1. Introduction
This paper will consider the defective renewal equation
(1)Z(x)=z(x)+q∫0xZ(x-y)F(dy), x≥0,
where F is a proper distribution on [0,∞), z(x)≥0 is a known and locally bounded function on [0,∞), and 0<q<1. The only solution Z(x) to (1) is given by
(2)Z(x)=(1-q)-1∫0xz(x-y)U0(dy), x≥0,
where
(3)U0(x)=(1-q)∑n=0∞qnF*n(x), x≥0
(see e.g., Asmussen [1, Chapter V]) and here F*n is the n-fold convolution of F with itself, n≥2, F*1=F, and F*0 is the distribution degenerate at zero.

Since in most cases it is not easy to calculate (2), more attention is paid to the asymptotics of the solution Z(x). When Z=F, the asymptotics of Z(x) have been investigated by many researchers, such as Embrechts et al. [2], Embrechts and Goldie [3], and Cline [4]. Asmussen [5] and Asmussen et al. [6] considered the case that z is a subexponential density. Yin and Zhao [7] obtained the asymptotics of Z(x) for the monotone function z. For the above case, K. Wang and Y. Wang [8] gave the local asymptotics of Z(x). Cui et al. [9] considered a new case that
(4)limx→∞z(x)F¯(x)=c,
where c is a positive constant. In Corollary 5.1 and Theorem 5.2 of Cui et al. [9], they obtained the asymptotics of Z(x) under the condition that F∈S and F∈S(γ) for some γ>0 with F^(γ)=∫-∞∞eγuF(du)<1, respectively. The classes S and (γ), γ>0, (the definitions of these distribution classes will be given below) are convolution equivalent distribution classes. But beyond the convolution equivalent distribution classes, there exist some other distributions. How to estimate the asymptotics of the solution Z(x) for the nonconvolution equivalent distribution F will be an interesting question. This paper will investigate this case. Under the conditions (4) and that F may not belong to the convolution equivalent distribution class, this paper obtains the asymptotics of the solution Z(x). In order to better illuminate our motivation and results, we will introduce some notions and notation.

Without special statement, in this paper a limit is taken as x→∞. For two nonnegative functions a(·) and b(·), we write a(x)≲b(x) if limsupa(x)/b(x)≤1, write a(x)≳b(x) if liminfa(x)/b(x)≥1, write a(x)~b(x) if lima(x)/b(x)=1, and write a(x)=o(b(x)) if lima(x)/b(x)=0. For a proper distribution V on (-∞,∞), the tail of V is V¯=1-V. For a real number γ, denote by V^(γ)=∫-∞∞eγuV(du) the moment generating function of V.

Firstly, we will introduce some heavy-tailed and light-tailed distribution classes. Say that a random variable (r.v.) ξ (or its corresponding distribution V) is heavy-tailed if for all λ>0, V^(γ)=∞; otherwise, say that it is light-tailed. Let V be a distribution on (-∞,∞). Say that the distribution V belongs to the class L(γ) for some γ≥0, if for any t∈(-∞,∞),
(5)V¯(x-t)~eγtV¯(x),
where, when γ>0 and V is a lattice distribution, x and t are both taken as a multiple of the lattice step. Say that the distribution V belongs to the class S(γ) for some γ≥0, if V∈L(γ), V^(γ)<∞, and
(6)V*2¯(x)~2V^(γ)V¯(x).
The class S(γ), γ≥0, is called the convolution equivalent distribution class and was introduced by [10] and Chover et al. [11, 12] for distributions on [0,∞) and by Pakes [13] for distributions on (-∞,∞). Especially, we call S(0) and L(0) the subexponential distribution class and the long-tailed distribution class, denoted by S and L, respectively.

This paper will mainly investigate the case that the distribution F may not be convolution equivalent. We will introduce another distribution class. Say that the distribution V belongs to the class 𝒪𝒮, if V¯(x)>0 for sufficiently large x and
(7)CV*=limsupx→∞V2*¯(x)V¯(x)<∞.
Clearly, if V∈S(γ) for some γ≥0 then CV*=2V^(γ). Therefore, for each γ≥0, S(γ)⊂𝒪𝒮. If V∈L(γ) for some γ≥0 then CV*≥2V^(γ), which can be obtained by Lemma 2.4 of Embrechts and Goldie [3] and Theorems 1.1 and 1.2 of Yu et al. [14]. The class 𝒪𝒮 is first introduced by Klüppelberg [15] and detailedly studied in Klüppelberg and Villasenor [16], Shimura and Watanabe [17], Watanabe and Yamamura [18], Lin and Wang [19], Yang and Wang [20], and Wang et al. [21], among others. This paper will consider the case that F∈L(γ)∩𝒪𝒮, γ≥0. As noted by Wang et al. [21], for each γ>0, S(γ)⊂L(γ)∩𝒪𝒮 and the class (L(γ)∩𝒪𝒮)∖S(γ) is nonempty.

We first present the main result for the heavy-tailed case.

Theorem 1.
For the renewal equation (1), assume that (4) holds. If F∈L∩𝒪𝒮 and CF*<1+q-1, then
(8)c1-qF¯(x)≲Z(x)≲(11-q+CU0*-21-q(CF*-1))cF¯(x).

Remark 2.
If F∈S then U0∈S by Theorem 1 of Cline [4]. Hence, CU0*=2. The result of Theorem 1 implies that
(9)Z(x)~c1-qF¯(x),
which is Corollary 5.1 (ii) of Cui et al. [9].

In the following, we give the result for the light-tailed case.

Theorem 3.
For the renewal equation (1), assume that (4) holds. If F∈L(γ)∩𝒪𝒮 for some γ>0 satisfying F^(γ)<1 and CF*<1+F^(γ), then
(10)c1F¯(x)≲Z(x)≲c2F¯(x),
where
(11)c1=c1-qF^(γ)+qI(1-qF^(γ))2,c2=c1-qF^(γ)+c(1-qF^(γ))1-q(CF*-F^(γ))(CU0*1-q-21-qF^(γ)) +qI(1-q(CF*-F^(γ)))(1-qF^(γ)),I=γ∫0∞eγyz(y)dy.

Remark 4.
If F∈S(γ) for some γ>0, by Theorem 1 of Cline [4], then CF*=2F^(γ) and U0∈S(γ). Therefore, CU0*=2U0^(γ)=2(1-q)/(1-qF^(γ)). Then we can obtain from Theorem 3 that
(12)Z(x)~c1F¯(x),
which is Theorem 5.2 (ii) of Cui et al. [9].

2. Proofs of Theorems
Before giving the proof of Theorems 1 and 3, we first give some lemmas. The first lemma comes from Lemma 2.2 of Yu and Wang [22], which will need the following notation. For a distribution V on (-∞,∞) and any γ≥0, define
(13)ℋV(γ)={h on [0,∞):h(x)↑∞, x-1h(x)⟶0,hhhV¯(x-y)~eγyV¯(x)hhhh on [0,∞):h(x)↑∞, x-1h(x)⟶0,uniformly for |y|≤h(x)}.

Lemma 5.
Suppose that V is a distribution on (-∞,∞) and belongs to the class ℒ(γ)∩𝒪𝒮 for some γ≥0. Then for any h∈ℋV(γ),
(14)limsup∫h(x)x-h(x)V¯(x-u)V¯(x)V(du)=CV*-2V^(γ).

Lemma 6.
For the random sum (3), assume that F∈ℒ(γ)∩𝒪𝒮 for some γ≥0. When γ=0, let CF*<1+q-1; when γ>0, let F^(γ)<1 and CF*<1+F^(γ). Then U0∈ℒ(γ)∩𝒪𝒮,
(15)liminfU0¯(x)F¯(x)=q(1-q)(1-qF^(γ))2,(16)limsupU0¯(x)F¯(x)≤q(1-q)(1-q(CF*-F^(γ)))(1-qF^(γ)).

Proof.
We first prove (15). Let τ be a r.v. with a distribution P(τ=n)=(1-q)qn, n=0,1,…. When γ=0 and since F is heavy-tailed and τ is light-tailed, by Theorem 2 of Denisov et al. [23], it holds that
(17)liminfU0¯(x)F¯(x)=q(1-q).

When γ>0 and since there exists ϵ1>0 such that
(18)E(max{F^(γ)+ϵ1,1})τ<∞,
by Theorem 1.2 of Yu et al. [14], it holds that
(19)liminfU0¯(x)F¯(x)≤Eτ(F^(γ))τ-1=q(1-q)(1-qF^(γ))2.
On the other hand, since F∈ℒ(γ), by Fatou’s lemma and Lemma 5.4 of Pakes [13], we have
(20)liminfU0¯(x)F¯(x)≥Eτ(F^(γ))τ-1=q(1-q)(1-qF^(γ))2.
This completes the proof of (15).

Now we prove (16). Since CF*<1+q-1 for γ=0 and CF*<1+F^(γ) for γ>0, there exists ϵ2>0 such that for γ≥0,
(21)E(CF*-F^(γ)+ϵ2)τ<∞.
Hence, by Corollary 1 of Yu and Wang [22], we get U0∈ℒ(γ)∩𝒪𝒮 and (16) holds.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.
Since F∈ℒ∩𝒪𝒮, we get U0∈ℒ∩𝒪𝒮 by Lemma 6. For any fixed positive constant M, when x is sufficiently large, we get
(22)Z(x)=(1-q)-1(∫0M+∫Mx-M+∫x-Mx)z(x-y)U0(dy)=:J1(x,M)+J2(x,M)+J3(x,M).
By (4) and F∈ℒ, we get
(23)J1(x,M)~c(1-q)-1∫0MF¯(x-y)U0(dy)~c(1-q)-1F¯(x)U0(M).
Hence,
(24)limM→∞ limx→∞J1(x,M)F¯(x)=c1-q.

For J3(x,M), since supt∈[0,M]z(t)<∞, by U0∈ℒ and Lemma 6, it holds that
(25)J3(x,M)≤(1-q)-1supt∈[0,M]z(t)(U0¯(x-M)-U0¯(x))=o(U0¯(x))=o(F¯(x)).

For J2(x,M), we first estimate the asymptotics of
(26)∫Mx-MU0¯(x-y)U0(dy)
as firstly letting x→∞ and then letting M→∞. For any x≥0, it holds that
(27)U*2¯(x)=2∫0MU0¯(x-y)U0(dy) +∫Mx-MU0¯(x-y)U0(dy) +U0¯(x-M)U0¯(M).
Since U0∈ℒ, we have
(28)limM→∞ limx→∞∫0MU0¯(x-y)U0¯(x)U0(dy)=1,limM→∞ limx→∞U0¯(x-M)U0¯(M)U0¯(x)=0.
Hence, U0∈𝒪𝒮 means that
(29)limsupM→∞ limsupx→∞∫Mx-MU0¯(x-y)U0¯(x)U0(dy)=CU0*-2,
which, combining with (4) and Lemma 6, yields that
(30)limsupM→∞ limsupx→∞J2(x,M)F¯(x) ≤c1-q(CF*-1) ×limsupM→∞ limsupx→∞∫Mx-MU0¯(x-y)U0¯(x)U0(dy) =c1-q(CF*-1)(CU0*-2).
Hence, (8) can be obtained by (22)–(25) and (30).

Proof of Theorem <xref ref-type="statement" rid="thm1.2">3</xref>.
It follows from Lemma 6 and F∈ℒ(γ)∩𝒪𝒮 that U0∈ℒ(γ)∩𝒪𝒮. Taking h∈ℋU0(γ), when x is sufficiently large, we get
(31)Z(x)=(1-q)-1(∫0h(x)+∫h(x)x-h(x)+∫x-h(x)x) ×z(x-y)U0(dy)=:I1(x)+I2(x)+I3(x).
By (4) and F∈ℒ(γ), we have
(32)I1(x)~c(1-q)-1∫0h(x)F¯(x-y)U0(dy)~c1-qF^(γ)F¯(x).
By (4) and Lemmas 5 and 6, it holds that
(33)I2(x)~c(1-q)-1∫h(x)x-h(x)F¯(x-y)U0(dy)≲c(1-qF^(γ))2q(1-q)2∫h(x)x-h(x)U0¯(x-y)U0(dy)≲c(1-qF^(γ))2q(1-q)2(CU0*-2U0^(γ))U0¯(x)≲c(1-qF^(γ))1-q(CF*-F^(γ))(CU0*1-q-21-qF^(γ))F¯(x).
For I3(x), using Lemma 6 and the way of dealing with Z3(x) in Theorem 5.2 of Cui et al. [9], we can get
(34) limsupI3(x)F¯(x)≤qI(1-q(CF*-F^(γ)))(1-qF^(γ)), liminfI3(x)F¯(x)≥qI(1-qF^(γ))2.
Hence, (10) can be obtained by (31)–(34).