }(x+{\Delta }_T)=M(V_{-\gamma },\gamma )e^{-\gamma x}\left(V(x+{\Delta }_T)-\gamma {\int }_0^T{e}^{-\gamma y}V(x+y,x+T]dy\right),x\ge 0.\end{array}$$

(14)

Further, we obtain the following inequality,

$$\begin{array}{c}\hfill e^{-\gamma T}V(x+{\Delta }_T)\le e^{\gamma x}{V}_{-\gamma }(x+{\Delta }_T)/M(V_{-\gamma },\gamma )\le V(x+{\Delta }_T),x\ge 0.\end{array}$$

(15)

If $V\in {\mathcal{OS}}_{{\Delta }_T}$, then according to Radon–Nikodym Theorem, by (15) and (4), we have

$$\begin{array}{ccc}& & V_{-\gamma }^{*2}(x+{\Delta }_T)={\int }_0^x{V}_{-\gamma }(x-y+{\Delta }_T)V_{-\gamma }\left(dy\right)+{\int }_x^{x+T}{V}_{-\gamma }(0,x-y+T]V_{-\gamma }\left(dy\right)\hfill \\ & \le & e^{-\gamma x}M(V_{-\gamma },\gamma ){\int }_0^{x}V(x-y+{\Delta }_T)V\left(dy\right)/M(V,-\gamma )+V_{-\gamma }(x+{\Delta }_T)\hfill \\ & \le & e^{-\gamma x}M(V_{-\gamma },\gamma )V^{*2}(x+{\Delta }_T)/M(V,-\gamma )+V_{-\gamma }(x+{\Delta }_T)\hfill \\ & \le & 2C_{{\Delta }_T}^{*}\left(V\right)M(V_{-\gamma },\gamma )e^{-\gamma x}V(x+{\Delta }_T)/M(V,-\gamma )+V_{-\gamma }(x+{\Delta }_T)\hfill \\ & \le & \left(2C_{{\Delta }_T}^{*}\left(V\right)M(V_{-\gamma },\gamma )e^{\gamma T}/M(V,-\gamma )+1\right)V_{-\gamma }(x+{\Delta }_T)\mathrm{for}\mathrm{large}\mathrm{enough}x>0,\hfill \end{array}$$

that is $V_{-\gamma }\in {\mathcal{OS}}_{{\Delta }_T}$. Conversely, if $V_{-\gamma }\in {\mathcal{OS}}_{{\Delta }_T}$, then we also get $V\in {\mathcal{OS}}_{{\Delta }_T}$ by the same approach.

The second conclusion comes from the arbitrariness of T.

If $V_{-\gamma }\in \mathcal{L}\left(\gamma \right)$, then $V_{-\gamma }^{*2}\in \mathcal{L}\left(\gamma \right)$. Thus, for each $0<T\le \infty$, by

$$V_{-\gamma }^{*k}(x+{\Delta }_T)\sim (1-e^{-\gamma T})\overline{V_{-\gamma }^{*k}}\left(x\right),k=1,2,$$

the third conclusion holds.

Proposition 1 and the third conclusion imply the final conclusion. □

Now, we continue to prove the theorem. According to Lemma 1 and Proposition 1, by $H\in \left(\mathcal{L}\left(\gamma \right)\cap \mathcal{OS}\right)\setminus \mathcal{S}\left(\gamma \right)$ and $F_{\left(\gamma \right)}\in \mathcal{OL}\setminus \left(\mathcal{L}\left(\gamma \right)\cup \mathcal{OS}\right)$, we know that $H_{\gamma }\in \left({\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}\right)\setminus {\mathcal{S}}_{loc}$, while $F_{\left(\gamma \right),\gamma }\in {\mathcal{OL}}_{loc}\setminus \left({\mathcal{L}}_{loc}\cup {\mathcal{OS}}_{loc}\right)$. Therefore, the class ${\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ is not closed under I.D.D. roots.

3.2. Proof of Theorem 2

To prove this theorem, we give two preliminary results. Firstly, we consider the closure under random convolution roots for the distribution class ${\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$. Clearly, this result and the following Theorem 5 not only play a key role in the proof of Theorems 2 and 3, but also have their own independent value.

Let V be a distribution and let $\tau$ be a nonnegative integer-valued random variable with masses $p_k=P(\tau =k)$ for all nonnegative integers k satisfying ${\sum }_{k=0}^{\infty }{p}_k=1$. Denoted by $V^{*\tau }$ is the random convolution or compound convolution generated by V and $\tau$, i.e.,

$$V^{*\tau }=\sum _{k=0}^{\infty }{p}_k{V}^{*k}.$$

Let $m=sup\{k:p_k>0\}$. In this paper, we consider the following two cases:

$$\mathrm{Case}1:p_k>0\mathrm{for}\mathrm{all}k\ge 1;\mathrm{Case}2:1\le m<\infty \mathrm{and}{p}_k>0\mathrm{for}\mathrm{all}1\le k\le m.$$

Theorem 4.

Assume that for any $0<\epsilon <1$ and some $0<T_0<\infty$, there exists an integer $n_0=n_0(V,\epsilon ,\tau ,T_0)\ge 1$ such that

$$\begin{array}{c}\hfill \sum _{k=n_0+1}^{\infty }{p}_k{V}^{*(k-1)}(x+{\Delta }_{T_0})\le \epsilon V^{*\tau }(x+{\Delta }_{T_0}),x\ge 0,\end{array}$$

(16)

and for each $k\ge 1$ in Case 1 or $1\le k\le m$ in Case 2,

$$\begin{array}{c}\hfill lim\; inf\overline{V_{-\gamma }^{*k}}(x-t)/\overline{V_{-\gamma }^{*k}}\left(x\right)\ge e^{\gamma t}foreacht>0.\end{array}$$

(17)

If $V^{*\tau }\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, then for the above two cases, there exists an integer $l_0\ge 1$ in Case 1 or $1\le l_0\le m$ in Case 2 such that $V^{*n}\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ for all $l_0\le n<m$ and $V^{*n}\notin {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ for all $1\le n\le l_0-1$. In particular, if $V\in {\mathcal{OS}}_{loc}$, then $V^{*n}\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ for all $n\ge 1$ in Case 1 or $1\le n\le m$ in Case 2.

Proof. 

We first prove the theorem for Case 1 that $m=\infty$.

In Lemma 1, we replace V with $V^{*\tau }$. Then by $V^{*\tau }\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, we know that

$${\left(V^{*\tau }\right)}_{-\gamma }\in \mathcal{L}\left(\gamma \right)\cap \mathcal{OS}.$$

In addition,

$$0<M(V^{*\tau },-\gamma )=\sum _{k=0}^{\infty }{p}_k{M}^k(V,-\gamma )=E{M}^{\tau }(V,-\gamma )<1$$

and

$$\begin{array}{c}\hfill \overline{{\left(V^{*\tau }\right)}_{-\gamma }}\left(x\right)=\sum _{k=1}^{\infty }\frac{p_k{M}^k(V,-\gamma )}{M(V^{*\tau },-\gamma )}\overline{V_{-\gamma }^{*k}}\left(x\right)=\sum _{k=1}^{\infty }{q}_k\overline{V_{-\gamma }^{*k}}\left(x\right)=\overline{{\left(V_{-\gamma }\right)}^{*\sigma }}\left(x\right),x\ge 0,\end{array}$$

(18)

where $\sigma$ is a random variable such that $P(\sigma =k)=q_k$ for all nonnegative integers k satisfying ${\sum }_{k=0}^{\infty }{q}_k=1$.

For any $0<\epsilon <1$, we denote ${\epsilon }_0=\epsilon e^{-\gamma T_0}$. By (15), (18) and (16) replaced $\epsilon$ with ${\epsilon }_0$, according to Fubini Theorem, for the corresponding $n_0=n_0(V,{\epsilon }_0,\tau ,T_0)$ large enough, we have

$$\begin{array}{cc}& \sum _{k=n_0+1}^{\infty }{q}_k\overline{V_{-\gamma }^{*(k-1)}}\left(x\right)=\sum _{k=n_0+1}^{\infty }{q}_k\sum _{m=0}^{\infty }{V}_{-\gamma }^{*(k-1)}(x+m{T}_0+{\Delta }_{T_0})\hfill \\ \le & \sum _{k=n_0+1}^{\infty }{p}_k{M}^{k-1}(V,-\gamma )M^{k-1}(V_{-\gamma },\gamma )\sum _{m=0}^{\infty }{e}^{-\gamma (x+m{T}_0)}{V}^{*(k-1)}(x+m{T}_0+{\Delta }_{T_0})/M(V^{*\tau },-\gamma )\hfill \\ =& \sum _{k=n_0+1}^{\infty }{p}_k\sum _{m=0}^{\infty }{e}^{-\gamma (x+m{T}_0)}{V}^{*(k-1)}(x+m{T}_0+{\Delta }_{T_0})/M(V^{*\tau },-\gamma )\hfill \\ =& \sum _{m=0}^{\infty }{e}^{-\gamma (x+m{T}_0)}\sum _{k=n_0+1}^{\infty }{p}_k{V}^{*(k-1)}(x+m{T}_0+{\Delta }_{T_0})/M(V^{*\tau },-\gamma )\hfill \\ \le & {\epsilon }_0\sum _{m=0}^{\infty }{e}^{-\gamma (x+m{T}_0)}{V}^{*\tau }(x+m{T}_0+{\Delta }_{T_0})/M(V^{*\tau },-\gamma )\hfill \\ \le & {\epsilon }_0{e}^{\gamma T_0}\sum _{m=0}^{\infty }{\left(V^{*\tau }\right)}_{-\gamma }(x+m{T}_0+{\Delta }_{T_0})\hfill \\ =& \epsilon \overline{{\left(V^{*\tau }\right)}_{-\gamma }}\left(x\right)\hfill \\ =& \epsilon \overline{{\left(V_{-\gamma }\right)}^{*\sigma }}\left(x\right),x\ge 0.\hfill \end{array}$$

(19)

Since ${\left(V^{*\tau }\right)}_{-\gamma }\in \mathcal{L}\left(\gamma \right)\cap \mathcal{OS}$, according to Theorem 2.1 with $\gamma >0$ of Cui et al. [29], by (19) and (17), we have $V_{-\gamma }^{*n}\in \mathcal{L}\left(\gamma \right)\cap \mathcal{OS}$ for all $n\ge n_0$. Thus, according to Lemma 1, $V^{*n}\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ for all $n\ge n_0$.

Let $l_0=min\{n:V^{*n}\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}\}$. Then $1\le l_0\le n_0$. According to Lemma 1, by $V^{*l_0}\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, we know that $V_{-\gamma }^{*l_0}\in \mathcal{L}\left(\gamma \right)\cap \mathcal{OS}$. Furthermore, according to Theorem 3 of Embrechts and Goldie [15] and Proposition 2.6 of Shimura and Watanabe [13], we have $V_{-\gamma }^{*n}\in \mathcal{L}\left(\gamma \right)\cap \mathcal{OS}$ for all $n\ge l_0$. Therefore, $V^{*n}\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ for all $n\ge l_0$ after using Lemma 1 again.

Similarly, we can prove $V^{*n}\notin {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ for all $1\le n\le l_0-1$.

In particular, if $V\in {\mathcal{OS}}_{loc}$, then $V_{-\gamma }\in \mathcal{OS}$. Thus, according to Theorem 2.1 with $\gamma >0$ of Cui et al. [29], we have $V_{-\gamma }\in \mathcal{L}\left(\gamma \right)$, which implies $V\in {\mathcal{L}}_{loc}$. Therefore, $l_0=1$, that is $V^{*n}\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ for all $n\ge 1$.

Next, we prove the theorem for the Case 2 that $1\le m<\infty$ and $p_m>0$.

Because

$$\overline{{\left(V^{*\tau }\right)}_{-\gamma }}\left(x\right)\ge q_m\overline{V_{-\gamma }^{*m}}\left(x\right)\ge q_m\overline{V_{-\gamma }^{*k}}\left(x\right),1\le k\le m-1,$$

$\overline{{\left(V^{*\tau }\right)}_{-\gamma }}\left(x\right)\asymp \overline{V_{-\gamma }^{*m}}\left(x\right)$. Then by ${\left(V^{*\tau }\right)}_{-\gamma }\in \mathcal{OS}$, we immediately get $V_{-\gamma }^{*m}\in \mathcal{OS}$. Consequently, there is an integer $l_0=min\{1\le n\le n_0:V_{-\gamma }^{*n}\in \mathcal{OS}\}$ such that $1\le l_0\le m$ and $V_{-\gamma }^{*l_0}\in \mathcal{OS}$. According to Proposition 2.6 of Shimura and Watanabe [13], $V_{-\gamma }^{*n}\in \mathcal{OS}$ and $\overline{{\left(V^{*\tau }\right)}_{-\gamma }}\left(x\right)\asymp \overline{V_{-\gamma }^{*n}}\left(x\right)$ for all $l_0\le n\le m$. Thus, for each $n\ge l_0$, there is a constant $D_n=D_n(V,\tau )>0$ such that

$$lim\; sup\overline{{\left(V^{*\tau }\right)}_{-\gamma }}\left(x\right)/\overline{V_{-\gamma }^{*n}}\left(x\right)=D_n<\infty .$$

Further, we prove $V_{-\gamma }^{*n}\in \mathcal{L}\left(\gamma \right)$ for each $l_0\le n\le m$. Since ${\left(V^{*\tau }\right)}_{-\gamma }\in \mathcal{L}\left(\gamma \right)$, for any $0<\epsilon <1$ and each $t>0$, there is a constant $x_1=x_1(V_{-\gamma },\tau ,\epsilon ,t)>t$ such that, for all $x>x_1$,

$$\begin{array}{ccc}& & \epsilon \overline{{\left(V^{*\tau }\right)}_{-\gamma }}\left(x\right)\ge \overline{{\left(V^{*\tau }\right)}_{-\gamma }}(x-t)-e^{\gamma t}\overline{{\left(V^{*\tau }\right)}_{-\gamma }}\left(x\right)\hfill \\ & =& \left(\sum _{1\le k\ne n\le m}+\sum _{k=n}\right)p_k\left(\overline{V_{-\gamma }^{*k}}(x-t)-e^{\gamma t}\overline{V_{-\gamma }^{*k}}\left(x\right)\right)\hfill \\ & \ge & -\epsilon e^{\gamma t}\sum _{1\le k\ne n\le n_0}{p}_k\overline{V_{-\gamma }^{*k}}\left(x\right)+p_n\left(\overline{V_{-\gamma }^{*n}}(x-t)-e^{\gamma t}\overline{V_{-\gamma }^{*n}}\left(x\right)\right)\hfill \\ & \ge & p_n\left(\overline{V_{-\gamma }^{*n}}(x-t)-e^{\gamma t}\overline{V_{-\gamma }^{*n}}\left(x\right)\right)-\epsilon e^{\gamma t}\overline{{\left(V^{*\tau }\right)}_{-\gamma }}\left(x\right),\hfill \end{array}$$

which implies that for all $x>x_1$,

$$\begin{array}{c}\hfill \overline{V_{-\gamma }^{*n}}(x-t)\le e^{\gamma t}\overline{V_{-\gamma }^{*n}}\left(x\right)+(1+e^{\gamma t})\epsilon \overline{{\left(V^{*\tau }\right)}_{-\gamma }}\left(x\right)/p_n.\end{array}$$

Hence,

$$\begin{array}{c}\hfill lim\; sup\overline{V_{-\gamma }^{*n}}(x-t)/\overline{V_{-\gamma }^{*n}}\left(x\right)\le e^{\gamma t}+(1+2e^{\gamma t})\epsilon D_n/p_n.\end{array}$$

(20)

Clearly, the fixed integer n is independent of $\epsilon$. Thus, combined with the arbitrariness of $\epsilon$, (20) and (17) lead to $V_{-\gamma }^{*n}\in \mathcal{L}\left(\gamma \right)$.

In particular, if $V_{-\gamma }\in \mathcal{OS}$, then by the same method, we can get $V_{-\gamma }^{*n}\in \mathcal{L}(O)\cap \mathcal{OS}$ for all $1\le n\le m$. □

Secondly, we consider the closure under convolution roots for the distribution class ${\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$.

Lemma 2.

Let $G_1$ be a distribution, $G_2=V^{*\tau }$ as above and $G=G_1*G_2$. Assume that for any $0<\epsilon <1$, there exists an integer $n_0=n_0(V,\epsilon ,\tau )\ge 1$ such that

$$\begin{array}{c}\hfill \sum _{k=n_0+1}^{\infty }{p}_k\overline{V_{-\gamma }^{*(k-1)}}\left(x\right)\le \epsilon \overline{{\left(V^{*\tau }\right)}_{-\gamma }}\left(x\right),x\ge 0.\end{array}$$

(21)

Further, suppose that (17) is satisfied for all $k\ge 1$ and

$$\begin{array}{c}\hfill \overline{G_{1,-\gamma }}\left(x\right)=o\left(\overline{G_{2,-\gamma }}\left(x\right)\right)\end{array}$$

(22)

If $G\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, then

$$G_2\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}\mathrm{and}{G}_2(x+{\Delta }_{T_0})\sim G(x+{\Delta }_{T_0}).$$

Proof. 

According to Lemma 1, by $G\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, we know that

$$G_{-\gamma }=G_{1,-\gamma }*G_{2,-\gamma }\in \mathcal{L}\left(\gamma \right)\cap \mathcal{OS}.$$

Thus, according to Lemma 3.1 of Cui et al. [29], by (17) for all $k\ge 1$, (21) for any given $0<\epsilon <1$ and (22), we have

$$G_{2,-\gamma }\in \mathcal{L}\left(\gamma \right)\cap \mathcal{OS}\mathrm{and}\overline{G_{-\gamma }}\left(x\right)\sim M(G_{1,-\gamma },\gamma )\overline{G_{2,-\gamma }}\left(x\right).$$

Therefore, according to Lemma 1 and Proposition 1, by (7), we can prove the lemma. □

Now, we prove Theorem 2.

$\left(i\right)$ Firstly, we prove

$$\begin{array}{c}\hfill \overline{H_{1,-\gamma }}\left(x\right)=o\left(\overline{H_{2,-\gamma }}\left(x\right)\right).\end{array}$$

(23)

To the end, we denote

$$\overline{H_{-\gamma }}\left(x\right)=\overline{H_{-\gamma }}(ln{e}^x)=\overline{H_{-\gamma }}(lny)=f_{-\gamma }\left(y\right).$$

According to Lemma 1, by $H\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, we have $H_{-\gamma }\in \mathcal{L}\left(\gamma \right)\cap \mathcal{OS}$. Thus, $f_{-\gamma }(·)$ is a regular variation function with index $\gamma$, which implies

$$\begin{array}{c}\hfill e^{\beta x}\overline{H_{-\gamma }}\left(x\right)\to \infty \mathrm{for}\mathrm{each}\beta >\gamma .\end{array}$$

(24)

By $\overline{H_1}\left(x\right)=O\left(e^{-\beta x}\right)$ for each $\beta >0$, we have

$$\begin{array}{ccc}\hfill & & e^{\beta x}\overline{H_{1,-\gamma }}\left(x\right)\le e^{\beta x}\overline{H_1}\left(x\right)/M(H_1,\gamma )\to 0\mathrm{for}\mathrm{each}\beta >0.\hfill \end{array}$$

(25)

For $i=1,2$, let $X_i$ be a random variable with distribution $H_{i,-\gamma }$. Then

$$\overline{H_{-\gamma }}\left(x\right)=\overline{H_{1,-\gamma }*H_{2,-\gamma }}\left(x\right)\le P\left(max\{X_1,X_2\}>x/2\right)\le \overline{H_{1,-\gamma }}(x/2)+\overline{H_{2,-\gamma }}(x/2).$$

Thus, by (24) and (25), we know that

$$\begin{array}{c}\hfill e^{\beta x}\overline{H_{2,-\gamma }}\left(x\right)=e^{\left(2^{-1}\beta \right)2x}\overline{H_{2,-\gamma }}(2x/2)\to \infty \mathrm{for}\mathrm{each}\beta >2\gamma .\end{array}$$

(26)

Combining with (25) and (26), we know that (23) holds.

Secondly, by (18), according to Proposition 6.1 of Watanabe and Yamamuro [33], we have

$$q_k=p_k{M}^k(F_{-\gamma },-\gamma )/M(H_2,-\gamma )=e^{-\mu }{\mu }^k{M}^k(F_{-\gamma },-\gamma )/\left(M(H_2,-\gamma )k!\right)\mathrm{for}\mathrm{all}k\ge 0.$$

Thus, for any $0<\epsilon <1$, there exists an integer $n_0=n_0(F_{-\gamma },H_{2,-\gamma },\epsilon )\ge 1$ such that

$$\begin{array}{c}\hfill \sum _{k=n_0+1}^{\infty }{q}_k\overline{F_{-\gamma }^{*(k-1)}}\left(x\right)\le \epsilon \overline{{\left(F^{*\tau }\right)}_{-\gamma }}\left(x\right)=\epsilon \overline{H_{2,-\gamma }}\left(x\right),x\ge 0.\end{array}$$

(27)

Finally, according to Lemma 2 replaced $G_i$ with $H_i,i=1,2$, combining with (8), (23) and (27), by $H\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, we know that

$$H_2\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}\mathrm{and}{H}_2(x+{\Delta }_{T_0})\sim H(x+{\Delta }_{T_0}).$$

$\left(ii\right)$ In Theorem 4, we take $V=F$ and $V^{*\tau }=H_2$. Clearly, (16) holds for each distribution $V=F$, any $0<\epsilon <1$ and some $n_0\ge 1$, see, for example, Watanabe and Yamamuro [33]. Further, according to Theorem 4, by $H_2\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, combined with (8) and (16), we obtain all the conclusions.

3.3. Proof of Theorem 3

In order to prove the theorem, we need the following two results. The first result is the local version of the half of Lemma 2.1 of Cui et al. [29].

Lemma 3.

Let $V^{*\tau }$ be a random convolution defined as above.

$\left(i\right)$ If $p_{k-1}\ge p_k>0$ for all $k\ge 2$, then the following proposition $\left(B\right)$ implies the proposition $\left(A\right)$ for some $0<T<\infty$.

$\left(A\right)$ For any $0<\epsilon <1$, there exists an integer $n_0=n_0(V,\tau ,\epsilon ,T)\ge 1$ such that

$$\begin{array}{c}\hfill \sum _{k=n_0+1}^{\infty }{p}_k{V}^{*k}(x+{\Delta }_T)\le \epsilon V^{*\tau }(x+{\Delta }_T),x\ge 0.\end{array}$$

(28)

$\left(B\right)$ For any $0<\epsilon <1$, there exists an integer $n_0=n_0(V,\tau ,\epsilon ,T)\ge 1$ such that (16) holds.

$\left(ii\right)$ If $V^{*\tau }\in {\mathcal{OS}}_{{\Delta }_T}$ for some $0<T<\infty$ with $p_1>0$, then the proposition $\left(B\right)$ implies the proposition $\left(A\right)$ replaced $x\ge 0$ by $x\ge x_1$ for some $x_1\ge x_0$.

Remark 3.$\left(i\right)$ In particular, if τ obeys a Poisson distribution, then for any $0<\epsilon <1$, (16) holds for some $n_0\ge 1$. Further, because $p_{k-1}\ge p_k>0$ for all $k\ge 2$, (28) holds for the same ε and $n_0$.

$\left(ii\right)$ The condition $0<p_k\le p_{k-1}$ for all $k\ge 2$ can be relaxed to the condition that $0<p_k\le C{p}_{k-1}$ for some $C>0$ and all $k\ge 2$.

Proof. 

$\left(i\right)$ If (16) holds, then by $p_{k-1}\ge p_k>0$ for all $k\ge 2$, we know that for any $n_0\ge 1$,

$$\sum _{k=n_0+1}^{\infty }{p}_k{V}^{*k}(x+{\Delta }_T)\le \sum _{k=n_0+1}^{\infty }{p}_{k-1}{V}^{*k}(x+{\Delta }_T),x\ge 0.$$

Therefore, (28) is implied by (16).

$\left(ii\right)$ Clearly, we only need to prove the lemma for Case 1. Because $V^{*\tau }\in {\mathcal{OS}}_{{\Delta }_T}$ for some $0<T<\infty$, there exists a constant $x_1=x_1(V,\tau ,T)\ge x_0$ such that

$$D^{*}(V^{*\tau },T)=\underset{x\ge x_1}{sup}{\left(V^{*\tau }\right)}^{*2}(x+{\Delta }_T)/V^{*\tau }(x+{\Delta }_T)<\infty .$$

For any $0<\epsilon <1$, we take

$${\epsilon }_0=p_1\epsilon /\left(1+D^{*}(V^{*\tau },T)\right),$$

then $0<{\epsilon }_0<1$.

For the above ${\epsilon }_0$, according to proposition $\left(B\right)$, by $m=\infty$, there exists an integer $n_0=n_0(V,\tau ,{\epsilon }_0,T)\ge 1$ such that $0<a_{n_0}={\displaystyle \sum _{k=n_0+1}^{\infty }}{p}_k<{\epsilon }_0$ and (16) holds, in which $\epsilon$ is replaced with ${\epsilon }_0$. Then ${\displaystyle \sum _{k=n_0+1}^{\infty }}{p}_k{V}^{*(k-1)}/a_{n_0}$ can be considered as a distribution. Therefore, by $p_1>0$ and $V^{*\tau }(x+{\Delta }_T)\ge p_{1}V(x+{\Delta }_T)$ for all $x\ge 0$, we have

$$\begin{array}{ccc}& & \sum _{k=n_0+1}^{\infty }{p}_k{V}^{*k}(x+{\Delta }_T)=a_{n_0}V*\left(\sum _{k=n_0+1}^{\infty }{p}_k{V}^{*(k-1)}/a_{n_0}\right)(x+{\Delta }_T)\hfill \\ & \le & a_{n_0}{\int }_{0-}^x\left(\sum _{k=n_0+1}^{\infty }{p}_k{V}^{*(k-1)}/a_{n_0}\right)(x-y+{\Delta }_T)V\left(dy\right)+a_{n_0}V(x+{\Delta }_T)\hfill \\ & \le & {\epsilon }_0\left({\int }_{0-}^x{V}^{*\tau }(x-y+{\Delta }_T)V^{*\tau }\left(dy\right)+V^{*\tau }(x+{\Delta }_T)\right)/p_1\hfill \\ & \le & {\epsilon }_0\left(V^{*2\tau }(x+{\Delta }_T)+V^{*\tau }(x+{\Delta }_T)\right)/p_1\hfill \\ & \le & {\epsilon }_0\left(1+D^{*}(V^{*\tau },T)\right)V^{*\tau }(x+{\Delta }_T)/p_1\hfill \\ & =& \epsilon V^{*\tau }(x+{\Delta }_T),x\ge x_1,\hfill \end{array}$$

that is (28) holds for any $0<\epsilon <1$, all $x\ge x_1$ and some $n_0\ge 1$. □

Theorem 5.

Assume that $V^{*\tau }\in {\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$ for some $0<\gamma ,T_0<\infty$ with $p_k>0$ for all $k\ge 1$ in Case 1 or $1\le k\le m$ in Case 2. If for any $0<\epsilon <1$, there exists an integer $n_0=n_0(V,\tau ,\epsilon ,T_0)\ge 1$ such that (16) holds, and for each the above k,

$$\begin{array}{c}\hfill lim\; inf{V}_{\gamma }^{*k}(x-t+{\Delta }_{T_0})/V_{\gamma }^{*k}(x+{\Delta }_{T_0})\ge 1foreacht>0,\end{array}$$

(29)

then there exists an integer $l_0\ge 1$ in Case 1 or $1\le l_0\le m$ in Case 2 such that $V^{*n}\in {\mathcal{TL}}_{{\Delta }_{T_0}}\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$ for all $n\ge l_0$ in Case 1 or $l_0\le n\le m$ in Case 2 and $V^{*n}\notin {\mathcal{TL}}_{{\Delta }_{T_0}}\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$ for all $1\le n\le l_0-1$. In particular, if $V\in {\mathcal{OS}}_{{\Delta }_{T_0}}$, then $V^{*n}\in {\mathcal{TL}}_{{\Delta }_{T_0}}\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$ for all $n\ge 1$ in Case 1 or $1\le n\le m$ in Case 2.

Proof. 

For case 1, we first prove $V^{*n}\in {\mathcal{OS}}_{{\Delta }_{T_0}}$ for all $n\ge n_0$, where $n_0$ fixed in (16). Because $V^{*\tau }\in {\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)$, $M(V^{*\tau },\gamma )<\infty$. Thus, $M(V^{*n},\gamma )<\infty$ for all $n\ge 1$. By (14), it holds that,

$$\begin{array}{c}\hfill V^{*\theta }(x+{\Delta }_{T_0})=\frac{M(V^{*\theta },\gamma )}{e^{\gamma x}}\left({\left(V^{*\theta }\right)}_{\gamma }(x+{\Delta }_{T_0})-\gamma {\int }_0^{T_0}\frac{{\left(V^{*\theta }\right)}_{\gamma }(x+y,x+T_0]}{e^{\gamma y}}dy\right),x\ge 0,\end{array}$$

(30)

where $\theta =k$ for each $k\ge 1$ or $\theta =\tau$. Thus, similar to (15), we have

$$\begin{array}{c}\hfill e^{-\gamma T_0}{\left(V^{*\theta }\right)}_{\gamma }(x+{\Delta }_{T_0})\le e^{\gamma x}{V}^{*\theta }(x+{\Delta }_{T_0})/M(V^{*\theta },\gamma )\le {\left(V^{*\theta }\right)}_{\gamma }(x+{\Delta }_{T_0}),x\ge 0.\end{array}$$

(31)

When $\theta =\tau$, just as (18), we denote

$$\begin{array}{c}\hfill {\left(V^{*\tau }\right)}_{\gamma }(x+{\Delta }_{T_0})=\sum _{k=1}^{\infty }{q}_k{V}_{\gamma }^{*k}(x+{\Delta }_{T_0})={\left(V_{\gamma }\right)}^{*\sigma }(x+{\Delta }_{T_0}),x\ge 0,\end{array}$$

(32)

where $q_k=p_k{M}^k(V,\gamma )/M(V^{*\tau },\gamma ),k\ge 1$. According to Proposition 1 and Lemma 1, since $V^{*\tau }\in {\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$, so ${\left(V^{*\tau }\right)}_{\gamma }\in {\mathcal{L}}_{{\Delta }_{T_0}}\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$. Furthermore, according to Lemma 3, by (31), (28) with $x_1\ge x_0$ and (32), for $0<\epsilon <1$ and $n_0$ in (16), we have

$$\begin{array}{ccc}\hfill & & \sum _{k=1}^{n_0}{q}_k{V}_{\gamma }^{*k}(x+{\Delta }_{T_0})\ge e^{\gamma x}\sum _{k=1}^{n_0}{p}_k{V}^{*k}(x+{\Delta }_{T_0})/M(V^{*\tau },\gamma )\hfill \\ & \ge & (1-\epsilon )e^{\gamma x}{V}^{*\tau }(x+{\Delta }_{T_0})/M(V^{*\tau },\gamma )\hfill \\ & \ge & (1-\epsilon )e^{-\gamma T_0}{\left(V^{*\tau }\right)}_{\gamma }(x+{\Delta }_{T_0}),x\ge x_1.\hfill \end{array}$$

(33)

Further, for each $n\ge 1$, using Fatou lemma, by (29), we have

$$\begin{array}{ccc}\hfill lim\; inf\frac{V_{\gamma }^{*(n+1)}(x+{\Delta }_{T_0})}{V_{\gamma }^{*n}(x+{\Delta }_{T_0})}& \ge & {\int }_0^{\infty }lim\; inf\frac{V_{\gamma }^{*n}(x-y+{\Delta }_{T_0})}{V_{\gamma }^{*n}(x+{\Delta }_{T_0})}{\mathbf{1}}_{[0,x]}\left(y\right)V_{\gamma }\left(dy\right)\ge 1.\hfill \end{array}$$

(34)

Combining with (33), (34) and ${\left(V^{*\tau }\right)}_{\gamma }\in {\mathcal{OS}}_{{\Delta }_{T_0}}$, we know that

$$\begin{array}{c}\hfill V_{\gamma }^{*n}(x+{\Delta }_{T_0})\asymp {\left(V^{*\tau }\right)}_{\gamma }(x+{\Delta }_{T_0})\mathrm{and}{V}_{\gamma }^{*n}\in {\mathcal{OS}}_{{\Delta }_{T_0}}\mathrm{for}\mathrm{all}n\ge n_0.\end{array}$$

(35)

Using Lemma 1 again, by (31), we have

$$V^{*n}(x+{\Delta }_{T_0})\asymp V^{*\tau }(x+{\Delta }_{T_0}).$$

Therefore, by $V^{*\tau }\in {\mathcal{OS}}_{{\Delta }_{T_0}}$, we know that $V^{*n}\in {\mathcal{OS}}_{{\Delta }_{T_0}}$ for all $n\ge n_0$.

Next, we prove that $V^{*n}\in {\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)$ for each $n\ge n_0$. According to Lemma 3 $\left(ii\right)$, by (31), (16) and (32), for the above $0<\epsilon <1,n_0$, $x_1\ge x_0$ and each $n\ge n_0$, there exists an integer $m_0=m_0(F,\tau ,\epsilon ,T_0,\gamma )\ge n$ such that

$$\begin{array}{ccc}\hfill & & \sum _{k=m_0+1}^{\infty }{q}_k{V}_{\gamma }^{*k}(x+{\Delta }_{T_0})\le e^{\gamma (T_0+x)}\sum _{k=m_0+1}^{\infty }{p}_k{V}^{*k}(x+{\Delta }_{T_0})/M(V^{*\tau },\gamma )\hfill \\ & \le & e^{\gamma (T_0+x)}\epsilon e^{-\gamma T_0}{V}^{*\tau }(x+{\Delta }_{T_0})/M(V^{*\tau },\gamma )\hfill \\ & \le & \epsilon {\left(V^{*\tau }\right)}_{\gamma }(x+{\Delta }_{T_0})\hfill \\ & =& \epsilon {\left(V_{\gamma }\right)}^{*\sigma }(x+{\Delta }_{T_0})\mathrm{for}\mathrm{all}x\ge x_1.\hfill \end{array}$$

(36)

Further, by ${\left(V^{*\tau }\right)}_{\gamma }\in {\mathcal{L}}_{{\Delta }_{T_0}}\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$, (29) and (36), for each $t>0$, there exists a constant $x_2=x_2(V,\tau ,\epsilon ,t,m_0,\gamma )\ge x_1$ such that, for all $x\ge x_2$,

$$\begin{array}{ccc}& & \epsilon {\left(V^{*\tau }\right)}_{\gamma }(x+{\Delta }_{T_0})\ge {\left(V^{*\tau }\right)}_{\gamma }(x-t+{\Delta }_{T_0})-{\left(V^{*\tau }\right)}_{\gamma }(x+{\Delta }_{T_0})\hfill \\ & =& \left(\sum _{1\le k\ne n\le m_0}+\sum _{k=n}+\sum _{k\ge m_0+1}\right)q_k\left(V_{\gamma }^{*k}(x-t+{\Delta }_{T_0})-V_{\gamma }^{*k}(x+{\Delta }_{T_0})\right)\hfill \\ & \ge & -\epsilon \sum _{1\le k\le m_0}{q}_k{V}_{\gamma }^{*k}(x+{\Delta }_{T_0})+q_n\left(V_{\gamma }^{*n}(x-t+{\Delta }_{T_0})-V_{\gamma }^{*n}(x+{\Delta }_{T_0})\right)-\epsilon {\left(V^{*\tau }\right)}_{\gamma }(x+{\Delta }_{T_0}),\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill V_{\gamma }^{*n}(x-t+{\Delta }_{T_0})\le V_{\gamma }^{*n}(x+{\Delta }_{T_0})+3\epsilon {\left(V^{*\tau }\right)}_{\gamma }(x+{\Delta }_{T_0})/q_n,x\ge x_2.\end{array}$$

Hence, by ${\left(V^{*\tau }\right)}_{\gamma }(x+{\Delta }_{T_0})\asymp V_{\gamma }^{*n}(x+{\Delta }_{T_0})$ and the arbitrariness of $\epsilon$, we can get

$$\begin{array}{c}\hfill lim\; sup{V}_{\gamma }^{*n}(x-t+{\Delta }_{T_0})/V_{\gamma }^{*n}(x+{\Delta }_{T_0})\le 1.\end{array}$$

(37)

Combined with (29) and (37), $V_{\gamma }^{*n}\in {\mathcal{L}}_{{\Delta }_{T_0}}$. Therefore, $V^{*n}\in {\mathcal{TL}}_{{\Delta }_{T_0}\left(\gamma \right)}$, for all $n\ge n_0$.

Similar to the proof of Theorem 4, the theorem can be proved.

For Case 2, by (34), we have $V_{\gamma }^{*m}(x+{\Delta }_{T_0})\asymp {\left(V^{*\tau }\right)}_{\gamma }(x+{\Delta }_{T_0})$. Then, it is easy to get that $V_{\gamma }^{*m}\in {\mathcal{OS}}_{{\Delta }_{T_0}}$.

Next, we prove that $V^{*m}\in {\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)$. For any $0<\epsilon <1$ and x large enough, by (29) for $1\le k\le m$, we can get

$$\begin{array}{ccc}& & \epsilon {\left(V^{*\tau }\right)}_{\gamma }(x+{\Delta }_{T_0})\ge {\left(V^{*\tau }\right)}_{\gamma }(x-t+{\Delta }_{T_0})-{\left(V^{*\tau }\right)}_{\gamma }(x+{\Delta }_{T_0})\hfill \\ & =& \left(\sum _{1\le k<m}+\sum _{k=m}\right)q_k\left(V_{\gamma }^{*k}(x-t+{\Delta }_{T_0})-V_{\gamma }^{*k}(x+{\Delta }_{T_0})\right)\hfill \\ & \ge & -\epsilon \sum _{1\le k<m}{q}_k{V}_{\gamma }^{*k}(x+{\Delta }_{T_0})+q_m\left(V_{\gamma }^{*m}(x-t+{\Delta }_{T_0})-V_{\gamma }^{*m}(x+{\Delta }_{T_0})\right)\mathrm{for}\mathrm{each}t>0.\hfill \end{array}$$

After the same simplification, we have

$$\begin{array}{c}\hfill V_{\gamma }^{*m}(x-t+{\Delta }_{T_0})\le V_{\gamma }^{*m}(x+{\Delta }_{T_0})+2\epsilon {\left(V^{*\tau }\right)}_{\gamma }(x+{\Delta }_{T_0})/q_m\mathrm{for}\mathrm{each}t>0.\end{array}$$

Hence, we can obtain the same conclusion as (37) for $n=m$ which implies $V^{*m}\in {\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)$.

We omit the proof of the remaining conclusion, which is similar to that of Theorem 4. □

Now, we prove Theorem 3.

$\left(i\right)$ Firstly, we prove that

$$\begin{array}{c}\hfill H_{1,\gamma }(x+{\Delta }_{T_0})=o\left(H_{2,\gamma }(x+{\Delta }_{T_0})\right).\end{array}$$

(38)

Its proof is slightly more difficult than that of (23). For this, we denote

$$H_{\gamma }(x+{\Delta }_{T_0})=H_{\gamma }(ln{e}^x+{\Delta }_{T_0})=H_{\gamma }(lny+{\Delta }_{T_0})=f_{\gamma }\left(y\right),x\ge 0.$$

According to Lemma 1, by $H\in {\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$, we have $H_{\gamma }\in {\mathcal{L}}_{{\Delta }_{T_0}}\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$. Thus, $f_{\gamma }(·)$ is a regular variation function with index 0, which implies

$$\begin{array}{c}\hfill e^{\beta x}{H}_{\gamma }(x+{\Delta }_{T_0})\to \infty \mathrm{for}\mathrm{each}\beta >0.\end{array}$$

(39)

By $\overline{H_1}\left(x\right)=O\left(e^{-\beta x}\right)$ for each $\beta >0$ and (15) with $V=H_{1,\gamma }$ and $T=T_0$, we have

$$\begin{array}{c}\hfill e^{\beta x}{H}_{1,\gamma }(x+{\Delta }_{T_0})\le e^{\gamma T_0}{M}^{-1}(H_1,\gamma )e^{(\beta +\gamma )x}{H}_1(x+{\Delta }_{T_0})\to 0\mathrm{for}\mathrm{each}\beta >0.\end{array}$$

(40)

Then by (39) and (40), we know that

$$\begin{array}{c}\hfill H_{1,\gamma }(x+{\Delta }_{T_0})=o\left(H_{\gamma }(x+{\Delta }_{T_0})\right).\end{array}$$

(41)

Furthermore, by (10), for each pair $m,k\ge 1$, we have

$$\begin{array}{c}\hfill lim\; inf{F}_{\gamma }^{*k}(x-j{T}_0+{\Delta }_{T_0})/F_{\gamma }^{*k}(x+{\Delta }_{T_0})\ge 1\mathrm{for}\mathrm{each}1\le j\le m.\end{array}$$

(42)

In addition, there exists an integer $n_1=n_1(H_{1,\gamma },T_0)$ large enough such that $H_{1,\gamma }(0,n_1{T}_0]>0$. Then by (41) and $H_{\gamma }\in {\mathcal{L}}_{{\Delta }_{T_0}}\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$, for any

$$0<\epsilon <H_{1,\gamma }(0,n_1{T}_0]/\left(2(n_1+1)C_{{\Delta }_{T_0}}^{*}\left(H_{\gamma }\right)\right),$$

there exists an integer $m_1=m_1(H_1,H_2,\epsilon ,T_0,\gamma )$ and a constant $x_3\ge x_2$ such that

$$\begin{array}{c}\hfill H_{1,\gamma }(x+{\Delta }_{T_0})<\epsilon H_{\gamma }(x+{\Delta }_{T_0}),x\ge m_1{T}_0,\end{array}$$

(43)

$$\begin{array}{c}\hfill \sum _{j=0}^{n_1}{H}_{\gamma }^{*2}(x+j{T}_0+{\Delta }_{T_0})\le 2(n_1+1)C_{{\Delta }_{T_0}}^{*}\left(H_{\gamma }\right)H_{\gamma }(x+{\Delta }_{T_0}),x\ge m_1{T}_0\end{array}$$

(44)

and

$$\begin{array}{c}\hfill \sum _{j=0}^{m_1}{F}_{\gamma }^{*n_0}(x-j{T}_0+{\Delta }_{T_0})\le 2(m_1+1)F_{\gamma }^{*n_0}(x-m_1{T}_0+{\Delta }_{T_0}),x\ge x_3,\end{array}$$

(45)

where the final inequality stems from (42) with $k=n_0$ in (35) and (16). In addition, by (35) with $V=F$, we know that for the above $m_1$ and $n_0$, there are

$$0<C_1=C_1(H_{2,\gamma },T_0,m_1,n_0)<C_2=C_2(H_{2,\gamma },T_0,m_1,n_0)<\infty$$

and $x_4=x_4(H_{2,\gamma },T_0,m_1,n_0)\ge x_3$ such that, for all $1\le j\le m_1$,

$$\begin{array}{c}\hfill C_1{H}_{2,\gamma }(x-j{T}_0+{\Delta }_{T_0})\le F_{\gamma }^{*n_0}(x-j{T}_0+{\Delta }_{T_0})\le C_2{H}_{2,\gamma }(x-j{T}_0+{\Delta }_{T_0}),x\ge x_4.\end{array}$$

(46)

For $i=1,2$, let $X_i$ be a random variable with distribution $H_{i,\gamma }$. Assume that $X_1$ is independent of $X_2$ and $(X_1^{*},X_2^{*})$ is an independent copy of $(X_1,X_2)$. Further, denote $A_0=\{X_1+X_2\in x+{\Delta }_{T_0}\}$ for all $x\ge 0$. We then divide $H_{\gamma }(x+{\Delta }_{T_0})=P\left(A_0\right)$ as follows:

$$\begin{array}{ccc}\hfill & & P\left(A_0\right)=P(A_0,0\le X_2\le x-m_1{T}_0)+P(A_0,x-m_1{T}_0<X_2\le x+T_0)\hfill \\ & =& P_1\left(x\right)+P_2\left(x\right),x\ge 0.\hfill \end{array}$$

(47)

For $P_1\left(x\right)$, by (43), (44) and $H_{\gamma }\in {\mathcal{L}}_{{\Delta }_{T_0}}\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$, we have

$$\begin{array}{ccc}\hfill & & P_1\left(x\right)={\int }_{0-}^{x-m_1{T}_0}{H}_{1,\gamma }(x-y+{\Delta }_{T_0})H_{2,\gamma }\left(dy\right)\hfill \\ & \le & \epsilon {\int }_{0-}^{x-m_1{T}_0}{H}_{\gamma }(x-y+{\Delta }_{T_0})H_{2,\gamma }\left(dy\right)\hfill \\ & =& \epsilon P(X_1^{*}+X_2^{*}+X_2\in x+{\Delta }_{T_0},0\le X_2\le x-m_1{T}_0)\hfill \\ & \le & \epsilon P(x<X_1^{*}+X_2^{*}+X_2\le x+T_0,0\le X_1\le n_2{T}_0)/H_{1,\gamma }\left((0,n_2{T}_0]\right)\hfill \\ & \le & \epsilon P(x<X_1+X_2+X_1^{*}+X_2^{*}\le x+T_0+n_2{T}_0)/H_{1,\gamma }(0,n_2{T}_0]\hfill \\ & =& \epsilon \sum _{j=0}^{n_2}{H}_{\gamma }^{*2}(x+j{T}_0+{\Delta }_{T_0})/H_{1,\gamma }(0,n_2{T}_0]\hfill \\ & \le & 2\epsilon (n_2+1)C_{{\Delta }_{T_0}}^{*}\left(H_{\gamma }\right)H_{\gamma }(x+{\Delta }_{T_0})/H_{1,\gamma }(0,n_2{T}_0],x\ge m_1{T}_0.\hfill \end{array}$$

(48)

For $P_2\left(x\right)$, by (45) and (46), we have

$$\begin{array}{ccc}\hfill & & P_2\left(x\right)\le P(x-x_0<X_2\le x+T_0)=\sum _{j=0}^{m_1}{H}_{2,\gamma }(x-j{T}_0+{\Delta }_{T_0})\hfill \\ & \le & \sum _{j=0}^{m_1}{F}_{\gamma }^{*n_0}(x-j{T}_0+{\Delta }_{T_0})/C_1\hfill \\ & \le & 2(m_1+1)F_{\gamma }^{*n_0}(x-m_1{T}_0+{\Delta }_{T_0})/C_1\hfill \\ & \le & 2C_2(m_1+1)H_{2,\gamma }(x-m_1{T}_0+{\Delta }_{T_0})/C_1,x\ge x_4.\hfill \end{array}$$

(49)

Combined with (47), (48) and (49), we have

$$\begin{array}{c}\hfill \left(1-\frac{2\epsilon (n_2+1)C_{{\Delta }_{T_0}}^{*}\left(H_{\gamma }\right)}{H_{1,\gamma }(0,n_2{T}_0]}\right)H_{\gamma }(x+{\Delta }_{T_0})\le \frac{2C_2(m_1+1)}{C_1}{H}_{2,\gamma }(x-m_1{T}_0+{\Delta }_{T_0})\end{array}$$

(50)

for $x\ge max\{m_1{T}_0,x_4\}$. Furthermore, by (50), (39) and $2\epsilon (n_2+1)C^{*}\left(H_{\gamma }\right)/H_{1,\gamma }(0,n_2{T}_0]<1$, we know that

$$\begin{array}{ccc}& & e^{\beta (x-m_1{T}_0)}{H}_{2,\gamma }(x-m_1{T}_0+{\Delta }_{T_0})=e^{-\beta m_1{T}_0}{e}^{\beta x}{H}_{2,\gamma }(x-m_1{T}_0+{\Delta }_{T_0})\to \infty \mathrm{for}\mathrm{each}\beta >0,\hfill \end{array}$$

that is

$$\begin{array}{c}\hfill e^{\beta x}{H}_{2,\gamma }(x+{\Delta }_{T_0})\to \infty \mathrm{for}\mathrm{each}\beta >0.\end{array}$$

(51)

Then by (15), (40) and (51), it holds that

$$\begin{array}{c}\hfill H_{1,\gamma }(x+{\Delta }_{T_0})/H_{2,\gamma }(x+{\Delta }_{T_0})=e^{\beta x}{H}_{1,\gamma }(x+{\Delta }_{T_0})/e^{\beta x}{H}_{2,\gamma }(x+{\Delta }_{T_0})\to 0\mathrm{for}\mathrm{each}\beta >0,\end{array}$$

and thus (38) holds.

Secondly, by $H_{\gamma }\in {\mathcal{L}}_{{\Delta }_{T_0}}\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$ and (50), we know that

$$H_{2,\gamma }(x+{\Delta }_{T_0})\asymp H_{\gamma }(x+{\Delta }_{T_0})\mathrm{and}{H}_{2,\gamma }\in {\mathcal{OS}}_{{\Delta }_{T_0}}.$$

Finally, we prove that $H_{2,\gamma }\in {\mathcal{L}}_{{\Delta }_{T_0}}$. On one hand, for any $0<\epsilon <1/2$, take $n_0$ in (33) and (16) with $V_{\gamma }=F_{\gamma }$, by (32) and (10), according to Lemma 3 $\left(i\right)$, for each $t>0$, there is a constant $x_5=x_5(F,\epsilon ,t,\gamma )\ge x_4$ such that

$$\begin{array}{ccc}\hfill & & H_{2,\gamma }(x-t+{\Delta }_{T_0})\ge \sum _{k=1}^{n_0}{q}_k{F}_{\gamma }^{*k}(x-t+{\Delta }_{T_0})\hfill \\ & \ge & (1-\epsilon )\sum _{k=1}^{n_0}{q}_k{F}_{\gamma }^{*k}(x+{\Delta }_{T_0})\hfill \\ & \ge & (1-\epsilon )\sum _{k=1}^{\infty }{q}_k{F}_{\gamma }^{*k}(x+{\Delta }_{T_0})-\sum _{k=n_0+1}^{\infty }{q}_k{F}_{\gamma }^{*k}(x+{\Delta }_{T_0})\hfill \\ & \ge & (1-2\epsilon )H_{2,\gamma }(x+{\Delta }_{T_0}),x\ge x_5.\hfill \end{array}$$

(52)

On the other hand, for any $0<\epsilon <1$, each $t>0$ and $n_0$ in (33) with $V_{\gamma }=F_{\gamma }$, by (36) and (10) for all $k\ge 1$, there is a constant $x_6=x_6(F,\epsilon ,t,\gamma )\ge x_5$ such that, when $x\ge x_6$,

$$\begin{array}{c}\hfill \frac{H_{2,\gamma }(x+{\Delta }_{T_0})-H_{2,\gamma }(x-t+{\Delta }_{T_0})}{H_{2,\gamma }(x-t+{\Delta }_{T_0})}\le \sum _{k=1}^{n_0}\left(\frac{F_{\gamma }^{*k}(x+{\Delta }_{T_0})}{F_{\gamma }^{*k}(x-t+{\Delta }_{T_0})}-1\right)+\epsilon \le \epsilon (n_0+1).\end{array}$$

(53)

Combining (52) and (53), with the arbitrariness of $\epsilon$, we know that $H_{2,\gamma }\in {\mathcal{L}}_{{\Delta }_{T_0}}$. Then $\left(i\right)$ holds by Lemma 1 and Proposition 1.

$\left(ii\right)$ In Theorem 5, we take $V=F,G=H,G_1=H_1$ and $G_2=H_2$. Because $p_k=e^{-\mu }{\mu }^k/k!k\ge 0$, according to Remark 3$\left(i\right)$, (16) holds for each distribution V, thus for F. Therefore, since $H_2\in {\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$, according to Theorem 5, by (16) for F and (10), we obtain all the results.

4. On the Condition (10)

In this section, we give some concise and convenient conditions to replace condition (10), see the following Theorem 6. To this end, we require three lemmas.

Lemma 4.

If a distribution $V\in {\mathcal{OL}}_{{\Delta }_T}$ for some $0<T<\infty$ satisfying

$$\begin{array}{c}\hfill lim\; infV(x-t+{\Delta }_T)/V(x+{\Delta }_T)\ge 1foreacht>0,\end{array}$$

(54)

then

$$\begin{array}{c}\hfill V(x-t+{\Delta }_T)\asymp V(x+{\Delta }_T)\asymp V(x+t+{\Delta }_T)foreacht>0\end{array}$$

(55)

and

$$\begin{array}{c}\hfill V(x+{\Delta }_{T_1})=O\left(V(x+{\Delta }_T)\right)foreachpair0<T_1\ne T<\infty .\end{array}$$

(56)

Proof. 

Firstly, by (54) and $V\in {\mathcal{OL}}_{{\Delta }_T}$, we know that, for each $t>0$,

$$V(x+{\Delta }_T)\lesssim V(x-t+{\Delta }_T)\lesssim C_{{\Delta }_T}(V,t)V(x+{\Delta }_T),$$

that is $V(x-t+{\Delta }_T)\asymp V(x+{\Delta }_T)$. Thus

$$V(x+{\Delta }_T)=V(x+t-t+{\Delta }_T)\asymp V(x+t+{\Delta }_T).$$

Therefore, (55) holds.

Secondly, for each $0<T_1\ne T<\infty$, there exists an integer $m\ge 1$ such that $(m-1)T<T_1\le mT$. Further, by $V\in {\mathcal{OL}}_{{\Delta }_T}$ and $V(x+{\Delta }_T)\asymp V(x+t+{\Delta }_T)$ for each $t>0$, we have

$$V(x+{\Delta }_{T_1})\le V(x+{\Delta }_{mT})=\sum _{k=0}^{m-1}V(x+kT+{\Delta }_T)\lesssim \sum _{k=0}^{m-1}{C}_{{\Delta }_T}^{*}(V,kT)V(x+{\Delta }_T),$$

that is (56) holds. □

Lemma 5.

For $i=1,2$, let $V_i$ be a distribution such that $V_i\in {\mathcal{OL}}_{{\Delta }_T}$ for some $0<T<\infty$ and

$$\begin{array}{c}\hfill lim\; inf{V}_i(x-t+{\Delta }_T)/V_i(x+{\Delta }_T)\ge 1foreacht>0.\end{array}$$

(57)

$$\begin{array}{c}\hfill \left(i\right)Then{V}_i(x+{\Delta }_T)=O\left(V_1\ast V_2(x+{\Delta }_T)\right),i=1,2.\end{array}$$

(58)

$\left(ii\right)$ If

$$\begin{array}{c}\hfill lim{C}_{{\Delta }_T}^{\ast }(V_1,x)V_2(x+{\Delta }_T)=0,\end{array}$$

(59)

then $V_1\ast V_2\in {\mathcal{OL}}_{{\Delta }_T}$ and

$$\begin{array}{c}\hfill C_{{\Delta }_T}^{*}(V_1*V_2,t)\le max\{C_{{\Delta }_T}^{*}(V_1,t),C_{{\Delta }_T}^{*}(V_2,t)\}foreacht\ge 0.\end{array}$$

(60)

Proof. 

$\left(i\right)$ For any $0<A<\infty$, according to Fatou lemma, by (57), we have

$$lim\; inf\frac{V_1*V_2(x+{\Delta }_T)}{V_i(x+{\Delta }_T)}\ge {\int }_{0-}^{A}lim\; inf\frac{V_i(x-y+{\Delta }_T)}{V_i(x+{\Delta }_T)}{V}_j\left(dy\right)\ge V_j\left([0,A]\right)\to 1,\mathrm{as}A\to \infty ,$$

for all $1\le i\ne j\le 2$, that is (58) holds.

$\left(ii\right)$ In order to prove (60), we perform some preparatory work.

For each $t>0$, any $0<\epsilon <1$ and $i=1,2$, by $V_i\in {\mathcal{OL}}_{{\Delta }_T}$, there exists $x_i=x_i(V_i,\epsilon ,T,t)>0$ such that

$$\begin{array}{c}\hfill V_i(x-t+{\Delta }_T)\le (1+\epsilon )C_{{\Delta }_T}^{*}(V_i,t)V_i(x+{\Delta }_T)\mathrm{for}\mathrm{all}x\ge x_i.\end{array}$$

(61)

For the above $\epsilon$, by (59), there exists $x_3=x_3(V_i,\epsilon ,T)>0$ such that when $x\ge x_3$,

$$\begin{array}{c}\hfill C_{{\Delta }_T}^{*}(V_1,x)V_2(x+{\Delta }_T)<\epsilon .\end{array}$$

(62)

For above $t>0$, by (56), (61), (62) and (58), we know that there is $x_0=x_0(V_1,V_2,\epsilon ,T,t)\ge max\{x_1,x_2,x_3\}$, $0<K_i=K_i(V_1,V_2,T)<\infty ,i=1,2$ and $m=m(V_1,V_2,T,\epsilon ,t)\ge 2$ such that, when $x\ge m{x}_0$,

$$\begin{array}{ccc}\hfill & & V_1(x-x_0-t-T+{\Delta }_{2T})V_2(x_0+{\Delta }_T)\le K_1{V}_1(x-x_0-t-T+{\Delta }_T)V_2(x_0+{\Delta }_T)\hfill \\ & \le & K_1{(1+\epsilon )}^3{C}_{{\Delta }_T}^{*}(V_1,T)C_{{\Delta }_T}^{*}(V_1,t)C_{{\Delta }_T}^{*}(V_1,x_0)V_1(x+{\Delta }_T)V_2(x_0+{\Delta }_T)\hfill \\ & \le & K_1{K}_2{(1+\epsilon )}^3{C}_{{\Delta }_T}^{*}(V_1,T)C_{{\Delta }_T}^{*}(V_1,t)C_{{\Delta }_T}^{*}(V_1,x_0)V_2(x_0+{\Delta }_T)V_1\ast V_2(x+{\Delta }_T)\hfill \\ & <& \epsilon {(1+\epsilon )}^{3}K{V}_1\ast V_2(x+{\Delta }_T),\hfill \end{array}$$

(63)

where $K=K_1{K}_2{C}_{{\Delta }_T}^{*}(V_1,T)C_{{\Delta }_T}^{*}(V_1,t)$. In addition, let X and Y be the two random variables with corresponding distributions $V_1$ and $V_2$. Suppose that X is independent of Y. Denote

$$A_t=\{X+Y\in x-t+{\Delta }_T\}\mathrm{for}t\ge 0.$$

In the following, we deal with $V_1\ast V_2(x-t+{\Delta }_T)$ in two cases where $T\le t<\infty$ and $0<t<T$. For $T\le t<\infty$, by (61) and $x\ge m{x}_0+T$, we have

$$\begin{array}{ccc}\hfill & & V_1\ast V_2(x-t+{\Delta }_T)=P(A_t,0\le X\le x-t-x_0)+P(A_t,x-t-x_0<X\le x-t+T)\hfill \\ & \le & P(A_t,0\le X\le x-t-x_0)+P(A_t,0<Y\le T+x_0)\hfill \\ & =& {\int }_{0-}^{x-t-x_0}{V}_2(x-t-y+{\Delta }_T)V_1\left(dy\right)+{\int }_0^{T+x_0}{V}_1(x-t-y+{\Delta }_T)V_2\left(dy\right)\hfill \\ & \le & (1+\epsilon )(C_{{\Delta }_T}^{*}(V_2,t){\int }_{0-}^{x-t-x_0}{V}_2(x-y+{\Delta }_T)V_1\left(dy\right)\hfill \\ & & +C_{{\Delta }_T}^{*}(V_1,t){\int }_0^{T+x_0}{V}_1(x-y+{\Delta }_T)V_2\left(dy\right))\hfill \\ & \le & (1+\epsilon )max\{C_{{\Delta }_T}^{*}(V_1,t),C_{{\Delta }_T}^{*}(V_2,t)\}{V}_1*V_2(x+{\Delta }_T).\hfill \end{array}$$

(64)

For $0<t<T$, we give a segmentation for $V_1\ast V_2(x-t+{\Delta }_T)$ which is different from (64) as follows. Further, by (61), (63) and $x\ge m{x}_0+T$, we have

$$\begin{array}{ccc}\hfill & & V_1\ast V_2(x-t+{\Delta }_T)\le P(A_t,0\le X\le x-t-x_0)+P(A_t,0<Y\le x_0)\hfill \\ & & +P(A_t,x_0<Y\le T+x_0)\hfill \\ & \le & {\int }_{0-}^{x-t-x_0}{V}_2(x-t-y+{\Delta }_T)V_1\left(dy\right)+{\int }_0^{x_0}{V}_1(x-t-y+{\Delta }_T)V_2\left(dy\right)\hfill \\ & & +V_1(x-x_0-t-T+{\Delta }_{2T})V_2(x_0+{\Delta }_T)\hfill \\ & \le & (1+\epsilon )\left(C_{{\Delta }_T}^{*}(V_2,t){\int }_{0-}^{x-t-x_0}{V}_2(x-y+{\Delta }_T)V_1\left(dy\right)+C_{{\Delta }_T}^{*}(V_1,t){\int }_0^{x_0}{V}_1(x-y+{\Delta }_T)V_2\left(dy\right)\right)\hfill \\ & & +\epsilon {(1+\epsilon )}^{2}K{V}_1\ast V_2(x+{\Delta }_T)\hfill \\ & \le & (1+\epsilon )\left(max\{C_{{\Delta }_T}^{*}(V_1,t),C_{{\Delta }_T}^{*}(V_2,t)\}+\epsilon {(1+\epsilon )}^{3}K\right)V_1\ast V_2(x+{\Delta }_T).\hfill \end{array}$$

(65)

Therefore, $V_1\ast V_2\in {\mathcal{OL}}_{{\Delta }_T}$ and (60) holds by (64), (65) and the arbitrariness of $\epsilon$. □

Lemma 6.

Let $V_1$ and $V_2$ be the two distributions belonging to the class ${\mathcal{OL}}_{{\Delta }_T}$ for some $0<T<\infty$. If conditions (57) and (59) are satisfied, then for each $t>0$,

$$\begin{array}{c}\hfill lim\; inf{V}_1\ast V_2(x-t+{\Delta }_T)/V_1\ast V_2(x+{\Delta }_T)\ge 1.\end{array}$$

(66)

Proof. 

In order to prove (66), we carry out some preparatory work. For each $t>0$, by (57) and $V_2\in {\mathcal{OL}}_{{\Delta }_T}$, we know that, for any $0<\epsilon <1$, there exists $x_1=x_1(V_1,V_2,\epsilon ,T,t)>0$ such that, when $x\ge x_1$,

$$\begin{array}{c}\hfill (1-\epsilon )V_i(x+{\Delta }_T)\le V_i(x-t+{\Delta }_T)\le (1+\epsilon )C_{{\Delta }_T}^{*}(V_i,t)V_i(x+{\Delta }_T),i=1,2.\end{array}$$

(67)

Furthermore, according to Lemma 5, there exists $x_2=x_2(V_1,V_2,T,t)>0$ and $C>0$ such that, when $x\ge x_2$,

$$\begin{array}{c}\hfill V_1(x+{\Delta }_T)\le C{V}_1\ast V_2(x+{\Delta }_T)\mathrm{and}{V}_{1\left(\mathrm{or}2\right)}\left(x+{\Delta }_t\left(\mathrm{or}{\Delta }_{t+T}\right)\right)\le C{V}_1(x+{\Delta }_T).\end{array}$$

(68)

Further, we denote $x_0=max\{x_1,x_2\}+t+T$ and $A_t=\{X+Y\in x-t+{\Delta }_T\}$, where X and Y are two random variables defined in Lemma 5.

Now, we prove (66) for each $t>0$. When $x\ge x_0$, by (67), we have

$$\begin{array}{ccc}\hfill & & V_1\ast V_2(x-t+{\Delta }_T)=P(A_t,0\le X\le x-t-x_0)+P(A_t,x-t-x_0<X\le x-t+T)\hfill \\ & =& {\int }_{0-}^{x-t-x_0}{V}_2(x-t-y+{\Delta }_T)V_1\left(dy\right)+P(A_t,x-t-x_0<X\le x-t+T)\hfill \\ & \ge & (1-\epsilon ){\int }_{0-}^{x-t-x_0}{V}_2(x-y+{\Delta }_T)V_1\left(dy\right)+P(A_t,x-t-x_0<X\le x-t+T)\hfill \\ & \ge & (1-\epsilon )(V_1*V_2(x+{\Delta }_T)-{\int }_{x-t-x_0}^{x-x_0}{V}_2(x-y+{\Delta }_T)V_1\left(dy\right)\hfill \\ & & -P(A_0,x-x_0<X\le x+T)+P(A_t,x-t-x_0<X\le x-t+T))\hfill \\ & =& (1-\epsilon )\left(V_1\ast V_2(x+{\Delta }_T)-P_1\left(x\right)-P_2\left(x\right)+P_3\left(x\right)\right).\hfill \end{array}$$

(69)

Firstly, we estimate $P_1\left(x\right)$. When $x-t-x_0<y\le x-x_0$,

$$V_2(x-y+{\Delta }_T)\le P(x_0<Y\le x_0+t+T).$$

Then, by (68), (67) and (59), we know that

$$\begin{array}{ccc}\hfill & & P_1\left(x\right)\le V_2(x_0+{\Delta }_{t+T})V_1(x-x_0-t+{\Delta }_t)\hfill \\ & \le & C^2{V}_2(x_0+{\Delta }_T)V_1(x-x_0-t+{\Delta }_T)\hfill \\ & \le & {(1+\epsilon )}^2{C}^2{V}_1(x+{\Delta }_T)C_{{\Delta }_T}^{*}(V_1,t)C_{{\Delta }_T}^{*}(V_1,x_0)V_2(x_0+{\Delta }_T)\hfill \\ & \le & {(1+\epsilon )}^2{C}^3{V}_1\ast V_2(x+{\Delta }_T)C_{{\Delta }_T}^{*}(V_1,t)C_{{\Delta }_T}^{*}(V_1,x_0)V_2(x_0+{\Delta }_T)\hfill \\ & =& o\left(V_1\ast V_2(x+{\Delta }_T)\right)\mathrm{as}{x}_0\to \infty .\hfill \end{array}$$

(70)

Secondly, we estimate $P_3\left(x\right)-P_2\left(x\right)$.

$$\begin{array}{ccc}\hfill & & P_3\left(x\right)-P_2\left(x\right)=P(A_t,x-t-x_0<X\le x-t+T,0<Y<x_0+T)\hfill \\ & & -P(A_0,x-x_0<X\le x+T,0\le Y<x_0+T)\hfill \\ & =& {\int }_{0-}^{x_0}\left(V_1(x-t-y+{\Delta }_T)-V_1(x-y+{\Delta }_T)\right)V_2\left(dy\right)\hfill \\ & & +{\int }_{x_0}^{x_0+T}P(X\in x-t-y+{\Delta }_T,x-t-x_0<X\le x-t+T)V_2\left(dy\right)\hfill \\ & & -{\int }_{x_0}^{x_0+T}P(X\in x-y+{\Delta }_T,x-x_0<X\le x+T)V_2\left(dy\right)\hfill \\ & =& P_{11}\left(x\right)+P_{12}\left(x\right)-P_{13}\left(x\right).\hfill \end{array}$$

(71)

By (67), we have

$$\begin{array}{ccc}\hfill \frac{P_{11}\left(x\right)}{V_1\ast V_2(x+{\Delta }_T)}& \ge & \frac{-\epsilon }{V_1*V_2(x+{\Delta }_T)}{\int }_{0-}^{x_0}{V}_1(x-y+{\Delta }_T)V_2\left(dy\right)\ge -\epsilon .\hfill \end{array}$$

(72)

Using the proof method of (70), we can get that

$$\begin{array}{ccc}\hfill & & P_{12}\left(x\right)={\int }_{x_0}^{x_0+T}P(x-t-x_0<X\le x-t-y+T)V_2\left(dy\right)\hfill \\ & \le & V_1(x-t-x_0+{\Delta }_T)V_2(x_0+{\Delta }_T)\hfill \\ & =& o\left(V_1\ast V_2(x+{\Delta }_T)\right),as{x}_0\to \infty .\hfill \end{array}$$

(73)

Similarly, we have

$$\begin{array}{ccc}\hfill & & P_{13}\left(x\right)={\int }_{x_0}^{x_0+T}P(x-x_0<X\le x-y+T)V_2\left(dy\right)\hfill \\ & \le & V_1(x-x_0+{\Delta }_T)V_2(x_0+{\Delta }_T)\hfill \\ & =& o\left(V_1*V_2(x+{\Delta }_T)\right),as{x}_0\to \infty .\hfill \end{array}$$

(74)

Combining with (69)–(74), we know that (66) holds. □

Theorem 6.

Suppose that $V\in {\mathcal{OL}}_{{\Delta }_T}$ for some $0<T<\infty$. If conditions (57) and (59) are satisfied for $V_1=V_2=V$, then for all $k\ge 2$, $V^{*k}\in {\mathcal{OL}}_{{\Delta }_T}$,

$$\begin{array}{c}\hfill C_{{\Delta }_T}^{*}(V^{*k},t)\le C_{{\Delta }_T}^{*}(V,t)\mathrm{for}\mathrm{each}t\ge 0.\end{array}$$

(75)

and

$$\begin{array}{c}\hfill lim\; inf{V}^{*k}(x-t+{\Delta }_T)/V^{*k}(x+{\Delta }_T)\ge 1\mathrm{for}\mathrm{each}t\ge 0.\end{array}$$

(76)

Proof. 

We use mathematical induction to prove the result.

Clearly, (75) and (76) hold for $k=1$. Assume that $V^{*k}\in {\mathcal{OL}}_{{\Delta }_T}$, (75) and (76) hold for some $k\ge 2$. Set $V_1=V^{*k}$ and $V_2=V$ in Theorem 6. By (59) and (75), we have $V^{*(k+1)}=V_1\ast V_2\in {\mathcal{OL}}_{{\Delta }_T}$ and (76) holds for $k+1$. Thus, (75) holds for $k+1$, too. □

In particular, we take $V=F_{\gamma }$ and $T=T_0$ in (76), then we obtain (10) in Theorem 3 of this paper.

5. Conclusions and Future Work

In this paper, we prove that the class ${\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, in addition to ${\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)\cap {\mathcal{OS}}_{{\Delta }_{T_0}}$ for some $0<\gamma ,T_0<\infty$ are not closed under the I.I.D. root. However, by adding certain conditions, the two classes become closed under the I.I.D. root. At the same time, we also provide the corresponding results under the random convolution roots.

In this section, we briefly introduce the theoretical significance and application value of the above results reported herein, in addition to some unresolved problems.

5.1. Theoretical Significance and Application Value

In complex practice, F is often in a “black box”, that is, it is unknown or partially unknown. For example, in Theorem 2, we only know that F has property (8) or (9), but we do not know whether it has property $F^{*k}\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ for some $k\ge 1$. Furthermore, the properties of H, as the external expression of F, can be estimated by some statistical methods. Therefore, it is of great theoretical significance and application value to use known H to estimate unknown F. This presents the research purpose of this paper.

In the following, we provide some specific examples to illustrate applications of the research findings herein.

Firstly, it is well known that the distribution of components of the Lévy process is I.I.D. Therefore, research on I.I.D. H is beneficial to the Lévy process.

Secondly, in the Cramér–Lundeberg risk model, the distributions F, $H_2$ and $H_1$ satisfying the conditions (2) and (3) can be regarded as the distributions of the claim, the total claim amount and the perturbation to the total claim amount, respectively, see SubSection 1.3.3 of Embrechts et al. [34]. If the disturbed distribution of total claim amount $H=H_1\ast H_2$ is an I.I.D. and $H\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, then according to Theorem 2, we have $H_2\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ and $H_2(x+{\Delta }_T)\sim H(x+{\Delta }_T)$ for each $0<T\le \infty$ under condition (8) or (9). Interestingly, F does not have to belong to class ${\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, but $F^{*k}$ belongs to that class for all $k\ge 2$, see Theorems 1 and 2 mentioned in this paper.

There are many similar examples, such as $H_2$ which is the distribution of proportional reinsurance or the claim in Poisson model, see Example 5.2 (i) of Klüppelberg and Mikosch [35] and the main theorems of Veraverbeke [36].

Therefore, the results of this paper undoubtedly play an important role in risk theory and other fields.

Finally, the results of this paper can offer a more complete and profound answer to the famous Embrechts–Goldie conjecture, see Section 5.2 below for details.

5.2. On the Embrechts–Goldie Conjecture

Let $\mathcal{X}$ be a distribution class, and let V be a distribution. If $V^{*2}\in \mathcal{X}$ implies $V\in \mathcal{X}$, then we say that the class $\mathcal{X}$ is closed under convolution roots. Clearly, the closure under I.D.D roots is the natural extension of the closure under convolution roots for some distribution class.

Theorem 2 of Embrechts et al. [2] shows that the class $\mathcal{S}$ is closed under convolution roots. The same conclusion also holds for the class $\mathcal{S}\left(\gamma \right)$ for some $\gamma >0$ if the distribution $V\in \mathcal{L}\left(\gamma \right)$, see Theorem 2.10 of Embrechts and Goldie [24]. Therefore, Embrechts and Goldie [15,24] put forward a famous conjecture:

$$\begin{array}{c}\hfill \mathrm{If}{V}^{*k}\in \mathcal{L}\left(\gamma \right)\mathrm{for}\mathrm{some}(\mathrm{even}\mathrm{for}\mathrm{all})k\ge 2\mathrm{and}\gamma \ge 0,\mathrm{then}V\in \mathcal{L}\left(\gamma \right).\end{array}$$

Many positive or negative conclusions related to the conjecture are then proposed. Some positive results can be found in Theorem 1.2 of Watanabe [11] for the class $\mathcal{S}\left(\gamma \right)$ for some $\gamma >0$, Theorem 6 of Xu et al. [31] for the classes $\mathcal{L}\left(\gamma \right)$ and $\mathcal{L}\left(\gamma \right)\cap \mathcal{OS}$. Of course, these outcomes are valid under certain restrictive conditions.

The following references provide us with the negative results.

Theorem 1.1 of Watanabe [11] shows that the class $\mathcal{S}\left(\gamma \right)$ for some $\gamma >0$ is not closed under the convolution roots in general.

Earlier, Shimura and Watanabe [37] showed that there is a distribution V such that $V^{*2}\in \mathcal{L}\left(\gamma \right)\setminus \mathcal{OS}$ for some $\gamma \ge 0$, while $V\in \mathcal{OL}\setminus \left({\cup }_{\gamma \ge 0}\mathcal{L}\left(\gamma \right)\cup \mathcal{OS}\right)$ and $\overline{V}\left(x\right)=o\left(\overline{V^{*2}}\left(x\right)\right)$.

Further, Theorem 1.1 of Xu et al. [31] points out that there is a distribution $V\in \mathcal{OL}\setminus \left({\cup }_{\gamma \ge 0}\mathcal{L}\left(\gamma \right)\cup \mathcal{OS}\right)$ and $\overline{V}\left(x\right)\ne o\left(\overline{V^{*2}}\left(x\right)\right)$ such that $V^{*2}\in \left(\mathcal{L}\left(\gamma \right)\cap \mathcal{OS}\right)\setminus \mathcal{S}\left(\gamma \right)$ for each $\gamma >0$.

For $\gamma =0$, Theorem 2.2 (1) of Xu et al. [30] shows that there is a distribution V such that $V\in \mathcal{OL}\setminus \left(\mathcal{L}\cup \mathcal{OS}\right)$ and $\overline{V}\left(x\right)\ne o\left(\overline{V^{*2}}\left(x\right)\right)$, while $V^{*k}\in \left(\mathcal{L}\left(\gamma \right)\cap \mathcal{OS}\right)\setminus \mathcal{S}$ for all $k\ge 2$. Then, Proposition 2.2 of Xu et al. [30] points out that there are two distributions $V_1$ and $V_2$ such that $V_1,V_2\notin \mathcal{OL}$, while $V_1^{*k}\in \left(\mathcal{L}\cap \mathcal{OS}\right)\setminus \mathcal{S}$ and $V_2^{*k}\in \mathcal{L}\setminus \mathcal{OS}$ for all $k\ge 2$.

This result reveals a surprising phenomenon that, although the properties of a distribution V are very poor, its convolution, and even its random convolution and the corresponding I.I.D., bear good properties.

Therefore, the Embrechts–Goldie conjecture has been denied for the class $\mathcal{L}\left(\gamma \right)$ and its subclasses $\mathcal{S}\left(\gamma \right)\setminus \mathcal{S}$, $\left(\mathcal{L}\right(\gamma )\cap \mathcal{OS})\setminus \mathcal{S}\left(\gamma \right)$ and $\mathcal{L}\left(\gamma \right)\setminus \mathcal{OS}$ for each $\gamma \ge 0$, where the corresponding distribution $V\in \mathcal{OL}\setminus \left({\cup }_{\gamma \ge 0}\mathcal{L}\left(\gamma \right)\cup \mathcal{OS}\right)$, and even $V\notin \mathcal{OL}$.

In this subsection, we mainly focus on the local closure under the convolution root.

For negative conclusions, Corollary 1.1 of Watanabe [11] shows that the classes ${\mathcal{S}}_{loc}$, ${\mathcal{L}}_{loc}$, ${\mathcal{S}}_{{\Delta }_T}$ and ${\mathcal{L}}_{{\Delta }_T}$ for some $0<T<\infty$ are not closed under convolution roots. Further, Theorem 1 of the paper and its proof show that the class ${\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ is not closed either.

In addition, Theorem 1.1 and Corollary 1.1 of Watanabe and Yamamuro [38] and Theorem 1.1 and Corollary of Watanabe [39] obtain some results corresponding to Corollary 1.1 of Watanabe [11] for the subexponential density classes and the subexponential lattice distribution classes, respectively. Clearly, the lattice distribution is a special local distribution, and the density is closely related to its local distribution.

As positive conclusions, Theorem 2.1 of Watanabe [39] shows that the subexponential lattice distribution classes are closed under convolution roots with a condition. However, other positive conclusions about the local closure in non-lattice cases are rare.

In this paper, according to Theorem 6 of Xu et al. [31], Proposition 1 and Lemma 1 of the paper, using the Esscher transform, we give a corresponding positive result for the class ${\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$ and omit the proof details.

Theorem 7.

Let V be a distribution, and let γ and T be two positive and finite constants.

$\left(i\right)$ Assume that $V\in {\mathcal{OS}}_{loc}$ and

$$\begin{array}{c}\hfill lim\; inf\overline{V_{-\gamma }}(x-t)/\overline{V_{-\gamma }}\left(x\right)\ge e^{\gamma t}foreacht>0\end{array}$$

(77)

or

$$\begin{array}{c}\hfill \overline{V_{-\gamma }}\left(x\right)=o\left(\overline{V_{-\gamma }^{*2}}\left(x\right)\right).\end{array}$$

(78)

If $V^{*2}\in {\mathcal{L}}_{loc}$, then $V\in {\mathcal{L}}_{loc}$.

$\left(ii\right)$ Assume that $V\in {\mathcal{L}}_{loc}$ with the mean ${\mu }_V<\infty$, the condition (77) is satisfied and

$$\begin{array}{c}\hfill C^{*}\left(V_{-\gamma }^{*2}\right)<6M(V_{-\gamma }^{*2},\gamma ).\end{array}$$

(79)

If $V^{*2}\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, then $V\in {\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$.

Using the Esscher transform, by (15), we can replace the condition (78) with a more immediate condition.

Proposition 2.

If $V^{*2}\in {\mathcal{L}}_{loc}$, then (78) is implied by the following condition:

$$\begin{array}{c}\hfill V(x+{\Delta }_T)=o\left(V^{*2}(x+{\Delta }_T)\right).\end{array}$$

5.3. Some Unresolved Problems

Clearly, for other local distribution classes, such as the class ${\mathcal{L}}_{loc}\setminus {\mathcal{OS}}_{loc}$ and the class ${\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)\setminus {\mathcal{OS}}_{{\Delta }_{T_0}}$ for some $0<\gamma ,T_0<\infty$, the following corresponding questions arise:

Are they closed under the I.I.D. root? If not, under what conditions are they closed under the I.I.D. root?

Perhaps we can first solve the corresponding problem of the global distribution class $\mathcal{L}\left(\gamma \right)\setminus \mathcal{OS}$ with some $\gamma >0$. In addition, the existing results, apart from Proposition 2.1 of Xu et al. [30], often assume that $F\in \mathcal{OL}$. Then, if $F\notin \mathcal{OL}$, what will we get?

Further, if F does not belong to the class ${\mathcal{L}}_{loc}\cap {\mathcal{OS}}_{loc}$, ${\mathcal{L}}_{loc}\setminus {\mathcal{OS}}_{loc}$ or ${\mathcal{TL}}_{{\Delta }_{T_0}}\left(\gamma \right)\setminus {\mathcal{OS}}_{{\Delta }_{T_0}}$ for some $0<\gamma ,T_0<\infty$, what kind of F can make $F^{*k}$ for all $k\ge l_0$ and some $l_0\ge 2$, $H_2$ and H belong to the same class? Even if $F\notin {\mathcal{OL}}_{loc}$, what will we get?

In our opinion, these questions are both interesting and difficult to solve. The theory will become more complete following the provision of solutions to these questions.