getdoced37. 244KB Jun 04 2011 12:05:09 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y}

Vol. 15 (2010), Paper no. 2, pages 44–74. Journal URL

http://www.math.washington.edu/~ejpecp/

Ratio of The Tail of An Infinitely

Divisible Distribution on The Line

to That of Its Lévy Measure

∗

Toshiro Watanabe

Kouji Yamamuro

Center for Mathematical Science,

Faculty of Engineering,

University of Aizu,

Gifu University,

Aizu-Wakamtsu 965-8580, Japan

Gifu 501-1193, Japan

t-watanb@u-aizu.ac.jp

yamamuro@gifu-u.ac.jp

Abstract

A necessary and sufficient condition for the tail of an infinitely divisible distribution on the real line to be estimated by the tail of its Lévy measure is found. The lower limit and the upper limit of the ratio of the right tailµ(r)of an infinitely divisible distributionµto the right tailν(r)of its Lévy measureνasr→ ∞are estimated from above and below by reviving Teugels’s classical method. The exponential class and the dominated varying class are studied in detail.

Key words: infinite divisibility, Lévy measure,O-subexponentiality, dominated variation, expo-nential class.

AMS 2000 Subject Classification:Primary 60E07; Secondary: 60F99.

Submitted to EJP on September 1, 2009, final version accepted December 23, 2009.

1

Introduction

The study of tails of distributions is important in theoretical and applied probability. In one direction, it has been developed in relation to infinitely divisible distributions and their Lévy measures since early works such as Feller [16; 17], Cohen [8], and Embrechts et al. [13]. It is known that the two-sided case is harder to analyze than the one-sided case (distributions on

R₊ = [0,∞)). Letη(r)be the right tail of a measureη, that is, η(r):= η(r,∞) =η((r,∞)). For positive functions f(r) and g(r), the relation f(r)∼ g(r) means that lim_r→∞f(r)/g(r) =1. We defineρ_b(s):=R_Res xρ(d x)fors∈R_{. Denote byρ∗ηthe convolution of distributionsρandη.}

Definition 1.1. Letρbe a distribution onR_{. Suppose thatρ(r)>0 for allr∈}R_{. Letγ≥0.}

(1) We sayρ∈ L(γ)ifρ(r+a)∼e−aγρ(r)for alla∈R_.

(2) We sayρ∈ S(γ)ifρ∈ L(γ),ρb(γ)<∞, andρ∗ρ(r)∼2ρb(γ)ρ(r).

(3) We say ρ ∈ S_D(γ) if, for some a > 0, ρ is a distribution on aZ := {0,±a,±2a, . . .} with ρ({na})>0 for all sufficiently largen∈Z_{, and}

lim

n→∞

ρ({(n+1)a}) ρ({na}) =e

−γa

and ifρb(γ)<∞and

lim

n→∞

ρ∗ρ({na})

ρ({na}) =2ρb(γ).

LetL andS denoteL(0)andS(0), respectively. The distributions inL andS, respectively, are called long-tailed andsubexponential. Those in S(γ) are called convolution equivalent. The class

L(γ)is often calledexponential class.

We say µ ∈ ID₊ if µ is an infinitely divisible distribution on R _{with Lévy measure ν satisfying}

ν(r)>0 for allr∈R. Letµ∈ID₊. We defineC∗andC∗as

C∗:=lim inf

r→∞ µ(r)

ν(r) and C

∗_{:=lim sup}

r→∞ µ(r) ν(r).

If 0<C∗≤C∗<∞, then we can estimate the tailµ(r)by the tailν(r)in a weak sense. That is, for anyε∈(0, 1), there isR>0 such that

(1−ε)C∗ν(r)≤µ(r)≤(1+ε)C∗ν(r) (1.1)

whenever r >R. To make the estimate (1.1) more meaningful, we should give the expression of

C_∗ andC∗. But the expression is known only in some special cases. See Theorem 1.3 of[27] for one-sided strictly semistable distributions. However, there is an important case where the tailµ(r) is estimated by the tailν(r)in a stronger sense. Namely, it is proved in Theorem 1.1 of[31](Lemma 5.3 of Section 5 below) that ifµ∈ S(γ)withγ≥0, then there is an explicitC∈(0,∞)such that

µ(r)∼Cν(r). (1.2)

γ≥0, then (1.2) holds with some C ∈(0,∞). See Theorem 2 of[14]. We do not know whether the converse is true. Namely, there might beµ6∈S_γ≥0(S(γ)∪ S_D(γ))such that (1.2) holds with

C ∈(0,∞). Thus we are led to the following two problems.

Problem 1. What is a necessary and sufficient condition in order that 0< C∗≤ C∗ <∞ ? In the case where 0<C_∗≤C∗<∞, what are the expressions ofC_∗andC∗or the lower and upper bounds of them ?

Problem 2. What is a necessary and sufficient condition in order that (1.2) holds with 0<C <∞? In the case where (1.2) holds with 0<C <∞, what is the expression ofC ?

We will give answers to Problem 1 and to the second question of Problem 2. A partial answer to the first question of Problem 2 will be given too. Our results are important from the viewpoint of the asymptotic estimates of the transition probabilities of Lévy processes.

In Section 2 we give definitions of classes such as O S, O L, H, and D and formulate our main results in Theorems 2.1, 2.2, and 2.3 and two corollaries. Relations with other works are given in detail. Section 3 discusses the meaning of O S in infinite divisibility. Section 4 studies a bound separatingC∗andC∗forµ∈ H. In Section 5 we prove Theorem 2.1 and its corollaries. In Sections 6 and 7, we prove Theorems 2.2 and 2.3, respectively.

2

Main results

For positive functions f(x)andg(x), the relation f(r)≍g(r)means that lim inf_r→∞f(r)/g(r)>0 and lim sup_r→∞f(r)/g(r)<∞. We introduce some basic classes of distributions onR in addition toL(γ),S(γ), andS_D(γ)in Definition 1.1.

Definition 2.1. Letρbe a distribution onR_{satisfyingρ(r)>0 for allr∈}R_.

(1) We sayρ∈ O S ifρ∗ρ(r)≍ρ(r). (2) We sayρ∈ H ifρb(s) =∞for alls>0. (3) We sayρ∈ Difρ(2r)≍ρ(r).

(4) We sayρ∈ O L ifρ(r+a)≍ρ(r)for alla∈R_.

The distributions inO S are calledO-subexponentialand those inH are sometimes called heavy-tailed. Those inDare calleddominatedly varying.

Remark 2.1. Let γ ≥ 0. The classes in Definitions 1.1 and 2.1 satisfy the following inclusion

relations:

(i)D ∩ L ⊂ S ⊂ O S ∩ L. (ii)D ∪ L ⊂ H.

(iii)D ∪S_γ≥0(S(γ)∪ S_D(γ))⊂ O S ⊂ O L. (iv)S_γ≥0L(γ)⊂ O L.

Letµ∈ID₊with Lévy measureν. Denote byµt∗ t-th convolution power ofµfor t>0. We define the normalized Lévy measureν(1)on(1,∞)by

ν(1)(d x):=c0−11{x>1}(x)ν(d x),

wherec0:=ν(1,∞). Define quantitiesd∗andγ∗as

d∗:=lim sup

r→∞

ν(1)∗ν(1)(r)

ν(1)(r)

,

and

γ∗:=sup{γ≥0 :µb(γ)<∞}.

Under the assumption thatν(1)∈ O L, we define the following. LetΛbe the totality of increasing

sequences{λn}∞n=1 with limn→∞λn=∞such that, for every x ∈R, the followingm(x;{λn})exists

and is finite:

m(x;{λn}):=_nlim_→∞

ν(λn−x)

ν(λn)

.

The idea of the use of the functionm(x;{λn})goes back to Teugels[29]. See Remark 5.1 in Section

5 for richness of the setΛ. Given a distributionρ, define

I_∗(ρ):= inf {λn}∈Λ

Z ∞

−∞

m(x;{λn})ρ(d x),

I∗(ρ):= sup {λn}∈Λ

Z ∞

−∞

m(x;{λn})ρ(d x).

Let B := d∗−I_∗(ν(1))−I∗(ν(1)). Note that 0≤ B <∞whenever ν(1)∈ O S, as will be shown in

Lemma 5.2 (iv) of Section 5. Defineµ1 as the compound Poisson distribution with Lévy measure 1{x>1}ν(d x)and letµ2be the infinitely divisible distribution satisfyingµ=µ1∗µ2. Then, under the assumption thatν(1)∈ O S, define

J(µ):=

  

I∗(µ2)exp(c0(I∗(ν(1))−1)) ifB=0,

I∗(µ2)exp(c0(I∗(ν(1))−1))

exp(c0B)−1

c0B

ifB>0.

We answer Problem 1 in the main theorem (Theorem 2.1 below). Shimura and Watanabe [27]

solved the first question of Problem 1 in the one-sided case. We will give a method to reduce the two-sided case to the one-sided case. Concerning the second question, the existing knowledge is that C∗ ≥1 in the one-sided case and that if moreoverµ∈ H, then C∗ =1. See Proposition 2 of

[11]and the proof of Theorem 7 of[9]. We can give general lower and upper bounds ofC_∗andC∗

by evolving the theory ofO-subexponentiality with the help of Teugels’s idea. The most involved part of our discussion is in finding the upper bound ofC∗.

Theorem 2.1. Letµbe a distribution inID+ with Lévy measureν.

(ii)Letν(1)∈ O S. Then0≤γ∗<∞and0<I∗(µ)≤µb(γ∗)≤J(µ)<∞. Moreover

I∗(µ)≤C∗≤µb(γ∗), (2.1)

and

b

µ(γ∗)≤C∗≤J(µ). (2.2)

Our results concerning Problem 2 are as follows. The first question in Problem 2 is hard to solve and our answer (Corollary 2.2 below) is only partial, which is an improvement of Corollary 1.3 of

[27]. It is a long-standing problem for thirty years since[13; 14]and is still open. Moreover, we do not know yet whetherµ∈ S_D(γ) impliesν(1)∈ SD(γ) in the two-sided case. The second question

is solved by Theorem 1.1 of [31]under the assumption thatν(1) ∈ L(γ) withγ≥ 0. We show in

Corollary 2.1 below that such an additional assumption is not needed.

Corollary 2.1. Letµbe a distribution inID+ with Lévy measureν. If(1.2)holds with0< C <∞,

then0≤γ∗<∞,µb(γ∗)<∞, and C =µb(γ∗).

Corollary 2.2. Letµbe a distribution inID+with Lévy measureν. Suppose that there are real numbers

a₁,a₂ with a₂ 6= 0 and a₁/a₂ being irrational such that, for a = 0,a₁,a₂, there is C(a) ∈(0,∞)

satisfying

µ(r+a)∼C(a)ν(r).

Then0≤γ∗<∞andµ∈ S(γ∗).

Remark 2.2. Suppose that all assumptions of Corollary 2.2 are satisfied except the irrationality of

a₁/a₂. Then 0≤γ∗<∞andµb(γ∗)<∞. In caseγ∗=0 we haveµ∈ S, but in caseγ∗>0 we may haveµ∈ S_D(γ∗)andµ /∈ S(γ∗).

We present explicit lower and upper bounds ofC_∗ and C∗ forµin (L(γ)∪ D)∩ID₊ in Theorems 2.2 and 2.3 below. The classL(γ)is extensively studied by [7; 10; 12]in the one-sided case and by[1; 5; 24]also in the two-sided case. Concerning Theorem 2.2 (i), a recent paper[1]of Albin contains an assertion that ifν(1)∈ L(γ), thenµ∈ L(γ). However, his proof forγ >0 depends on

an incorrect lemma, as will be explained in Remark 6.1 of Section 6. Braverman [5]also proved that if ν(1)∈ L(γ) andνb(1)(γ) =∞, thenµ∈ L(γ) forγ >0. Applications of the class L(γ)to

Lévy processes are found in[2; 5].

Theorem 2.2. Letµbe a distribution inID₊ with Lévy measureν. Letγ≥0.

(i)Ifν(1)∈ L(γ), thenµt∗∈ L(γ)for every t>0. In the converse direction, ifµt∗∈ L(γ)for every

t>0andν(1)∈ O S, thenν(1)∈ L(γ).

(2)If 2ν_b(1)(γ)<d∗<∞, thenν(1)∈(L(γ)∩ O S)\S(γ)and

b

µ(γ)<C∗≤µ_b(γ)exp(c0(d ∗₋₂

b

ν(1)(γ)))−1

c₀(d∗_−2νb

(1)(γ))

<∞. (2.3)

(3)If d∗=∞, thenν(1)∈ L(γ)\O S and C∗=∞.

Remark 2.3. Letγ≥0. There existsµin ID₊ such that ν(1)∈ L/ (γ)butµt∗∈ L(γ)for all t >0.

We shall show this in a forthcoming paper. Embrechts and Goldie[10; 12] conjectured that the class L(γ) is closed under convolution roots. However, Shimura and Watanabe [28] disproved their conjecture for all γ≥0. We do not know yet whether the classL(γ)∩ID₊ is closed under convolution roots.

Remark 2.4. Letγ≥0. We see from Lemma 5.4 of[24]thatν(1)∈ L(γ)impliesd∗≥2νb(1)(γ).

The class(O S ∩ L(γ))\S(γ)is not empty. We know that there areρ andρ′ both inS(γ)such thatρ∗ρ′is not inS(γ). As the classesO S andL(γ) are closed under convolution, thisρ∗ρ′ is inO S ∩ L(γ). See Lemma 3.1 (iii) of Section 3 and Lemma 2.5 of[31]. Hence thisρ∗ρ′is in (O S ∩ L(γ))\S(γ). For example, in the case ofγ=0, we can take the distributions in Section 6 of[23]asρandρ′. In the case ofγ >0, we can take the distributions in the proof of Theorem 2 of

[22]asρandρ′. The classL(γ)\O S is not empty. For example, in the case ofγ=0, a distribution onR_{+in Section 3 of[10]belongs toL \O S. In the case ofγ >0, any distributionρ∈ L(γ)with}

b

ρ(γ) =∞can be taken asρ∈ L(γ)\O S, becauseρ∈ O S ∩ L(γ) impliesρb(γ)<∞by Lemma 6.4 of[31]. Thus none of the cases (1)–(3) in Theorem 2.2 (ii) is empty.

Feller [16] started the study of dominated variation of infinitely divisible distributions. But his assertion thatν(1)∈ D impliesµ(r)∼ν(r)is not true. In the following Theorem 2.3 we clarify the

role of dominated variation in our problems. In the one-sided case, Watanabe[30]proved assertion (i) by preparing a Tauberian theorem similar to Theorem 1 of [20] and Shimura and Watanabe

[27]gave an alternative proof by employingO-subexponentiality. However, they did not discuss the values ofC∗ andC∗ forµin D ∩ID+. A result weaker than assertion (ii) is given in Theorem 1 of

Yakimiv[33]. An application of the classDto selfsimilar processes with independent increments is found in[30].

Define constantsQ∗andQ∗as

Q_∗:= lim

N→∞lim infr→∞

ν(r+N) ν(r) ,

Q∗:= lim

N→∞lim sup_r→∞

ν(r−N)

ν(r) = (Q∗) −1_.

Theorem 2.3. Letµbe a distribution inID+ with Lévy measureν.

(i)µ∈ Dif and only ifν(1)∈ D.

(ii)Ifν(1)∈ D, then0<Q∗≤1≤Q∗<∞,

1−(1−Q_∗)µ(−∞, 0)≤C_∗≤1, (2.4)

and

Remark 2.5. The classD ∩S contains all distributions with regularly varying tails. The lognormal distributions are infinitely divisible and belong to the classS \ D. On the other hand, the Peter and Paul distribution belongs to the classD \ S. One-sided strictly semistable distributions with discrete Lévy measures are infinitely divisible distributions in the classD \ S withC∗=1 andC∗=Q∗>1. See Theorem 1.3 of[27]. We show in Example 7.1 of Section 7 that there existsµ∈ D ∩ID₊such thatQ∗<C∗<1<C∗<Q∗.

3

Class

O S

and Infinite Divisibility

The classO S was introduced by Shimura and Watanabe[27]. They studied the asymptotic relation between an infinitely divisible distribution onR_{+and its Lévy measure by usingO-subexponentiality.}

In this section, we extend their results to the two-sided case. Letδa(d x) be the delta measure at

a∈R. For a distributionρonR, we define

ρ+(d x):=ρ(−∞, 0]δ0(d x) +1(0,∞)(x)ρ(d x).

Denote byρn∗ n-th convolution power of ρ with the understanding thatρ0∗(d x) :=δ0(d x). Let µ∈ID₊with Lévy measureν. In what follows, defineµ3as the compound Poisson distribution with Lévy measure 1_{x>c}ν(d x) withc > 1 and let µ4 be the infinitely divisible distribution satisfying µ=µ3∗µ4. We define the distributionν(c)by

ν(c)(d x):= (¯ν(c))−11{x>c}(x)ν(d x).

We choose sufficiently largec>1 such thatµ4(0,∞)>0.

Lemma 3.1. Letρandηbe distributions onR_.

(i)Ifρ(r)≍η(r)for someη∈ O S , thenρ∈ O S.

(ii)ρ∈ O S if and only ifρ+∈ O S.

(iii)Ifρ,η∈ O S, thenρ∗η∈ O S. In particular, ifρ∈ O S, thenρn∗∈ O S for all n≥1.

Proof. First we prove (i). Suppose thatρ(r)≍η(r)for someη∈ O S. Then we have

ρ∗ρ(r) =

Z

R

ρ(r−y)ρ(d y)

≍

Z

R

η(r− y)ρ(d y) =

Z

R

ρ(r−y)η(d y)

≍

Z

R

η(r− y)η(d y) =η∗η(r)

≍ η(r)≍ρ(r). (3.1)

Thus we haveρ∈ O S . We haveρ(r) =ρ+(r)forr>0. It follows from (i) that (ii) holds. Suppose

thatρ,η∈ O S . We see as in (3.1) that

Thus we obtain from (i), (ii), and Proposition 2.5 of [27] that ρ+∗η+ ∈ O S and thereby

ρ∗η∈ O S. The second assertion of (iii) is clear. ƒ

Lemma 3.2. Letρbe a distribution onR_.

(i)Ifρ∈ O S, thenρ∈ O L.

(ii)Ifρ∈ O L, then there isγ1∈(0,∞)such thatρb(γ1) =∞. (iii)Ifρ∈ O S, there is K>0such that, for all n≥1and r∈R_,

ρn∗_(r)≤Kn_ρ(r).

Proof. Suppose thatρ∈ O S. Thenρ+∈ O S. Letr>0. By Proposition 2.1 (ii) of[27], we have

ρ+∈ O L andρ(r+a) =ρ+(r+a)≍ρ+(r) =ρ(r)for anya≥0 and therebyρ∈ O L. Thus we

have proved (i). We see from Proposition 2.2 of[27]that ifρ∈ O L, then there isγ1∈(0,∞)such that ρb+(γ1) =∞, that is, ρb(γ1) = ∞. Thus assertion (ii) is true. Finally assertion (iii) is due to

Lemma 6.3 (ii) of[31]. ƒ

Lemma 3.3. Letµbe a distribution inID+. Thenµ∈ O S if and only ifµ1 ∈ O S. Furthermore, if µ∈ O S, thenµ(r)≍µ1(r)andµ2(r) =o(µ1(r)) =o(µ(r)).

Proof. Suppose that µ1 ∈ O S. We see from Lemma 3.2 (i) that µ1(logr) is in OR. As regards

the definition of the class OR, see [3]. By virtue of Theorem 2.2.7 of [3], there is c₁ > 0 such that ec1r_µ

1(r) → ∞as r → ∞. Furthermore, by Theorem 26.8 of [26], there is c2 > 0 such that µ2(r) = o(e−c2rlogr). Thus µ2(r) = o(µ1(r)) and there isc3 > 0 such that µ2(r)≤ c3µ1(r). We obtain that

µ(r) =

Z

R

µ2(r−y)µ1(d y)

≤ c₃ Z

R

µ1(r−y)µ1(d y) =c3µ1∗µ1(r)≍µ1(r).

Chooseb∈R_{such thatc4:=µ2(b,∞)>0. Then we have}

µ(r)≥

Z ∞

b+

µ1(r−y)µ2(d y)≥c4µ1(r−b). (3.3)

Hence by Lemma 3.2 (i) we haveµ(r)≍µ1(r). It follows from Lemma 3.1 (i) thatµ∈ O S.

Conversely, suppose that µ∈ O S. It follows thatµ2(r) = o(µ(r−b)) in the same way as above. Letε >0. There isa>0 such thatµ2(r)≤εµ(r−b)forr≥a. Hence we have

µ(r+a) ≤

Z r+

−∞

µ2(r+a−y)µ1(d y) +µ1(r)

≤ ε

Z r+

−∞

≤ εµ∗µ1(r+a−b) +µ1(r)

= ε

Z

R

µ1(r+a−b−y)µ(d y) +µ1(r)

≤ εc−1₄ Z

R

µ(r+a− y)µ(d y) +µ1(r)

= εc−1₄ µ∗µ(r+a) +µ1(r).

Here we used (3.3) in the last inequality. Sinceµ∈ O S, there isc5>0 such thatµ∗µ(r)≤c5µ(r). Hence we have(1−εc₄−1c₅)µ(r+a)≤µ1(r). Here we can takeεsatisfyingεc4−1c5<1. Therefore we see from Lemma 3.2 (i) that there is c₆ > 0 such that µ(r + b) ≤ c₆µ1(r). Combining this inequality with (3.3), we haveµ(r)≍µ(r+b)≍µ1(r). Furthermore, we haveµ1∈ O S by Lemma 3.1 (i). Finally, we see thatµ(r)≍µ1(r)andµ2(r) =o(µ1(r)) =o(µ(r)). ƒ

Lemma 3.4.(Theorem 1.1 of[27]) Letµbe a distribution onR_{+inID+with Lévy measureν.}

(i)µ(r)≍ν(r)if and only ifν(1)∈ O S.

(ii)The following statements are equivalent:

(1)µ∈ O S;

(2)(ν(1))n∗∈ O S for some n≥1;

(3)µ(r)≍(ν(1))n∗(r)for some n≥1.

Next we present the main result of this section.

Propsition 3.1. Letµbe a distribution inID+with Lévy measureν.

(i)We have0<C_∗.

(ii)C∗<∞if and only ifν(1)∈ O S.

(iii)The following statements are equivalent:

(1)µ∈ O S;

(2)(ν(1))n∗∈ O S for some n≥1;

(3)µ(r)≍(ν(1))n∗(r)for some n≥1.

Remark 3.1. In Proposition 3.1 (iii), we cannot replace statement (2) by the statementν(1)∈ O S.

The classO S is not closed under convolution roots. See Remark 1.3 and Proposition 1.1 of[27]

for further details. We see from Proposition 3.1 (iii) thatO S ∩ID+is closed under convolution roots.

Proof of Proposition 3.1. We prove (i). Notice that, forr>0,

µ3(r) =e−a ∞

X

n=1

an(n!)−1(ν(c))n∗(r),

witha:=ν(c,∞). Then we have

µ(r) ν(r) ≥

Z

R

e−aν(c)(r−y)

ν(c)(r)

≥ e−aµ4(0)>0.

HenceC∗>0.

Next we show (ii). We see from Lemma 3.4 that ν(1)∈ O S if and only if µ1(r) ≍ν(r). We find from (i) that C∗<∞if and only ifµ(r)≍ν(r). Suppose thatν(1)∈ O S. Then µ1(r)≍ν(r)and µ1∈ O S. We see from Lemma 3.3 thatµ∈ O S andµ(r)≍µ1(r). It follows thatµ(r)≍ν(r), that is,C∗<∞. Conversely, suppose thatC∗<∞, that is,µ(r)≍ν(r). Letr>0 andc₇:=µ4(0,∞)>0. We have

µ(r)≥

Z

(0,∞)

µ3(r−y)µ4(d y)≥c7µ3(r). (3.4)

Asµ3 is compound Poisson, there isc8 >0 such thatµ3(r)≥ c8ν(r). Hence we obtain from (3.4) thatc₇µ3(r)≤µ(r)≍ν(r)≤c8−1µ3(r). This implies thatµ3(r)≍ν(r). Hence we see from Lemmas 3.1 (i) and 3.4 (i) that(ν(c))(1)=ν(c)∈ O S and therebyν(1)∈ O S.

Lastly, we prove (iii). Lemma 3.3 states thatµ∈ O S if and only ifµ1∈ O S. We see from Lemma 3.4 thatµ1 ∈ O S if and only if(ν(1))n∗∈ O S for some n≥ 1. Thus we have proved that (1) is

equivalent to (2). Suppose that µ(r)≍(ν(1))n∗(r). Since µ3 is compound Poisson, there isc9 >0 such thatµ3(r)≥c9(ν(c))n∗(r). Hence we obtain from (3.4) that

c₇µ3(r)≤µ(r)≍(ν(1))n∗(r)≍(ν(c))n∗(r)≤c9−1µ3(r).

This implies that µ3(r) ≍ (ν(c))n∗(r) and thereby we see from Lemma 3.4 that µ3 ∈ O S and (ν(c))n∗∈ O S. It follows that(ν(1))n∗∈ O S and therebyµ1∈ O S. Thus we find from Lemma 3.3 thatµ∈ O S . Conversely, suppose thatµ∈ O S. We obtain from Lemmas 3.3 and 3.4 (ii) that

µ(r)≍µ1(r)≍(ν(1))n∗(r)

for somen≥1. We have proved that (1) is equivalent to (3). ƒ

4

Bound separating

C

_∗

and

C

∗

for

µ

in

H

The study of the lower and upper limits of ratios of tails of distributions on R_{+ was initiated by}

[11; 25]and progressed by[9; 18]. Watanabe and Yamamuro[32]discussed them for distributions onR_{and showed basic diferrences between the one-sided and two-sided cases. In this section, we}

study a bound separatingC∗andC∗forµ∈ H. We need the following lemma of Denisov et al.[9].

Lemma 4.1. (Lemma 2 of [9]) If a distributionρ onR_{+ is in} H, then there exists an increasing

concave function h:R_+→R_{+such that}

Z

R₊

eh(x)ρ(d x)<∞ and Z

R₊

Now we present the main result of this section.

Proposition 4.1. Letµbe a distribution inID+ with Lévy measureν.

(i)µ∈ H if and only ifν(1)∈ H.

(ii)Ifν(1)∈ H, then0<C∗≤1≤C∗≤ ∞.

Proof. Assertion (i) is obvious by virtue of Theorem 25.17 of[26]. We prove only (ii). Suppose

thatν(1)∈ H. First we show thatC∗≥1. We have

Hence we see from (4.1) and (4.2) that

concave function h_b(x) := min{h(x),b x}. Hence there is x₁ > 0 such that h_b(x) = h(x) for all

Here we see that, for any positive integern,

Therefore we obtain from (4.3) and (4.4) that

Thus it follows from (4.3) and (4.4) that

lim sup

Let δ1 > 0. Consequently, we obtain from (4.6) and (4.7) that, for sufficiently small b and suffi-ciently larget,

= lim inf

t→∞

Z ∞

x0+

µ(x)d(x[t]ehb(x)₎

Z ∞

x0+

ν(c)(x)d(x[t]ehb(x))

≥ ν(c) +δ1.

This contradicts (4.8). Hence we have lim inf_r→∞µ(r)/ν(c)(r)≤ν(c) +δ1, because x0 is arbitrary. Lettingδ1↓0, we obtain thatC∗≤1. We see from Proposition 3.1 (i) that 0<C∗. ƒ

5

Proofs of Theorem 2.1 and Its Corollaries

Let γ∈ R_{. We define the class M(γ) as the totality of distributionsρ satisfyingρb(γ) < ∞. For}

ρ∈ M(γ), we define the exponential tiltρ〈γ〉 ofρas

ρ〈γ〉(d x):= 1

b

ρ(γ)e

γx_{ρ(d x). (5.1)}

Note that the exponential tilt conserves convolution. That is, forρ,η∈ M(γ),(ρ∗η)〈γ〉=ρ〈γ〉∗η〈γ〉.

Let {Xj}∞j=0 be i.i.d. random variables with distribution ν(1). Let Y be a random variable with

distributionµ2 independent of{Xj}. Define a random walk{Sn}∞n=0asSn:=

Pn

j=1Xj forn≥1 and

S0:=0. Recall thatc0:=ν(1,∞).

Lemma 5.1. Letµbe a distribution inID+with Lévy measureν.

(i)If0≤γ∗<∞andµ_b(γ∗) =∞, then C∗=∞.

(ii)If0≤γ∗<∞andµb(γ∗)<∞, then C∗≤µb(γ∗)≤C∗.

Proof. Suppose that 0< γ∗<∞andµb(γ∗) =∞. Sinceµ∈ M(γ)for 0< γ < γ∗, µ〈γ〉 exists. We haveµ〈γ〉 = (µ3)〈γ〉∗(µ4)〈γ〉 andµb3(γ∗) =∞. Defineν1(d x):=eγxν(d x). Note thatν1 is the Lévy measure ofµ〈γ〉. Since(µ3)〈γ〉 is one-sided, we have

lim inf

r→∞

(µ3)〈γ〉(r)

ν1(r) ≥1.

By using integration by parts, we have

¯

µ〈γ〉(r) =

eγrµ¯(r)

b

µ(γ) + γ

b

µ(γ)

Z ∞

r

eγtµ¯(t)d t, (5.2)

and

ν1(r) =eγrν¯(r) +γ

Z ∞

r

eγtν¯(t)d t. (5.3)

Thus we see from Fatou’s lemma that

C∗ b

µ(γ) ≥ lim infr→∞

≥

Thus assertion (i) is true. It is clear from Proposition 4.1 that assertion (ii) is true for γ∗ = 0. Suppose that 0≤γ∗<∞andµ_b(γ∗)<∞. Then we see thatµ〈γ∗_〉∈ H. Defineν₂(d x):=eγ

∗_x ν(d x). Thus we obtain from Proposition 4.1 and (5.2) and (5.3) with replacingγbyγ∗that

C∗

Thus we have proved the lemma. ƒ

Proposition 5.1. Letµbe a distribution inID+with Lévy measureν. Suppose thatν(1)∈ O S . Then

Since{T_n(y)}∞_n=1is a sequence of increasing functions, uniformly bounded on all finite intervals, by the selection principle (see Chap. VIII of[17]) there exists an increasing subsequence{λn}of{xn}

Proof. Suppose thatν(1) ∈ O L. Defineh∗(x):=lim infr→∞ν(r−x)/ν(r). Since ν(1)∈ O L, we

We see from Lemma 3.3 that

lim

k→∞P(Y > λk)/P(X0> λk) =0. (5.8)

Thus we established from (5.4)-(5.8) that

≥

Thus we have proved the proposition. ƒ

By virtue of the dominated convergence theorem, we see that

Thus we obtain from (5.9)-(5.12) that, forn≥1,

= KnP(S1+Y >x) P(X0>x)

≤Kn(d₁+δ),

with some positive constantsK andδ. Thus we can use Fatou’s lemma and establish from (5.14) that

C∗ = lim sup

x→∞

e−c0

∞

X

n=1

c₀n−1

n!

P(S_n+Y >x)

P(X0>x)

≤ e−c0

∞

X

n=1

c₀n−1

n! dn

≤ e−c0

∞

X

n=1

c₀n−1 n! I

∗_(µ 2)

n−1

X

k=0

I∗(ρ)k(d∗−I∗(ρ))n−1−k=J(µ).

We have proved the proposition. ƒ

Proof of Theorem 2.1. Assertion (i) is due to Proposition 3.1. Assertion (ii) is due to Propositions

5.1-5.3. ƒ

Proof of Corollary 2.1. We see from Theorem 2.1 (ii) that 0 ≤ γ∗ < ∞, µb(γ∗) < ∞ and

C =C∗=C∗=µb(γ∗). ƒ

The following is due to Theorem 1 of[13]and Theorem 3.1 of[24], and conclusively to Theorem 1.1 of [31]. An interesting history of the establishment of this result is found in [31]. It has an application to the local subexponentiality of an infinitely divisible distribution. See [32]. Applications of the classS(γ)to Lévy processes are found in[4; 6; 21].

Lemma 5.3. Letγ≥0.Letµbe a distribution inID+with Lévy measureν. Then the following are

equivalent:

(1)µ∈ S(γ).

(2)ν(1)∈ S(γ).

(3)ν(1)∈ L(γ),µb(γ)<∞, andµ¯(x)∼µb(γ)ν¯(x).

(4)ν(1)∈ L(γ)and, for some C ∈(0,∞),µ¯(x)∼Cν¯(x).

Remark 5.2. Letγ≥0. We see from the above lemma thatS(γ)∩ID+is closed under convolution

roots. The classS is closed under convolution roots. Refer to Theorem 2 of[13]in the one-sided case and see Prposition 2.7 of[31]in the two-sided case. However, we do not know whether the classS(γ) is closed under convolution roots for γ >0, so far. We find from Theorem 2.1 of[32]

thatS(γ)onR_{+is closed under convolution roots for some (equivalently for all)γ >0 if and only}

if so is the locally subexponential class onR_+.

Proof of Corollary 2.2. Let a ∈R_{. Note that µ}∗δ−a is also an infinitely divisible distribution on

Corollary 2.1 that 0≤γ∗<∞,µ_b(γ∗)<∞andC(a_j) =µ_b(γ∗)exp(−γ∗a_j)for j=1, 2. Define a set

EasE:={ma1+na2:m, n∈Z}. Then we have, for anya∈E,

µ(x+a)∼exp(−γ∗a)µ(x). (5.15)

Note thatEis a dense set inR_{becausea1/a2is irrational. Thus we have (5.15) for anya∈E}¯₌R_{. It}

follws thatµ∈ L(γ∗)andµ(x)∼µ_b(γ∗)ν(x). Thus we conclude from Lemma 5.3 thatµ∈ S(γ∗). ƒ

6

Proof of Theorem 2.2

Albin [1] asserted that ifν(1) ∈ L(γ), then µ ∈ L(γ). His proof in the case γ = 0 is complete.

However, his proof in the case γ > 0 depends on an incomplete lemma which is stated without precise proof.

Assertion (Lemma 2.1 of[1]) Letρ∈ L(γ)be supported on[0,∞)for someγ≥0. Given constants

ε >0and t∈R_{, pick a constant x0∈}R_{such that}

ρ(x−t)

ρ(x) ≤(1+ε)e

γt _{for x≥x}

0. (6.1)

Then

ρn∗_(x−t)

ρn∗_(x) ≤(1+ε)e

γt _{for x≥n(x}

0−t) +t. (6.2)

Remark 6.1. The assertion above is correct forγ=0. But, we can make the following

counterex-ample of this assertion forγ >0. Indeed, letρ(x):=R∞

x e

−u_{du. Thenρ∈ L(1)and the condition}

(6.1) is satisfied for x₀ =0. Notice thatρn∗_{(x) = ((n−1)!)}−1R∞

x u

n−1_e−u_{du. Let t < 0 and put}

c:=1−t. We have

lim

n→∞

ρn∗_(cn−t)

ρn∗_(cn) = nlim→∞

R∞

0 (cn−t+u)

n−1_e−(cn−t+u)_du

R∞

0 (cn+u)

n−1_e−(cn+u)_du

= et lim

n→∞

R∞ 0 (1+

−t+u cn )

n−1_e−u_du

R∞ 0 (1+

u cn)

n−1_e−u_du =e

−c−1_t

·et.

Takeε satisfying 1+ε < e−c−1t. The inequality (6.2) does not hold for sufficiently largen. The example for a generalγ >0 is analogous and is omitted.

Lemma 6.1. Letρandηbe distributions onR_{. Letγ≥0.}

(i) (Lemma 2.5 of [31]) If ρ,η ∈ L(γ), then ρ∗η ∈ L(γ). In particular, if ρ ∈ L(γ), then

(ii) (Lemma 1 of[5]) If ρ ∈ L(γ), then, for every a∈R_{, there is a positive constant M(a) which}

depends only on a such that, for every x∈R_{and every n}≥1,

ρn∗_{(x−a)≤M(a)ρ}n∗_{(x). (6.3)}

The following lemma is suggested by an argument in[5].

Lemma 6.2. Letγ≥0, a∈R_{. Suppose thatρ∈ L(γ). For anyε with0< ε <1, there is b >0}

such that for all n≥1and for all x∈R,

ρ(n+1)∗_{(x+a)≤(1+ε)e}−γa_ρ(n+1)∗_{(x) +ρ}n∗_{(x+a−b), (6.4)}

and

ρ(n+1)∗_{(x+a)≥(1−ε)e}−γa_(ρ(n+1)∗_(x)−ρn∗_{(x−b)). (6.5)}

Proof.We have

ρ(n+1)∗_{(x) =}

Z (x−b)+

−∞

ρ(x−y)ρn∗(d y) +

Z ∞

(x−b)+

ρ(x−y)ρn∗(d y)

= K1(x) +K2(x).

Let 0< ε <1. Sinceρ∈ L(γ), it follows that

(1−ε)e−γa≤ ρ(x−y+a)

ρ(x−y) ≤(1+ε)e −γa

for sufficiently large b>0 and y≤ x−b. Notice that

K1(x+a) =

Z (x−b)+

−∞

ρ(x−y+a)

ρ(x−y) ρ(x−y)ρ

n∗_{(d y).}

Hence we obtain that, for sufficiently largeb>0,

(1−ε)e−γaK1(x)≤K1(x+a)≤(1+ε)e−γaK1(x).

Here we have K₁(x)≤ρ(n+1)∗_{(x)and K}

2(x)≤ρn∗(x−b). Hence we get (6.4). Furthermore, we have

K₁(x) =ρ(n+1)∗_(x)−K

2(x)≥ρ(n+1)∗(x)−ρn∗(x−b).

Hence we get (6.5). ƒ

Proof. As in the proof of Theorem 1.1 of[31], we see from Lemma 2.5 of [24]that ifµ∈ L(γ), thenµ1∈ L(γ). Conversely, ifµ1∈ L(γ), then we haveµ∈ L(γ)by Lemma 2.1 of[24]. ƒ

Proposition 6.1. Letµbe a distribution inID+with Lévy measureν. Letγ≥0. Ifν(1)∈ L(γ), then

µt∗ ∈ L(γ)for all t >0. In the converse direction, ifµt∗∈ L(γ) for all t>0andν(1)∈ O S, then

ν(1)∈ L(γ).

Proof. We prove the first assertion. Suppose that ν(1) ∈ L(γ). Without loss of generality, we

can assume that t = 1. Denote ρ := ν(1) and λn := e−c0c0n(n!)

Furthermore, we obtain from Lemmas 6.1 (ii) and 6.2 that, for all x∈R_,

J_N(x+a) ≥ (1−ε)e−γa

In consequence, it follows that, for sufficiently large x,

= €(1+ε)e−γa+εM(b−a)Šµ1(x),

and

µ1(x+a) ≥ (1−ε)e−γaIN(x) + (1−ε)e−γa(JN(x)−εM(b)µ1(x)) = €(1−ε)e−γa−εM(b)Šµ1(x).

Thus we obtain that

(1−ε)e−γa−εM(b) ≤ lim inf

x→∞

µ1(x+a) µ1(x)

≤ lim sup

x→∞

µ1(x+a) µ1(x)

≤(1+ε)e−γa+εM(b−a).

Lettingε→0, we can showµ1∈ L(γ). Hence we haveµ∈ L(γ)by Lemma 6.3.

Next we prove the second assertion. Suppose thatµt∗∈ L(γ) for any t > 0 andν(1) ∈ O S. We

obtain from Lemma 6.3 thatµt₁∗∈ L(γ)for anyt>0. DefineC_∗(t)andC∗(t)fort>0 as

C_∗(t):=lim inf

r→∞

µt₁∗(r)

tν(r),

and

C∗(t):=lim sup

r→∞

µ₁t∗(r)

tν(r).

Sinceµ₁t∗is one-sided, we find from Proposition 2 of[11]that 1≤C∗(t). We see from Theorem 2.1 (ii) thatC∗(t)≤J(µt₁∗). SinceI∗(δ0(d x)) =1, we can representJ(µ1t∗), forB=0, as

J(µt₁∗) =exp(c₀t(I∗(ν(1))−1)),

and, forB>0, as

J(µ₁t∗) =exp(c0t(I∗(ν(1))−1))

exp(c₀B t)−1

c₀B t .

Thus we have lim_t→0J(µ1t∗) =1. Hence we obtain that

lim

t→0C∗(t) =limt→0C

∗_{(t) =1.}

Thus we see that, for anya∈R_,

e−γa=lim

t→0lim infr→∞

µ₁t∗(r+a)

µt∗ 1 (r)

=lim inf

r→∞

ν(r+a) ν(r)

≤ lim sup

r→∞

ν(r+a)

ν(r) =limt→0lim sup_r→∞ µt∗

1 (r+a)

µ₁t∗(r)

=e−γa.

Now we prove Theorem 2.2. We can obtain the same upper bound ofC∗as in (2.3) again by using Corollary 2.6 (ii) of[7].

Proof of Theorem 2.2. Note thatγ∗=γ. Assertion (i) is due to Proposition 6.1.

Next we prove (ii). Sinceν(1)∈ L(γ)⊂ O L, we see that m(x;{λn}) =eγx for any{λn} ∈Λand

thereby obtain from Proposition 5.2 that

C∗≥I∗(µ) =µb(γ).

Hence we see from Lemma 5.1 (ii) thatC_∗=µ_b(γ)∈(0,∞].

Lastly, we prove assertions (1)-(3) in (ii). If d∗ = 2νb(1)(γ) < ∞, then ν(1) ∈ S(γ). It follows

from Lemma 5.3 that C_∗ = C∗=µ_b(γ). Suppose that 2ν_b(1)(γ) <d∗< ∞. Thenν(1)∈ O S \S(γ).

We have C∗ ≥ µ_b(γ) by Theorem 2.1. Next suppose that C∗ = µ_b(γ). Since C∗ = µb(γ), we have µ(r)∼ µ_b(γ)ν(r). As we haveν(1)∈ L(γ), we obtain from Lemma 5.3 thatν(1)∈ S(γ). This is a

contradiction. Hence C∗ > µb(γ). Since I_∗(ν(1)) =I∗(ν(1)) =νb(1)(γ) and I∗(µ2) =µb2(γ), we have

B=d∗−2ν_b(1)(γ)and

J(µ) =µ_b(γ)exp(c0(d ∗₋₂

b

ν(1)(γ)))−1

c₀(d∗−2ν_b(1)(γ))

.

Thus we have (2.3) by (2.2). If d∗ = ∞, then ν(1)6∈ O S. We see from Proposition 3.1 (ii) that

C∗=∞. ƒ

Remark 6.2. Let µbe a distribution inID₊with Lévy measure ν. Suppose thatν(1)∈ O S. Then

µ∈ L if and only ifν(1)∈ L.

Proof. Suppose thatµ∈ L andν(1)∈ O S. Letρ:=ν(1). Then we haveµ1 ∈ L by Lemma 6.3.

Thus we see that

0= lim

r→∞

µ1(r)−µ1(r+1) µ1(r)

≥e−c0_c

0lim sup

r→∞

ρ(r)−ρ(r+1) µ1(r)

≥0. (6.6)

Sinceρ:=ν(1)∈ O S, we haveρ(r)≍µ1(r)by Proposition 3.1. Hence we obtain from (6.6) that

lim

r→∞

ρ(r)−ρ(r+1) ρ(r) =0,

that is,ν(1)=ρ∈ L. Conversely, we see from Proposition 6.1 that ifν(1)∈ L, thenµ∈ L. ƒ

7

Proof of Theorem 2.3.

Among the classes in Definitions 1.1 and 2.1, only the classesDandH are closed under convolution and under convolution roots, simultaneously. The proof forH is easy and that forD is as follows.

Lemma 7.1. Letρandηbe distributions onR_.

(ii)Ifρn∗∈ D for some n≥1, thenρ∈ D.

(iii)Ifρ,η∈ D, thenρ∗η∈ D. In particular, ifρ∈ D, thenρn∗∈ Dfor all n≥1.

As we have

ρn∗_{(r) =P}

  

n

X

j=1

Xj>r

 

≤

n

X

j=1

P(Xj>n−1r) =nρ(n−1r),

it follows from (7.4) that (ii) holds. ƒ

Lemma 7.3. Letµbe a distribution inID₊with Lévy measureν.

(i)Ifν(1)∈ D, then0<Q∗≤1≤Q∗<∞.

(ii)Ifν(1)∈ D andµis a compound Poisson distribution onR+, then

C∗≤ lim

N→∞lim supr→∞

µ(r−N) ν(r) ≤Q

∗_{. (7.5)}

Proof. We prove (i). Suppose thatν(1)∈ D. Then we have

1≥Q∗≥lim inf_r→∞ ν(2r)

ν(r) >0,

and

1≤Q∗= (Q_∗)−1<∞.

Next we prove (ii). Let r > 0. Notice thatµ(r) = e−aP∞_n=1an(n!)−1ρn∗_{(r), where a:=ν(R}

+) =

ν(0)andρ:=a−1ν. Suppose thatν(1)∈ D. LetN1>0. Now we can use Fatou’s lemma by virtue of Lemma 7.2 (ii). Thus we see from Lemma 7.2 (i) that

C∗ ≤ lim sup

r→∞

µ(r−N₁) ν(r)

≤ e−a

∞

X

n=1

an

n!lim supr→∞

ρn∗_(r−N 1)

aρ(r) ≤Q

∗_.

AsN₁→ ∞, we get (7.5). ƒ

Now we prove Theorem 2.3.

Proof of Theorem 2.3. We prove (i). Ifν(1)∈ D ⊂ O S, we see from Theorem 2.1 and Lemma 7.1 (i)

thatµ(r)≍ν(r)andµ∈ D. Conversely, suppose thatµ∈ D ⊂ O S. Then we see from Proposition 3.1 (iii) that(ν(1))k∗∈ O S for some k≥1 andµ(r)≍(ν(1))k∗(r). It follows from Lemma 7.1 that

(ν(1))k∗∈ Dandν(1)∈ D.

We show (ii). LetY₁ andY₂ be independent random variables with distributionµ1 andµ2, respec-tively. Suppose that ν(1) ∈ D. It follows from Lemma 7.3 (i) that 0< Q∗ ≤ 1≤Q∗ < ∞. Since D ⊂ H, we see from Proposition 4.1 thatC∗≤1≤C∗. Letr>2N >0. Then we have

µ(r) =P(Y₁+Y₂>r)≤ 3

X

k=1

whereJ₁= P(Y₁+Y₂>r, |Y₂| ≤N),J₂=P(Y₁+Y₂>r, |Y₁| ≤N), andJ₃=P(Y₁+Y₂>r, |Y₁|>

Using Lemma 7.3 (ii) again, we obtain that

lim

Example 7.1. Letρbe the Peter and Paul distribution, that is,

≤ lim

n→∞ ¯ ρ(2n₎

¯ ρ(2n_−ε

n)

lim sup

n→∞ ¯ µ(2n₎

λρ¯(2n₎

≤ 2−1C∗. (7.6)

Let{λn} ∈Λ. Note thatm(x;{λn})andI∗(µ2)do not depend on the value ofλ. Sincem(x;{λn})≤

1 forx ≤0, we see thatI∗(µ2)≤1. Thus we have, forB=0,

lim

λ→0J(µ) =λ→0limI ∗_(µ

2)exp(λ(I∗(ρ)−1)) =I∗(µ2)≤1,

and, forB>0,

lim

λ→0J(µ) =λ→0limI ∗_(µ

2)exp(λ(I∗(ρ)−1))

exp(λB)−1

λB =I

∗_(µ 2)≤1.

Hence we obtain from Theorem 2.1 (ii) that

1≤ lim λ→0C

∗_{≤ lim}

λ→0J(µ) =I ∗_(µ

2)≤1,

that is, I∗(µ2) =1 and limλ→0C∗=1. Thus we conclude from (7.6) and Theorem 2.3 (ii) that, for sufficiently smallλ >0,

2−1<1−2−1µ(−∞, 0)≤C∗≤2−1C∗<1<2C∗≤C∗<2.

We have proved all the assertion. ƒ

Acknowledgement.The authors are grateful to Ken-iti Sato for his valuable advice on improvement

of the early draft.

References

[1] J. M. P. Albin. A note on the closure of convolution power mixtures (random sums) of expo-nential distributions.J. Austral. Math. Soc.84(2008), 1-7. MR2469263

[2] J. M. P. Albin, M. Sundén. On the asymptotic behaviour of Lévy processes. I. Subexponential and exponential processes.Stoch. Process. Appl.119(2009), 281-304. MR2485028

[3] N. H. Bingham, C. M. Goldie, J. L. Teugels. (1987).Regular Variation. Cambridge University Press, Cambridge 1987. MR0898871

[4] M. Braverman. Suprema and sojourn times of Lévy processes with exponential tails. Stoch.

Process. Appl.68(1997), 265-283. MR1454836

[5] M. Braverman. On a class of Lévy processes. Statist. Probab. Lett. 75 (2005), 179-189. MR2210548

[7] D. B. H. Cline. Convolutions of distributions with exponential and subexponential tails. J.

Austral. Math. Soc.43(1987), 347-365. MR0904394

[8] J. W. Cohen. Some results on regular variation for distributions in queueing and fluctuation theory.J. Appl. Probab.10(1973), 343-353. MR0353491

[9] D. Denisov, S. Foss, D. Korshunov. On lower limits and equivalences for distribution tails of randomly stopped sums,Bernoulli14(2008), 391-404. MR2544093

[10] P. Embrechts, C.M. Goldie. On closure and factorization properties of subexponential and re-lated distributions.J. Austral. Math. Soc. (Series A)29(1980), 243-256. MR0566289

[11] P. Embrechts, C.M. Goldie. Comparing the tail of an infinitely divisible distribution with inte-grals of its Lévy measure.Ann. Probab.9(1981), 468-481. MR0614631

[12] P. Embrechts, C.M. Goldie. On convolution tails. Stoch. Process. Appl. 13 (1982), 263-278. MR0671036

[13] P. Embrechts, C.M. Goldie, N. Veraverbeke. Subexponentiality and infinite divisibility. Zeit.

Wahrsch. Verw. Gebiete49(1979), 335-347. MR0547833

[14] P. Embrechts, J. A. Hawkes. A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory.J. Austral. Math. Soc. (Series A) 32(1982), 412-422. MR0652419

[15] P. Embrechts, C. Klüppelberg, T. Mikosch.Modelling Extremal Events for Insurance and Finance.

Applications of Mathematics 33, Springer, Berlin 1999. MR1458613

[16] W. Feller. One-sided analogues of Karamata’s regular variation.Enseignement Math.15(1969), 107-121. MR0254905

[17] W. Feller.An Introduction to Probability Theory and its Applications, Vol. II, 2nd ed. Wiley, New York 1971 . MR0270403

[18] S. Foss, D. Korshunov. Lower limits and equivalences for convolution tails. Ann. Probab. 35

(2007), 366-383. MR2303954

[19] C. M. Goldie. Subexponential distributions and dominated-variation tails.J. Appl. Probab.15

(1978), 440-442. MR0474459

[20] L. de Haan, U. Stadtmüller. Dominated variation and related concepts and Tauberian theorems for Laplace transforms.J. Math. Anal. Appl.108(1985), 344-365. MR0793651

[21] C. Küppelberg, A. E. Kyprianou, R. A. Maller. Ruin probabilities and overshoots for general Lévy insurance risk processes.Ann. Appl. Probab.14(2004),1766-1801. MR2099651

[22] C. Küppelberg, J. A. Villasenor. The full solution of the convolution closure problem for convolution-equivalent distributions.J. Math. Anal. Appl.160(1991), 79-92. MR1124078

[23] J. R. Leslie. On the non-closure under convolution of the subexponential family. J. Appl.

[24] A. G. Pakes. Convolution equivalence and infinite divisibility.J. Appl. Probab.41(2004), 407-424. MR2052581

[25] W. Rudin. Limits of ratios of tails of measures.Ann. Probab. 1(1973), 982-994. MR0358919

[26] K. Sato.Lévy Processes and Infinitely Divisible Distributions.Cambridge University Press, Cam-bridge 1999.

[27] T. Shimura, T. Watanabe. Infinite divisibility and generalized subexponentiality. Bernoulli11

(2005), 445-469. MR2146890

[28] T. Shimura, T. Watanabe. On the convolution roots in the convolution-equivalent class. The Institute of Statistical Mathematics Cooperative Research Report 175 (2005), pp1-15.

[29] J. L. Teugels. The class of subexponential distributions. Ann. Probab. 3 (1975), 1000-1011. MR0391222

[30] T. Watanabe. Sample function behavior of increasing processes of class L. Probab. Theory

Related Fields104(1996), 349-374. MR1376342

[31] T. Watanabe. Convolution equivalence and distributions of random sums.Probab. Theory

Re-lated Fields142(2008), 367-397. MR2438696

[32] T. Watanabe, K. Yamamuro. Local subexponentiality and self-decomposability. to appear inJ.

Theoret. Probab.(2010).

[33] A L. Yakimiv. Asymptotic behavior of a class of infinitely divisible distributions.Theory Probab.