- Email: [email protected]
Second order asymptotics for ruin probabilities in a renewal risk model with heavy-tailed claims
Insurance: Mathematics and Economics 51 (2012) 422–429
Contents lists available at SciVerse ScienceDirect
Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime
Second order asymptotics for ruin probabilities in a renewal risk model with heavy-tailed claims Jianxi Lin Mathematics School of Xiamen University, Xiamen, Fujian 361005, China
article
info
Article history: Received March 2012 Received in revised form July 2012 Accepted 3 July 2012 Keywords: Heavy-tailed distributions The convergence rate Ruin probability Renewal risk model Ladder height Random walk
abstract In this paper, we establish the second order asymptotics of ruin probabilities of a renewal risk model under the condition that the equilibrium distribution of claim sizes belongs to a rather general heavy-tailed distribution subclass—the class of second order subexponential distributions with finite mean. What is more, this requirement is proved to be necessary. Furthermore, a rather general sufficient condition on the claim size distribution itself is presented. Moreover, an extension to the case of random walk is also included. © 2012 Elsevier B.V. All rights reserved.
1. Introduction
then the ruin probability of the insurance company with initial capital x can be expressed as
In this paper, we consider a renewal risk model, which satisfies the following conditions.
ψ(x) = P sup Sn > x .
(a) The gross risk premium rate is c > 0, i.e., the insurance company receives (deterministically) c units of money per unit time. (b) The successive claim sizes Zn , n ≥ 1, form a sequence of nonnegative i.i.d. rv’s with a common distribution F which has a finite mean µ. (c) The successive claim inter-occurrence times Yn , n ≥ 1, form a sequence of positive i.i.d. rv’s with a finite mean α . (d) The relative safety loading ρ := (c α − µ)/µ > 0.
Many papers are devoted to the asymptotic estimate of ψ(x) as x tends to ∞ when the claim size distribution F is heavytailed. A rather general heavy-tailed subclass is the so-called subexponential class S , which was introduced independently by Chistyakov (1964) and Chover et al. (1973). By definition, a distribution G on [0, ∞), with its tail denoted by G(x) = 1 − G(x), is said to belong to S , if
n≥0
G∗2 (x) ∼ 2G(x),
x → ∞,
In particular, when the inter-occurrence time distribution is exponential, it reduces to the Cramér–Lundberg model. Denote the net loss over the nth period by
where and throughout this paper, ∼ means that the ratio of two sides tends to 1 as x → ∞ and where G∗2 denotes the 2-fold convolution of distribution G. The following well-known result by Embrechts and Veraverbeke (1982) provides a first order asymptotics of ψ(x).
Xn := Zn − cYn ,
Theorem 1.1 (Embrechts and Veraverbeke (1982)). If Fe ∈ S , then
(1.1)
then Xn , n ≥ 1, form a sequence of i.i.d rv’s, whose distribution is denoted by A. Let S0 = 0 and for n ≥ 1, Sn :=
n
Xk ,
k=1
E-mail address: [email protected]. 0167-6687/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2012.07.001
ψ(x) ∼
1
ρ
Fe (x),
x → ∞,
(1.3)
where Fe denotes the equilibrium distribution of F , i.e., (1.2) Fe (x) :=
1
µ
x
F (t )dt .
0
From Theorem 1 of Korshunov (1997), it is easy to see that the condition Fe ∈ S is also necessary for the relation (1.3) to hold.
J. Lin / Insurance: Mathematics and Economics 51 (2012) 422–429
To provide some condition on F such that Fe ∈ S , Klüppelberg (1988) introduced the class S∗ . By definition, a distribution G on [0, ∞) with finite mean µg is said to belong to S∗ , if x
G(x − y)G(y)dy ∼ 2µg G(x),
x → ∞.
0
The importance of S∗ can be seen from the following result. Theorem 1.2 (Klüppelberg (1988)). If F ∈ S∗ , then Fe ∈ S . The second order asymptotics of ψ(x) in the heavy-tailed case ¯ has been considered by several authors; see for example Baltrunas (1999) and Ale˘skevi˘ciene˙ et al. (2009). In this paper, the sufficient and necessary condition for the second order asymptotics of ψ(x) is provided. Our condition requires Fe to belong to the class S2 of the so-called second order subexponential distributions with finite means, introduced by Lin (2010) (for its definition, see Section 2 below). The main results in this respect are presented in Section 3.1 below. Our method is to express ψ(x) as the tail of a subordinated distribution and then apply the results by Lin (2010), in which a flexible framework was provided to establish the second order asymptotics of subordinated distributions. The main difficulty of this procedure is to establish the second order asymptotics for the strict ascending ladder height distribution of {Sn }n≥0 . Such a difficulty is settled by Lemmas 4.4 and 4.5 below. For related works about the second order asymptotics, we refer the readers to ¯ ¯ ¯ Baltrunas (2005), Baltrunas and Omey (1998, 2002), Baltrunas et al. (2006), Borovkov (2002), Geluk and Pakes (1991), Geluk (1992, 1996), Grübel (1987), Mikosch and Nagaev (2001), Omey (1994), Omey and Willekens (1986, 1987), among others. For higher-order asymptotic expansions, see Albrecher et al. (2010), Barbe et al. (2007a,b), Barbe and McCormick (2009), among others. The results in this part are stated in the context of insurance; however they can be easily extended to the case of general random walk, results for which are presented in Section 5. As the second contribution of this paper, we present some condition on F such that Fe ∈ S2 . We find that this condition also make F form a subclass of distributions with finite second order moment, which we denote by H . The properties of H are presented in Section 3.2. We show that H is a subclass of S∗ and a similar result as that of Theorem 1.2 holds. Moreover, some easily verified conditions on F such that F ∈ H are provided, from which we can see that H is large enough to include the following distributions. Pareto distributions: G(x) = cx−β , where c > 0 and β > 2; 2 2 Lognormal distributions: G has the density g (x) = e−(ln x−a) /2σ √ /x 2π σ 2 , x > 0, where a ∈ R, σ > 0.
β Weibull distributions: G(x) = e−x , β ∈ (0, 1).
It should be noted that all these typical heavy-tailed distributions are included in S , S∗ and S2 , too. Before stating the main results, some preliminaries are first presented in Section 2. Proofs of the main results are presented in Section 4. 2. Preliminaries Denote the right and left Wiener–Hopf factors of the distribution A by A+ and A− respectively. The right and left Wiener–Hopf factors can be seen as the distributions of the strict ascending ladder height and the weak descending ladder height of the underlying random walk respectively. For details we refer the readers to Chapter XII of Feller (1971), Chapter VI of Rolski et al. (1998), and Embrechts and Veraverbeke (1982).
423
It is well known that the following Wiener–Hopf factorization relation holds: A = A+ + A− − A+ ∗ A− .
(2.1)
Denoting the characteristic functions of A, A+ and A− by φ(s), φ+ (s) and φ− (s) respectively, then the relation (2.1) can be rewritten as 1 − φ(s) = [1 − φ+ (s)][1 − φ− (s)].
(2.2)
Since EX1 < 0, A− is a proper distribution while A+ is defective, i.e. A+ (∞) < 1. Define L(x) := A+ (x)/A+ (∞), then L is a proper distribution. It is well known that
ψ(x) = (1 − A+ (∞))
∞ (A+ (∞))n L∗n (x),
(2.3)
n=0
where L∗0 is the unit mass at zero, L∗n denotes the n-fold convolution of distribution L for n ≥ 1, and that (see Chapter XII of Feller (1971)) L(x) = −(A+ (∞))−1
0
A(x − t )dU− (t ),
(2.4)
−∞ ∗n where the renewal function U− (t) := n=0 A− [t , 0], t ≤ 0. Denote the expectations of A+ and A− by µ+ and µ− respectively. Denote the second order moments of A− , Z1 and Y1 (2) by µ− , µ2 and α2 respectively. It follows from Theorem 1 of Wolff (1984) that if µ2 < ∞, then µ+ < ∞. Now we introduce two other well-known heavy-tailed subclasses. Let D be the class of distributions with dominatedly varying tails, which consists of all the distributions G on [0, ∞) satisfying
∞
lim
x→∞
G(x/2) G(x)
< ∞.
Let L be the class of distributions with long tails, which consists of all the distributions G on [0, ∞) satisfying for every fixed y > 0, lim
x→∞
G(x − y) G(x)
= 1.
(2.5)
For convenience, a distribution G on (−∞, ∞) is still said to belong to L if the relation (2.5) is satisfied. It is well known that if G ∈ D ∩ L , then G ∈ S . Furthermore, if in addition, G has a finite mean, then G ∈ S∗ . Moreover, the following relation also holds: S∗ ⊂ S ⊂ L .
For more properties of these classes we refer the readers to Embrechts et al. (1997) and Rolski et al. (1998). Let G be a distribution on [0, ∞). For t ∈ (0, ∞], we write ∆(t ) = (0, t ], x + ∆(t ) = (x, x + t ] and G(x + ∆(t )) = G(x, x + t ] = G(x + t ) − G(x). The so-called local subexponential class, as well as the local longtailed class, is introduced by Asmussen et al. (2003). By definition, a distribution G on [0, ∞) is said to belong to the local long-tailed class L∆(t ) , if the relation G(x + y + ∆(t )) ∼ G(x + ∆(t )),
x→∞
holds uniformly in y ∈ [0, 1]. Furthermore, G is said to belong to the local subexponential class S∆(t ) , if G ∈ L∆(t ) and G∗2 (x + ∆(t )) ∼ 2G(x + ∆(t )),
x → ∞.
(2.6)
The second order subexponential class S2 is introduced by Lin (2010).
424
J. Lin / Insurance: Mathematics and Economics 51 (2012) 422–429
Definition 2.1. We say that a distribution G on [0, ∞) with finite mean µg belongs to the second order subexponential class S2 , if for all t ∈ (0, ∞), G ∈ L∆(t ) and G∗2 (x) − 2G(x) ∼ 2µg G(x, x + 1],
x → ∞.
(2.7)
Remark 2.1. The original definition of S2 in Lin (2010) requires G ∈ S∆(t ) , which is stronger than G ∈ L∆(t ) here. However these two definitions are equivalent, which is shown as follows. To this end, we only need to show that if G satisfies Definition 2.1 then G ∈ S∆(t ) for all t ∈ (0, ∞). Note that G∗2 (x + ∆(t )) − 2G(x + ∆(t )) then by (2.7) and in view of G ∈ L∆(t ) , it is easy to see that (2.8)
where and throughout this paper, by a(x) = o(b(x)) we mean lim
x→∞
a(x)
= 0.
b(x)
G(x + ∆(t )) ∼ tG(x, x + 1]. As has been pointed out before, the class S2 is large enough to include all the typical heavy-tailed distributions, such as the Pareto distributions with β > 1, the lognormal distributions and the Weibull distributions with β ∈ (0, 1). For more properties of S2 , we refer the readers to Lin (2010).
µ2 · F (x), (c α − µ)2
Fe (x) ∼
3.1. Second order asymptotics for ruin probabilities
Theorem 3.1. Assume µ2 , α2 < ∞, F ∈ L and the distribution A is non-arithmetic. Then the relation
Proof. For the Cramér–Lundberg model, Yi has the exponential distribution with mean α ; hence from Chapter XII of Feller (1971), we have A− (x) =
ex/c α , 1,
x<0 x≥0
x 1 F (t )dt , A+ (x) = c α 0 0,
x≥0 x < 0.
µ− = −c α,
µ+ =
µ2 2c α
and α2 = 2α 2 ,
which implies K =
µ2 . (c α − µ)2
Fe (x) ∼ K · F (x),
In this section, sufficient conditions on F such that Fe ∈ S2 are provided. To this end, we introduce a new class H . By definition, we say that the distribution F belongs to H , if
x/2 lim
x → ∞,
(3.1)
0
yF (x − y)F (y)dy F (x)
x→∞
=
where
Remark 3.3. It is easy to see
µ2 + c 2 α2 − 2c 2 α 2 − 2µ+ µ− K := , 2(c α − µ)2
µ2
holds if and only if Fe ∈ S2 . Remark 3.1. Theorem 3.1 is taken from Lin (2008). Since this result is not easily available we will provide a proof here. Further note that by (2.3), it is easy to see
sup Sn
(3.2)
3.2. Sufficient conditions for Fe ∈ S2
In this section, the second order asymptotics of ψ(x) is provided.
E
x→∞
Since the distribution of Yi is absolute continuous, so is A, then by Theorem 3.1, we obtain the desired result.
3. Main results
ρ
1
ρ
From this, it is easy to obtain
From this and (2.8), we obtain (2.6), as required.
ψ(x) −
ψ(x) −
and
Since G ∈ L∆(t ) for all t ∈ (0, ∞), then by Lemma 3.1 of Lin (2010), we have
1
Corollary 3.1. For the Cramér–Lundberg model, assume µ2 < ∞ and F ∈ L . Then the relation
holds if and only if Fe ∈ S2 .
= {G∗2 (x) − 2G(x)} − {G∗2 (x + t ) − 2G(x + t )}; G∗2 (x + ∆(t )) − 2G(x + ∆(t )) = o(G(x, x + 1]),
Remark 3.2. By Theorem 2.1.4 of Lukacs (1970), we know that a distribution is non-lattice if and only if its characteristic function, say m(t ), satisfies |m(t )| < 1 for all t ̸= 0. From this, it is easy to see that if any distribution of Z1 or Y1 is non-lattice, so is A and thus, A is non-arithmetic. Hence Theorem 3.1, as well as Corollary 3.1 below, unifies and improves the corresponding results in Ale˘skevi˘ciene˙ ¯ et al. (2009), and Baltrunas (1999, 2005).
= µ+ /(1 − A+ (∞)),
n≥0
which, combined with (4.14) below, gives
µ+ µ− = (µ − c α)E sup Sn . n≥0
From this, it is easy to see that the relation (3.1) is the same as that in Theorem 1 of Ale˘skevi˘ciene˙ et al. (2009). However note that by Corollaries 3.2 and 3.3 and Remark 3.4 below our result is more general than the one in Ale˘skevi˘ciene˙ et al. (2009).
2
µ2 2
< ∞.
(3.3)
∞
yF (y)dy.
=
(3.4)
0
Theorem 3.2. (1) H ⊂ S∗ . (2) If F ∈ H , then Fe ∈ S2 . Corollary 3.2. Suppose α2 < ∞, F ∈ H and the distribution A is non-arithmetic. Then the relation (3.1) holds. Corollary 3.3. For the Cramér–Lundberg model, if F ∈ H , then the relation (3.2) holds. Next we provide some conditions for F ∈ H . Similar conditions ¯ for S∗ can be found in Klüppelberg (1988, 1989) and Baltrunas et al. (2006). For two distributions F and G, they are said to be weakly tail-equivalent, if there exist two constants c1 , c2 ∈ (0, ∞) such that c1 ≤ lim inf x→∞
G(x) F (x)
≤ lim sup x→∞
G(x) F (x)
≤ c2 .
J. Lin / Insurance: Mathematics and Economics 51 (2012) 422–429
In particular, they are said to be tail-equivalent, if there exists some constant c ∈ (0, ∞) such that lim
x→∞
G(x) F (x)
= c.
425
Lemma 4.2 (Theorem 2.1 of Lin (2010)). Let G be a distribution on
[0, ∞) with finite mean µg . Define H (x) :=
∞
pn G∗n (x),
n=0
Proposition 3.1. If G and F are weakly tail-equivalent and F , G ∈ L , then F ∈H ⇔G∈H.
(3.5)
where {pn }n≥0 is a probability distribution on {0, 1, 2, . . .}. (1) If G ∈ S2 and
In particular, if G and F are tail-equivalent, then the relation (3.5) holds.
H (x) −
n=0
∼ µg
The hazard function of F is defined as
G(x)
npn
∞
n(n − 1)pn
q(x) := [Q (x)]′ = f (x)/F (x). The following result presents a Pitman-type condition for F ∈ H . Proposition 3.3. Assume the hazard rate q(x) of F exists and is eventually decreasing to 0 as x tends to ∞. If
Lemma 4.3. Assume g1 (x) is a bounded measurable function on (−∞, 0], g1 (x) → 0 as x → −∞, and G2 ∈ L . Then
x exp{xq(x)}F (x)dx < ∞,
(3.6)
∞
then F ∈ H . In the case that q(x) is not necessarily eventually decreasing, the following result may be useful. Define r := lim sup xq(x)/Q (x).
g1 (x − t )dG2 (t ) = o(G2 (x)),
(4.2)
Proof. For any y > 0, ∞
g1 (x − t )dG2 (t ) =
x +y
x
g1 (x − t )dG2 (t ) x
∞
x→∞
g1 (x − t )dG2 (t )
+ x+y
Proposition 3.4. If r < 1 and there exists ε > 0 such that r (ε) := r + ε ∈ (0, 1) and
:= I1 + I2 . It is easy to see that
(y)dy < ∞,
(3.7)
0
then F ∈ H . Corollary 3.4. If r < 1 and xq(x) → ∞ as x tends to ∞, then F ∈H.
|I1 | ≤ M (G2 (x) − G2 (x + y)), where M := supx<0 |g1 (x)| < ∞. Hence by G2 ∈ L , we obtain for every fixed y > 0, I1 = o(G2 (x)),
x → ∞.
(4.3)
Remark 3.4. By Theorem 1.8 of Borovkov (2002), Corollary 3.4 and Proposition 3.1, it is easy to see that the so-called strong semiexponential class (see Borovkov, 2002; Ale˘skevi˘ciene˙ et al., 2009) is a proper subclass of H .
On the other hand, we have
From Proposition 3.2 the Pareto distributions with β > 2 belong to H . From Proposition 3.3 and by standard calculations, both the Weibull distributions with β ∈ (0, 1) and the lognormal distributions belong to H . What is more, by Proposition 3.1, all the distributions which are tail-equivalent to the above distributions belong to H . Hence by Corollary 3.2, for all these distributions, the relation (3.1) holds.
Since g1 (x) → 0 as x → −∞, then
|I2 | ≤ sup |g1 (x)| · G2 (x + y). x<−y
sup |g1 (x)| → 0,
x<−y
4. Proofs
lim sup lim sup
Lemma 4.1 (Proposition 2.3 of Lin (2010)). Let G and H be two distributions. If G ∈ S2 and there exist constants k > 0, c ∈ R such that G(x, x + 1]
x→∞
|I2 | G2 (x)
= 0,
which, combined with (4.3), gives (4.2).
We first prove the results in Section 3.1. The following two lemmas are from Lin (2010).
H (x) − k · G(x)
y → ∞,
and thus by G2 ∈ L , we obtain
y→∞
then H ∈ S2 .
x → ∞.
x
0
1−r (ε)
(4.1)
(2) Suppose G ∈ L∆(t ) for all t ∈ (0, ∞), and G (x) = o(G(x, x+1]). If the relation (4.1) holds and there exists some n ≥ 2 such that pn > 0, then G ∈ S2 .
∞
yF
x → ∞. 2
If F has a density f , the hazard rate of F is defined as
∞
G(x, x + 1],
n =0
Q (x) := − ln(F (x)).
pn z n is analytic at z = 1, then
n =0
Proposition 3.2. Assume µ2 < ∞. If F ∈ D ∩ L , then F ∈ H .
∞
∞
→ c,
Denote X− = min{X1 , 0}. 2 Lemma 4.4. If E (X− ) < ∞, A ∈ L and the distribution A is nonarithmetic, then
L(x) = −
x → ∞,
+
1 A+ (∞)µ−
∞
A(t )dt
x
µ(−2) A(x) + o(A(x)), 2A+ (∞)µ2−
x → ∞.
426
J. Lin / Insurance: Mathematics and Economics 51 (2012) 422–429
Since F ∈ L , then by the dominated convergence theorem, as x → ∞,
Proof. Note that A+ ∗ A− (x) =
∞
A− (x − y)dA+ (y) ≤ A+ (∞)A− (x).
∞
η(y)F (x + y)dy ∼
0 0
Substituting this into (2.1), it is easy to see that
(2)
and thus, µ− < ∞. Since the distribution A is non-arithmetic, it follows from the proof of Lemma 1 of Sgibnev (2007) that A− is also non-arithmetic. Hence by the renewal theorem (see Feller (1971), Chapter XI, Section 3, Theorem 1) and in view of µ− < 0, we have U− (t) −
t
−
µ−
→ 0,
2µ2−
t → −∞.
From (4.5) it follows that A ∈ L . Thus, by (4.5), (4.7), (4.8) and Lemma 4.4, we obtain the desired result. Lemma 4.6. If E (Z12 ) < ∞, then 2
Fe (x) = o(F (x)).
E (Z1 1{Z1 >x} ) =
tdF (t )
= xF (x) +
−∞
µ(−2) 1 + 2 µ− µ− 2µ− −∞ ∞ (2) x−t µ =− U− (x − t ) − − −2 dA(t ) µ− 2µ− x ∞ (2) 1 µ− + A(t )dt − A(x). µ− x 2µ2−
0
t
A(x − t )d U− (t ) −
=
∞
A(t )dt
−
x
x → ∞.
(4.4)
∞
F (x + y)dη(y).
Since F ∈ L , then by the dominated convergence theorem, we have x → ∞.
A(t )dt = x
∞
F (t + y)dη(y)dt
x ∞ 0 ∞ = 0
∞
F (t + y)dtdη(y)
0
∞
F (t )dtdη(y).
(4.6)
x +y
A(t )dt =
∞
F (t )dt − x
≤
µ2
1 1
µ2
{E ([Z1 1{Z1 >x} ]1{Z1 >x} )}2 E (Z12 1{Z1 >x} )E (1{Z1 >x} ) F (x)E (Z12 1{Z1 >x} ),
Fe (x, x + 1] =
1
x +1
µ
F (t )dt ∼ x
1
µ
F (x),
x → ∞;
hence by Fe ∈ S2 , it follows from Lemmas 4.1 and 4.5 that L ∈ S2 . Thus by (2.3) and Lemma 4.2(1), we obtain
ψ(x) −
A+ (∞) 1 − A+ (∞)
L(x) ∼
2A+ (∞) · µ+
{1 − A+ (∞)}2
L(x, x + 1],
By Lemma 4.5, for every fixed t > 0, L( x , x + t ] ∼ −
η(y)F (x + y)dy. 0
t A+ (∞)µ−
F (x),
x → ∞.
−4µ+ µ− + {1 − A+ (∞)}µ(−2) + 2c α{1 − A+ (∞)}µ− F (x). 2{1 − A+ (∞)}2 µ2−
(4.11)
(4.7)
(4.12)
Now we aim to simplify the coefficients in the relation (4.12). Differentiating both sides of (2.2) with respect to s gives
Letting s = 0 in (4.13), we obtain
∞
(4.10)
φ ′ (s) = φ+′ (s){1 − φ− (s)} + φ−′ (s){1 − φ+ (s)}.
By using the partial integration in (4.6), we get ∞
µ2
where in the third step, the Cauchy–Schwarz inequality was used. From this we obtain (4.9) since E (Z12 1{Z1 >x} ) → 0 as x → ∞ in the case E (Z12 ) < ∞.
∼
x
=
1
{E (Z1 1{Z1 >x} )}2
Substituting (4.4) and (4.11) with t = 1 into (4.10), we obtain that as x → ∞, ∞ 1 ψ(x) + F (t )dt {1 − A+ (∞)}µ− x
∞
µ2
=
=
(4.5)
On the other hand, by Fubini’s theorem, we obtain ∞
F (t )dt . x
x → ∞.
0
A(x) ∼ F (x),
∞
Proof of Theorem 3.1. We first prove that the condition Fe ∈ S2 is sufficient for the relation (3.1). Since F ∈ L , we have
2 Proof. Since α2 < ∞, EX− < ∞ and thus Lemma 4.4 can be applied. Denote the distribution of cY1 by η. Obviously,
A(x) =
1
2
µ F e (x) A+ (∞)µ−
µ(2) + 2c αµ− + − F (x) + o(F (x)), 2A+ (∞)µ2−
F (t )dt ≥
Hence F e ( x) ≤
Lemma 4.5. If α2 < ∞, F ∈ L and the distribution A is nonarithmetic, then L(x) = −
∞
x
By Lemma 4.3, the first integral on the right-hand side is o(A(x)). Then by (2.4), we obtain the desired result.
x
∞ x
A(x − t )dU− (t )
(4.9)
0
(4.8)
Proof. By integrating by parts, we have
By integrating by parts, we obtain
η(y)dy · F (x) 0
= E (cY1 )F (x) = c α F (x).
A− (x) ≤ A(x)/(1 − A+ (∞)),
µ(−2)
∞
φ ′ (0) = φ−′ (0)(1 − φ+ (0)),
(4.13)
J. Lin / Insurance: Mathematics and Economics 51 (2012) 422–429
i.e., c α − µ = −{1 − A+ (∞)}µ− .
(4.14)
Differentiating (4.13), then letting s = 0 and using the equation 1 − φ− (0) = 0, we have EX12
= {1 −
(2) A+ (+∞)}µ−
− 2µ+ µ− .
(4.15)
Substituting (4.14) and (4.15) into (4.12) and in view of EX12
2
Proof. For any v > 0 and x > 2v ,
which, combined with (4.18), gives F (x − y)F (y)dy F (x)
F (x) v
≥
v
+
x
x/2
F (x − y)F (y)dy =
yF (y)dy +
F (x)
0
yF (x − y)F (y)dy
x/ 2
v
x/ 2
yF (y)dy.
x/4
F (x)
x/ 2 x/4
yF (x−y)F (y)dy F (x)
v
v
lim
F (x − v) F ( x)
−
v 0
yF (y)dy
yF (y)dy
F ( x)
yF (x − y)F (y)dy ≥
=
.
(4.16)
[Fe (x/2) − Fe (x)] = 2
= 1,
yF (x − y)F (y)dy F (x)
≤
v
→
yF (y)dy
(4.17)
0
0
F (x − y)F (y)dy F (x)
v
→
F (y)dy.
(4.18)
1
yF (x − y)F (y)dy F (x)
≤
x → ∞.
1 4 1 4
x/2
xF (x/2)
x/ 4
(4.22)
F (x − y)dy
3x/4
xF (x/2)
x/ 2
1
F (y)dy.
(4.23)
µ2 1 2µ
x/2
F (y)dy
xF (x/2) 2
1
µ
2
x
xF (x/2) 2
x
x/2
F (y)dy
3x/4
x/ 2
Combining (4.22)–(4.24) gives (4.20).
Fe (x + ∆(t )) =
0
F (y)dy.
(4.24)
∞
yF (y)dy,
→
1
µ
x +t
F (y)dy ∼
x
t
µ
F (x),
x → ∞.
From this, it is easy to see that Fe ∈ L∆(t ) for all t ∈ (0, ∞). Since µ2 < ∞, then by Lemma 4.6, Lemma 4.8 and in view of
By (3.3) and (4.17), we obtain
x/2
(4.21)
Proof of Theorem 3.2(2). Since F ∈ H , then by Lemma 4.7, F ∈ L . Hence for every fixed t > 0,
and
v
(4.20)
= 0.
yF (x − y)F (y)dy = o(F (x)),
Proof of Theorem 3.2(1). Let F ∈ H . Then by Lemma 4.7, F ∈ L and thus for every fixed v > 0, as x → ∞, 0
x → ∞.
On the other hand, it is easy to see that
which completes the proof.
v
yF (x − y)F (y)dy
Let x → ∞ in (4.16) and by (3.3), we obtain
x→∞
F (x − y)F (y)dy,
Obviously,
x/2
x/2
it is easy to see that F ∈ S∗ .
F (x − v)
0
(4.19)
Hence
Hence we have
x/2
x → ∞.
0
v→∞ x→∞
F (x)
F (x − v)
F (y)dy, 0
From this and in view of
lim lim
yF (x − y)F (y)dy
0
∞
→
Proof. By (3.3) and (4.17), we have
F (x)
=
x → ∞,
1
x/ 2 x/2
F (y)dy,
→
F (x)
yF (x − y)F (y)dy
v
∞
[Fe (x/2) − Fe (x)]2 = o(F (x)),
Lemma 4.7. If F ∈ H , then F ∈ L .
≤
F (x − y)F (y)dy
Lemma 4.8. If F ∈ H , then
Now we prove the results in Section 3.2.
1 ≤
1
0
have L (x) = o(L(x, x + 1]). Hence Lemma 4.2(2) can be applied to (2.3) and thus, the proof of the first part can be reverted to obtain Fe ∈ S2 .
0
x/ 2
2
we obtain (3.1), as required. Next we prove that the condition Fe ∈ S2 is necessary for the relation (3.1). Since F ∈ L , then by (4.11), we know that L ∈ L∆(t ) for all t ∈ (0, ∞). Moreover, by (4.11), Lemmas 4.5 and 4.6, we
x/2
then in view of F ∈ L and by Pratt’s lemma (Pratt, 1960), which can be seen as an improved version of the dominated convergence theorem, we obtain
x/ 2
= E (Z1 − cY1 ) = µ2 + c α2 − 2c αµ, 2
427
1
x → ∞.
2
2
Fe (x/2) ≤ 2Fe (x) + 2[Fe (x/2) − Fe (x)]2 ,
Since for every fixed y ∈ (1, x/2),
we obtain
F (x − y)F (y) ≤ yF (x − y)F (y),
Fe (x/2) = o(F (x)),
2
x → ∞.
428
J. Lin / Insurance: Mathematics and Economics 51 (2012) 422–429
Note that
Thus, by (3.6) and the dominated convergence theorem, we obtain x/ 2
Fe∗2 (x) =
∞
2
Fe (x − y)dFe (y) + Fe (x/2)
v
yF (x − y)F (y)dy F (x)
0 2
= 2Fe (x)(1 − Fe (x/2)) + Fe (x/2) x/ 2 [Fe (x − y) − Fe (x)]dFe (y). +
Fe (x)dx = 0
µ2 2µ
x/ 2
x→∞
µ2 2µ2
F (x),
x → ∞.
(4.25)
Note that x/ 2
¯ Lemma 4.9 (Lemma 4(ii) of Baltrunas et al. (2006)). If r < 1, then F ∈ L and s(x) is nonincreasing for x large enough. Proof of Proposition 3.4. By Lemma 4.9, there exists v > 0 such that s(x) is nonincreasing on [v, ∞) and for x > v , xq(x)
[Fe (x − y) − Fe (x)]dFe (y) x/2 x 1 F (t )dt F (y)dy = 2 µ 0 x−y
≤ r (ε),
Q (x)
0
and thus, by the mean value theorem, we have for x − y > y > v , Q (x) − Q (x − y) = yq(z ) ≤ r (ε)ys(z ) ≤ r (ε)ys(y) = r (ε)Q (y),
and
where z ∈ (x − y, x). It follows that for x > 2v ,
x
F (t )dt ≤ yF (x − y);
x/2
x −y
v
yF (x − y)F (y)dy F (x)
hence 1
µ2
F ( x)
x/ 2
yF (y)dy ≤
0
x/2
≤
[Fe (x − y) − Fe (x)]dFe (y)
µ
x/ 2
1 2
= v
yF (x − y)F (y)dy.
v
From this and in view of (3.3) and (3.4), we obtain (4.25), as required.
Proof of Proposition 3.2. Note that for all y ∈ (0, x/2), F (x − y) ≤ F (x/2), then in view of F ∈ D ∩ L and by the dominated convergence theorem, we obtain (3.3), i.e., F ∈ H .
x/ 2
≤
0
Proof of Proposition 3.1. Assume F ∈ H , then F ∈ L and the relation (4.21) holds. Since G and F are weakly tail-equivalent, it is easy to see that G also satisfies (4.21) with F replaced by G. Since G ∈ L , G also satisfies (4.17) with F replaced by G, and thus, G∈H. If G and F are tail-equivalent, then the result follows from the facts that H ⊂ L and F ∈ L ⇔ G ∈ L .
x/ 2
0
y exp{Q (x) − Q (x − y) − Q (y)}dy y exp{(r (ε) − 1)Q (y)}dy
x/ 2
=
x
v
Proof of Corollary 3.4. By Proposition 3.4, we only need to verify the condition (3.7). Since xq(x) → ∞, then for every fixed ε ∈ (0, 1 − r ), there exists v > 0 such that for x > v , q(x) ≥
3
yF (x − y)F (y)dy F ( x)
Q (x) − Q (v) ≥
∞
x/2
=
x x −y
x/ 2
≤ v
q(t )dt F (y)dy
y exp v
.
By integration we obtain
yF
1
1 − r (ε) x
3 1 − r (ε)
ln(x/v).
Hence
q(t )dt .
Then F ∈ L since q(x) converges to 0 as x tends to ∞, and thus the relation (4.17) holds. Since q(x) is decreasing eventually, that is to say, there exists a constant v > 0 such that q(x) is decreasing on [v, ∞), then for x > 2v , v
(y)dy.
Then by (3.7) and in view of F ∈ L (by Lemma 4.9), the dominated convergence theorem can be applied to obtain (4.26), which, combined with (4.17), gives (3.3), as required.
0
x/ 2
1−r (ε)
yF
Proof of Proposition 3.3. It is easy to see that F (x) = exp −
r := lim sup xq(x)/Q (x).
[Fe (x − y) − Fe (x)]dFe (y) ∼
yF (x) ≤
(4.26)
Recall that
,
0
yF (y)dy,
s(x) := Q (x)/x.
we only need to prove
v
∞
Define
Thus, in order to prove Fe ∈ S2 , in view of ∞
→
which, combined with (4.17), gives (3.3), as required.
0
y exp{yq(y)}F (y)dy.
1−r (ε)
v
∞
= v ∞ ≤ v
(y)dy
y exp{−(1 − r (ε))Q (y)}dy y exp{−(1 − r (ε))Q (v) − 3 ln(y/v)}dy
= exp{−(1 − r (ε))Q (v) + 3 ln(v)}
v
which is equivalent to (3.7), as required.
∞
y−2 dy < ∞,
J. Lin / Insurance: Mathematics and Economics 51 (2012) 422–429
5. Extensions to the case of random walk Notice that Theorem 3.1 is stated in the context of insurance, that is to say, the relation (1.1) is assumed. However, such a result can be easily extended to the case of general random walk without assuming (1.1). In the following discussion, let {Xn }n≥1 be a sequence of i.i.d. rv’s with finite mean EX1 < 0 and common distribution A. The related random walk is defined as in (1.2). Then by examining the proof of Theorem 3.1 and using Lemma 4.4 in place of Lemma 4.5, it is easy to obtain the following result. Theorem 5.1. Assume E (X12 ) < ∞, A ∈ L and A is non-arithmetic. Then the relation
P
sup Sn > x
+
n ≥0
∞
1 EX1
A(t )dt ∼ Ka · A(x),
x → ∞,
x
where Ka :=
E (X12 ) − 2µ+ µ− 2(EX1 )2
,
holds if and only if Ae ∈ S2 , where the distribution Ae has support on [0, ∞) and is defined as Ae (x) = ∞ 0
1 A(t )dt
x
A(t )dt ,
x ≥ 0.
0
Remark 5.1. The first order result was established by Veraverbeke (1977) and Korshunov (1997). Our result above improves Theorem 5.1 of Borovkov (2002). Acknowledgment The author is thankful to the anonymous referee for his/her helpful comments. References Albrecher, H., Hipp, C., Kortschak, D., 2010. Higher-order expansions for compound distributions and ruin probabilities with subexponential claims. Scandinavian Actuarial Journal 2, 105–135. ˘ ˙ A., Leipus, R., Siaulys, Ale˘skevi˘ciene, J., 2009. Second-order asymptotics of ruin probabilities for semiexponential claims. Lithuanian Mathematical Journal 49, 364–371. Asmussen, S., Foss, S., Korshunov, D., 2003. Asymptotics for sums of random variables with local subexponential behaviour. Journal of Theoretical Probability 16, 489–518. ¯ Baltrunas, A., 1999. Second order behaviour of ruin probabilities. Scandinavian Actuarial Journal 2, 120–133. ¯ Baltrunas, A., 2005. Second order behaviour of ruin probabilities in the case of large claims. Insurance: Mathematics & Economics 36, 485–498. ¯ Baltrunas, A., Omey, E., 1998. The rate of convergence for subexponential distributions. Liet. Matem. Rink. 38 (1), 1–18. ¯ Baltrunas, A., Omey, E., 2002. The rate of convergence for subexponential distributions and densities. Lithuanian Mathematical Journal 42, 1–14.
429
¯ Baltrunas, A., Omey, E., Van Gulck, S., 2006. Hazard rates and subexponential distributions. In: Nouvelle série, vol. 80. Publications de L’ Institut Mathématique, Tome, pp. 29–46. Barbe, P., McCormick, W.P., 2009. Asymptotic Expansions for Infinite Weighted Convolutions of Heavy Tail Distributions and Applications. Memoirs of the American Mathematical Society, 197. Barbe, P., McCormick, W.P., Zhang, C., 2007. Asymptotic expansions for distributions of compound sums of random variables with rapidly varying subexponential distribution. Journal of Applied Probability 44, 670–684. Barbe, P., McCormick, W.P., Zhang, C., 2007. Tail expansions for the distribution of the maximum of a random walk with negative drift and regularly varying increments. Stochastic Processes and their Applications 117, 1835–1847. Borovkov, A.A., 2002. On subexponential distributions and asymptotics of the distribution of the maximum of sequential sums. Siberian Mathematical Journal 43, 995–1022. Chistyakov, V.P., 1964. A theorem on sums of independent positive random variables and its application to branching random processes. Theory of Probability and its Applications 9, 640–648. Chover, J., Ney, P., Wainger, S., 1973. Functions of probability measures. Journal of D’Analyse. 26, 255–302. Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling Extremal Events for Insurance and Finance. Springer, Berlin. Embrechts, P., Veraverbeke, N., 1982. Estimates for probability of ruin with special emphasis on the possibility of large claims. Insurance: Mathematics & Economics 1, 55–72. Feller, W., 1971. An Introduction to Probability Theory and its Applications, vol. 2. Wiley, New York. Geluk, J.L., 1992. Second order tail behaviour of a subordinated probability distribution. Stochastic Processes and their Applications 40, 325–337. Geluk, J.L., 1996. Tails of subordinated laws: the regularly varying case. Stochastic Processes and their Applications 61, 147–161. Geluk, J.L., Pakes, A.G., 1991. Second order subexponential distributions. Journal of the Australian Mathematical Society (Series A) 51, 73–87. Grübel, R., 1987. On subordinated distributions and generalized renewal measures. Annals of Probability 15 (1), 394–415. Klüppelberg, C., 1988. Subexponential distributions and integrated tails. Journal of Applied Probability 25, 132–141. Klüppelberg, C., 1989. Estimation af ruin probabilities by means of hazard rates. Insurance: Mathematics & Economics 8, 279–285. Korshunov, D., 1997. On distribution tail of the maximum of a random walk. Stochastic Processes and their Applications 72, 97–103. Lin, J., 2010. Second order subexponential distributions with finite mean and their applications to subordinated distributions. Journal of Theoretical Probability Online first. Lin, J., 2008. Studies of some probability problems associated with heavy-tailed risk. Ph.D. Thesis. Xiamen University (in Chinese). Lukacs, E., 1970. Characteristic Functions. Charles Griffin, London, Hafner Publ., New York. Mikosch, T., Nagaev, A., 2001. Rates in approximations to ruin probabilities for heavy-tailed distributions. Extremes 4 (1), 67–78. Omey, E., 1994. On the difference between the product and the convolution product of distribution functions. In: Nouvelle série, vol. 55. Publications de L’ Institut Mathématique, Tome, pp. 111–145. (69). Omey, E., Willekens, E., 1986. Second order behaviour of the tail of a subordinated probability distribution. Stochastic Processes and their Applications 21, 339–351. Omey, E., Willekens, E., 1987. Second-order behaviour of distributions subordinate to distribution with finite mean. Communications in Statistics Stochastic Models 3, 311–342. Pratt, J.W., 1960. On interchanging limits and integrals. Annals of Mathematics Statistics 31, 74–77. Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.L., 1998. Stochastic Processes for Insurance and Finance. Wiley, Chichester. Sgibnev, M.S., 2007. Homogeneous conservative Wiener–Hopf equation. Sbornik: Mathematics 198 (9), 1341–1350. Veraverbeke, N., 1977. Asymptotic behavior of Wiener–Hopf factors of a random walk. Stochastic Processes and their Applications 5, 27–37. Wolff, R.W., 1984. Conditions for finite ladder height and delay moments. Operations Research 32 (4), 909–916.
Contents lists available at SciVerse ScienceDirect
Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime
Second order asymptotics for ruin probabilities in a renewal risk model with heavy-tailed claims Jianxi Lin Mathematics School of Xiamen University, Xiamen, Fujian 361005, China
article
info
Article history: Received March 2012 Received in revised form July 2012 Accepted 3 July 2012 Keywords: Heavy-tailed distributions The convergence rate Ruin probability Renewal risk model Ladder height Random walk
abstract In this paper, we establish the second order asymptotics of ruin probabilities of a renewal risk model under the condition that the equilibrium distribution of claim sizes belongs to a rather general heavy-tailed distribution subclass—the class of second order subexponential distributions with finite mean. What is more, this requirement is proved to be necessary. Furthermore, a rather general sufficient condition on the claim size distribution itself is presented. Moreover, an extension to the case of random walk is also included. © 2012 Elsevier B.V. All rights reserved.
1. Introduction
then the ruin probability of the insurance company with initial capital x can be expressed as
In this paper, we consider a renewal risk model, which satisfies the following conditions.
ψ(x) = P sup Sn > x .
(a) The gross risk premium rate is c > 0, i.e., the insurance company receives (deterministically) c units of money per unit time. (b) The successive claim sizes Zn , n ≥ 1, form a sequence of nonnegative i.i.d. rv’s with a common distribution F which has a finite mean µ. (c) The successive claim inter-occurrence times Yn , n ≥ 1, form a sequence of positive i.i.d. rv’s with a finite mean α . (d) The relative safety loading ρ := (c α − µ)/µ > 0.
Many papers are devoted to the asymptotic estimate of ψ(x) as x tends to ∞ when the claim size distribution F is heavytailed. A rather general heavy-tailed subclass is the so-called subexponential class S , which was introduced independently by Chistyakov (1964) and Chover et al. (1973). By definition, a distribution G on [0, ∞), with its tail denoted by G(x) = 1 − G(x), is said to belong to S , if
n≥0
G∗2 (x) ∼ 2G(x),
x → ∞,
In particular, when the inter-occurrence time distribution is exponential, it reduces to the Cramér–Lundberg model. Denote the net loss over the nth period by
where and throughout this paper, ∼ means that the ratio of two sides tends to 1 as x → ∞ and where G∗2 denotes the 2-fold convolution of distribution G. The following well-known result by Embrechts and Veraverbeke (1982) provides a first order asymptotics of ψ(x).
Xn := Zn − cYn ,
Theorem 1.1 (Embrechts and Veraverbeke (1982)). If Fe ∈ S , then
(1.1)
then Xn , n ≥ 1, form a sequence of i.i.d rv’s, whose distribution is denoted by A. Let S0 = 0 and for n ≥ 1, Sn :=
n
Xk ,
k=1
E-mail address: [email protected]. 0167-6687/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2012.07.001
ψ(x) ∼
1
ρ
Fe (x),
x → ∞,
(1.3)
where Fe denotes the equilibrium distribution of F , i.e., (1.2) Fe (x) :=
1
µ
x
F (t )dt .
0
From Theorem 1 of Korshunov (1997), it is easy to see that the condition Fe ∈ S is also necessary for the relation (1.3) to hold.
J. Lin / Insurance: Mathematics and Economics 51 (2012) 422–429
To provide some condition on F such that Fe ∈ S , Klüppelberg (1988) introduced the class S∗ . By definition, a distribution G on [0, ∞) with finite mean µg is said to belong to S∗ , if x
G(x − y)G(y)dy ∼ 2µg G(x),
x → ∞.
0
The importance of S∗ can be seen from the following result. Theorem 1.2 (Klüppelberg (1988)). If F ∈ S∗ , then Fe ∈ S . The second order asymptotics of ψ(x) in the heavy-tailed case ¯ has been considered by several authors; see for example Baltrunas (1999) and Ale˘skevi˘ciene˙ et al. (2009). In this paper, the sufficient and necessary condition for the second order asymptotics of ψ(x) is provided. Our condition requires Fe to belong to the class S2 of the so-called second order subexponential distributions with finite means, introduced by Lin (2010) (for its definition, see Section 2 below). The main results in this respect are presented in Section 3.1 below. Our method is to express ψ(x) as the tail of a subordinated distribution and then apply the results by Lin (2010), in which a flexible framework was provided to establish the second order asymptotics of subordinated distributions. The main difficulty of this procedure is to establish the second order asymptotics for the strict ascending ladder height distribution of {Sn }n≥0 . Such a difficulty is settled by Lemmas 4.4 and 4.5 below. For related works about the second order asymptotics, we refer the readers to ¯ ¯ ¯ Baltrunas (2005), Baltrunas and Omey (1998, 2002), Baltrunas et al. (2006), Borovkov (2002), Geluk and Pakes (1991), Geluk (1992, 1996), Grübel (1987), Mikosch and Nagaev (2001), Omey (1994), Omey and Willekens (1986, 1987), among others. For higher-order asymptotic expansions, see Albrecher et al. (2010), Barbe et al. (2007a,b), Barbe and McCormick (2009), among others. The results in this part are stated in the context of insurance; however they can be easily extended to the case of general random walk, results for which are presented in Section 5. As the second contribution of this paper, we present some condition on F such that Fe ∈ S2 . We find that this condition also make F form a subclass of distributions with finite second order moment, which we denote by H . The properties of H are presented in Section 3.2. We show that H is a subclass of S∗ and a similar result as that of Theorem 1.2 holds. Moreover, some easily verified conditions on F such that F ∈ H are provided, from which we can see that H is large enough to include the following distributions. Pareto distributions: G(x) = cx−β , where c > 0 and β > 2; 2 2 Lognormal distributions: G has the density g (x) = e−(ln x−a) /2σ √ /x 2π σ 2 , x > 0, where a ∈ R, σ > 0.
β Weibull distributions: G(x) = e−x , β ∈ (0, 1).
It should be noted that all these typical heavy-tailed distributions are included in S , S∗ and S2 , too. Before stating the main results, some preliminaries are first presented in Section 2. Proofs of the main results are presented in Section 4. 2. Preliminaries Denote the right and left Wiener–Hopf factors of the distribution A by A+ and A− respectively. The right and left Wiener–Hopf factors can be seen as the distributions of the strict ascending ladder height and the weak descending ladder height of the underlying random walk respectively. For details we refer the readers to Chapter XII of Feller (1971), Chapter VI of Rolski et al. (1998), and Embrechts and Veraverbeke (1982).
423
It is well known that the following Wiener–Hopf factorization relation holds: A = A+ + A− − A+ ∗ A− .
(2.1)
Denoting the characteristic functions of A, A+ and A− by φ(s), φ+ (s) and φ− (s) respectively, then the relation (2.1) can be rewritten as 1 − φ(s) = [1 − φ+ (s)][1 − φ− (s)].
(2.2)
Since EX1 < 0, A− is a proper distribution while A+ is defective, i.e. A+ (∞) < 1. Define L(x) := A+ (x)/A+ (∞), then L is a proper distribution. It is well known that
ψ(x) = (1 − A+ (∞))
∞ (A+ (∞))n L∗n (x),
(2.3)
n=0
where L∗0 is the unit mass at zero, L∗n denotes the n-fold convolution of distribution L for n ≥ 1, and that (see Chapter XII of Feller (1971)) L(x) = −(A+ (∞))−1
0
A(x − t )dU− (t ),
(2.4)
−∞ ∗n where the renewal function U− (t) := n=0 A− [t , 0], t ≤ 0. Denote the expectations of A+ and A− by µ+ and µ− respectively. Denote the second order moments of A− , Z1 and Y1 (2) by µ− , µ2 and α2 respectively. It follows from Theorem 1 of Wolff (1984) that if µ2 < ∞, then µ+ < ∞. Now we introduce two other well-known heavy-tailed subclasses. Let D be the class of distributions with dominatedly varying tails, which consists of all the distributions G on [0, ∞) satisfying
∞
lim
x→∞
G(x/2) G(x)
< ∞.
Let L be the class of distributions with long tails, which consists of all the distributions G on [0, ∞) satisfying for every fixed y > 0, lim
x→∞
G(x − y) G(x)
= 1.
(2.5)
For convenience, a distribution G on (−∞, ∞) is still said to belong to L if the relation (2.5) is satisfied. It is well known that if G ∈ D ∩ L , then G ∈ S . Furthermore, if in addition, G has a finite mean, then G ∈ S∗ . Moreover, the following relation also holds: S∗ ⊂ S ⊂ L .
For more properties of these classes we refer the readers to Embrechts et al. (1997) and Rolski et al. (1998). Let G be a distribution on [0, ∞). For t ∈ (0, ∞], we write ∆(t ) = (0, t ], x + ∆(t ) = (x, x + t ] and G(x + ∆(t )) = G(x, x + t ] = G(x + t ) − G(x). The so-called local subexponential class, as well as the local longtailed class, is introduced by Asmussen et al. (2003). By definition, a distribution G on [0, ∞) is said to belong to the local long-tailed class L∆(t ) , if the relation G(x + y + ∆(t )) ∼ G(x + ∆(t )),
x→∞
holds uniformly in y ∈ [0, 1]. Furthermore, G is said to belong to the local subexponential class S∆(t ) , if G ∈ L∆(t ) and G∗2 (x + ∆(t )) ∼ 2G(x + ∆(t )),
x → ∞.
(2.6)
The second order subexponential class S2 is introduced by Lin (2010).
424
J. Lin / Insurance: Mathematics and Economics 51 (2012) 422–429
Definition 2.1. We say that a distribution G on [0, ∞) with finite mean µg belongs to the second order subexponential class S2 , if for all t ∈ (0, ∞), G ∈ L∆(t ) and G∗2 (x) − 2G(x) ∼ 2µg G(x, x + 1],
x → ∞.
(2.7)
Remark 2.1. The original definition of S2 in Lin (2010) requires G ∈ S∆(t ) , which is stronger than G ∈ L∆(t ) here. However these two definitions are equivalent, which is shown as follows. To this end, we only need to show that if G satisfies Definition 2.1 then G ∈ S∆(t ) for all t ∈ (0, ∞). Note that G∗2 (x + ∆(t )) − 2G(x + ∆(t )) then by (2.7) and in view of G ∈ L∆(t ) , it is easy to see that (2.8)
where and throughout this paper, by a(x) = o(b(x)) we mean lim
x→∞
a(x)
= 0.
b(x)
G(x + ∆(t )) ∼ tG(x, x + 1]. As has been pointed out before, the class S2 is large enough to include all the typical heavy-tailed distributions, such as the Pareto distributions with β > 1, the lognormal distributions and the Weibull distributions with β ∈ (0, 1). For more properties of S2 , we refer the readers to Lin (2010).
µ2 · F (x), (c α − µ)2
Fe (x) ∼
3.1. Second order asymptotics for ruin probabilities
Theorem 3.1. Assume µ2 , α2 < ∞, F ∈ L and the distribution A is non-arithmetic. Then the relation
Proof. For the Cramér–Lundberg model, Yi has the exponential distribution with mean α ; hence from Chapter XII of Feller (1971), we have A− (x) =
ex/c α , 1,
x<0 x≥0
x 1 F (t )dt , A+ (x) = c α 0 0,
x≥0 x < 0.
µ− = −c α,
µ+ =
µ2 2c α
and α2 = 2α 2 ,
which implies K =
µ2 . (c α − µ)2
Fe (x) ∼ K · F (x),
In this section, sufficient conditions on F such that Fe ∈ S2 are provided. To this end, we introduce a new class H . By definition, we say that the distribution F belongs to H , if
x/2 lim
x → ∞,
(3.1)
0
yF (x − y)F (y)dy F (x)
x→∞
=
where
Remark 3.3. It is easy to see
µ2 + c 2 α2 − 2c 2 α 2 − 2µ+ µ− K := , 2(c α − µ)2
µ2
holds if and only if Fe ∈ S2 . Remark 3.1. Theorem 3.1 is taken from Lin (2008). Since this result is not easily available we will provide a proof here. Further note that by (2.3), it is easy to see
sup Sn
(3.2)
3.2. Sufficient conditions for Fe ∈ S2
In this section, the second order asymptotics of ψ(x) is provided.
E
x→∞
Since the distribution of Yi is absolute continuous, so is A, then by Theorem 3.1, we obtain the desired result.
3. Main results
ρ
1
ρ
From this, it is easy to obtain
From this and (2.8), we obtain (2.6), as required.
ψ(x) −
ψ(x) −
and
Since G ∈ L∆(t ) for all t ∈ (0, ∞), then by Lemma 3.1 of Lin (2010), we have
1
Corollary 3.1. For the Cramér–Lundberg model, assume µ2 < ∞ and F ∈ L . Then the relation
holds if and only if Fe ∈ S2 .
= {G∗2 (x) − 2G(x)} − {G∗2 (x + t ) − 2G(x + t )}; G∗2 (x + ∆(t )) − 2G(x + ∆(t )) = o(G(x, x + 1]),
Remark 3.2. By Theorem 2.1.4 of Lukacs (1970), we know that a distribution is non-lattice if and only if its characteristic function, say m(t ), satisfies |m(t )| < 1 for all t ̸= 0. From this, it is easy to see that if any distribution of Z1 or Y1 is non-lattice, so is A and thus, A is non-arithmetic. Hence Theorem 3.1, as well as Corollary 3.1 below, unifies and improves the corresponding results in Ale˘skevi˘ciene˙ ¯ et al. (2009), and Baltrunas (1999, 2005).
= µ+ /(1 − A+ (∞)),
n≥0
which, combined with (4.14) below, gives
µ+ µ− = (µ − c α)E sup Sn . n≥0
From this, it is easy to see that the relation (3.1) is the same as that in Theorem 1 of Ale˘skevi˘ciene˙ et al. (2009). However note that by Corollaries 3.2 and 3.3 and Remark 3.4 below our result is more general than the one in Ale˘skevi˘ciene˙ et al. (2009).
2
µ2 2
< ∞.
(3.3)
∞
yF (y)dy.
=
(3.4)
0
Theorem 3.2. (1) H ⊂ S∗ . (2) If F ∈ H , then Fe ∈ S2 . Corollary 3.2. Suppose α2 < ∞, F ∈ H and the distribution A is non-arithmetic. Then the relation (3.1) holds. Corollary 3.3. For the Cramér–Lundberg model, if F ∈ H , then the relation (3.2) holds. Next we provide some conditions for F ∈ H . Similar conditions ¯ for S∗ can be found in Klüppelberg (1988, 1989) and Baltrunas et al. (2006). For two distributions F and G, they are said to be weakly tail-equivalent, if there exist two constants c1 , c2 ∈ (0, ∞) such that c1 ≤ lim inf x→∞
G(x) F (x)
≤ lim sup x→∞
G(x) F (x)
≤ c2 .
J. Lin / Insurance: Mathematics and Economics 51 (2012) 422–429
In particular, they are said to be tail-equivalent, if there exists some constant c ∈ (0, ∞) such that lim
x→∞
G(x) F (x)
= c.
425
Lemma 4.2 (Theorem 2.1 of Lin (2010)). Let G be a distribution on
[0, ∞) with finite mean µg . Define H (x) :=
∞
pn G∗n (x),
n=0
Proposition 3.1. If G and F are weakly tail-equivalent and F , G ∈ L , then F ∈H ⇔G∈H.
(3.5)
where {pn }n≥0 is a probability distribution on {0, 1, 2, . . .}. (1) If G ∈ S2 and
In particular, if G and F are tail-equivalent, then the relation (3.5) holds.
H (x) −
n=0
∼ µg
The hazard function of F is defined as
G(x)
npn
∞
n(n − 1)pn
q(x) := [Q (x)]′ = f (x)/F (x). The following result presents a Pitman-type condition for F ∈ H . Proposition 3.3. Assume the hazard rate q(x) of F exists and is eventually decreasing to 0 as x tends to ∞. If
Lemma 4.3. Assume g1 (x) is a bounded measurable function on (−∞, 0], g1 (x) → 0 as x → −∞, and G2 ∈ L . Then
x exp{xq(x)}F (x)dx < ∞,
(3.6)
∞
then F ∈ H . In the case that q(x) is not necessarily eventually decreasing, the following result may be useful. Define r := lim sup xq(x)/Q (x).
g1 (x − t )dG2 (t ) = o(G2 (x)),
(4.2)
Proof. For any y > 0, ∞
g1 (x − t )dG2 (t ) =
x +y
x
g1 (x − t )dG2 (t ) x
∞
x→∞
g1 (x − t )dG2 (t )
+ x+y
Proposition 3.4. If r < 1 and there exists ε > 0 such that r (ε) := r + ε ∈ (0, 1) and
:= I1 + I2 . It is easy to see that
(y)dy < ∞,
(3.7)
0
then F ∈ H . Corollary 3.4. If r < 1 and xq(x) → ∞ as x tends to ∞, then F ∈H.
|I1 | ≤ M (G2 (x) − G2 (x + y)), where M := supx<0 |g1 (x)| < ∞. Hence by G2 ∈ L , we obtain for every fixed y > 0, I1 = o(G2 (x)),
x → ∞.
(4.3)
Remark 3.4. By Theorem 1.8 of Borovkov (2002), Corollary 3.4 and Proposition 3.1, it is easy to see that the so-called strong semiexponential class (see Borovkov, 2002; Ale˘skevi˘ciene˙ et al., 2009) is a proper subclass of H .
On the other hand, we have
From Proposition 3.2 the Pareto distributions with β > 2 belong to H . From Proposition 3.3 and by standard calculations, both the Weibull distributions with β ∈ (0, 1) and the lognormal distributions belong to H . What is more, by Proposition 3.1, all the distributions which are tail-equivalent to the above distributions belong to H . Hence by Corollary 3.2, for all these distributions, the relation (3.1) holds.
Since g1 (x) → 0 as x → −∞, then
|I2 | ≤ sup |g1 (x)| · G2 (x + y). x<−y
sup |g1 (x)| → 0,
x<−y
4. Proofs
lim sup lim sup
Lemma 4.1 (Proposition 2.3 of Lin (2010)). Let G and H be two distributions. If G ∈ S2 and there exist constants k > 0, c ∈ R such that G(x, x + 1]
x→∞
|I2 | G2 (x)
= 0,
which, combined with (4.3), gives (4.2).
We first prove the results in Section 3.1. The following two lemmas are from Lin (2010).
H (x) − k · G(x)
y → ∞,
and thus by G2 ∈ L , we obtain
y→∞
then H ∈ S2 .
x → ∞.
x
0
1−r (ε)
(4.1)
(2) Suppose G ∈ L∆(t ) for all t ∈ (0, ∞), and G (x) = o(G(x, x+1]). If the relation (4.1) holds and there exists some n ≥ 2 such that pn > 0, then G ∈ S2 .
∞
yF
x → ∞. 2
If F has a density f , the hazard rate of F is defined as
∞
G(x, x + 1],
n =0
Q (x) := − ln(F (x)).
pn z n is analytic at z = 1, then
n =0
Proposition 3.2. Assume µ2 < ∞. If F ∈ D ∩ L , then F ∈ H .
∞
∞
→ c,
Denote X− = min{X1 , 0}. 2 Lemma 4.4. If E (X− ) < ∞, A ∈ L and the distribution A is nonarithmetic, then
L(x) = −
x → ∞,
+
1 A+ (∞)µ−
∞
A(t )dt
x
µ(−2) A(x) + o(A(x)), 2A+ (∞)µ2−
x → ∞.
426
J. Lin / Insurance: Mathematics and Economics 51 (2012) 422–429
Since F ∈ L , then by the dominated convergence theorem, as x → ∞,
Proof. Note that A+ ∗ A− (x) =
∞
A− (x − y)dA+ (y) ≤ A+ (∞)A− (x).
∞
η(y)F (x + y)dy ∼
0 0
Substituting this into (2.1), it is easy to see that
(2)
and thus, µ− < ∞. Since the distribution A is non-arithmetic, it follows from the proof of Lemma 1 of Sgibnev (2007) that A− is also non-arithmetic. Hence by the renewal theorem (see Feller (1971), Chapter XI, Section 3, Theorem 1) and in view of µ− < 0, we have U− (t) −
t
−
µ−
→ 0,
2µ2−
t → −∞.
From (4.5) it follows that A ∈ L . Thus, by (4.5), (4.7), (4.8) and Lemma 4.4, we obtain the desired result. Lemma 4.6. If E (Z12 ) < ∞, then 2
Fe (x) = o(F (x)).
E (Z1 1{Z1 >x} ) =
tdF (t )
= xF (x) +
−∞
µ(−2) 1 + 2 µ− µ− 2µ− −∞ ∞ (2) x−t µ =− U− (x − t ) − − −2 dA(t ) µ− 2µ− x ∞ (2) 1 µ− + A(t )dt − A(x). µ− x 2µ2−
0
t
A(x − t )d U− (t ) −
=
∞
A(t )dt
−
x
x → ∞.
(4.4)
∞
F (x + y)dη(y).
Since F ∈ L , then by the dominated convergence theorem, we have x → ∞.
A(t )dt = x
∞
F (t + y)dη(y)dt
x ∞ 0 ∞ = 0
∞
F (t + y)dtdη(y)
0
∞
F (t )dtdη(y).
(4.6)
x +y
A(t )dt =
∞
F (t )dt − x
≤
µ2
1 1
µ2
{E ([Z1 1{Z1 >x} ]1{Z1 >x} )}2 E (Z12 1{Z1 >x} )E (1{Z1 >x} ) F (x)E (Z12 1{Z1 >x} ),
Fe (x, x + 1] =
1
x +1
µ
F (t )dt ∼ x
1
µ
F (x),
x → ∞;
hence by Fe ∈ S2 , it follows from Lemmas 4.1 and 4.5 that L ∈ S2 . Thus by (2.3) and Lemma 4.2(1), we obtain
ψ(x) −
A+ (∞) 1 − A+ (∞)
L(x) ∼
2A+ (∞) · µ+
{1 − A+ (∞)}2
L(x, x + 1],
By Lemma 4.5, for every fixed t > 0, L( x , x + t ] ∼ −
η(y)F (x + y)dy. 0
t A+ (∞)µ−
F (x),
x → ∞.
−4µ+ µ− + {1 − A+ (∞)}µ(−2) + 2c α{1 − A+ (∞)}µ− F (x). 2{1 − A+ (∞)}2 µ2−
(4.11)
(4.7)
(4.12)
Now we aim to simplify the coefficients in the relation (4.12). Differentiating both sides of (2.2) with respect to s gives
Letting s = 0 in (4.13), we obtain
∞
(4.10)
φ ′ (s) = φ+′ (s){1 − φ− (s)} + φ−′ (s){1 − φ+ (s)}.
By using the partial integration in (4.6), we get ∞
µ2
where in the third step, the Cauchy–Schwarz inequality was used. From this we obtain (4.9) since E (Z12 1{Z1 >x} ) → 0 as x → ∞ in the case E (Z12 ) < ∞.
∼
x
=
1
{E (Z1 1{Z1 >x} )}2
Substituting (4.4) and (4.11) with t = 1 into (4.10), we obtain that as x → ∞, ∞ 1 ψ(x) + F (t )dt {1 − A+ (∞)}µ− x
∞
µ2
=
=
(4.5)
On the other hand, by Fubini’s theorem, we obtain ∞
F (t )dt . x
x → ∞.
0
A(x) ∼ F (x),
∞
Proof of Theorem 3.1. We first prove that the condition Fe ∈ S2 is sufficient for the relation (3.1). Since F ∈ L , we have
2 Proof. Since α2 < ∞, EX− < ∞ and thus Lemma 4.4 can be applied. Denote the distribution of cY1 by η. Obviously,
A(x) =
1
2
µ F e (x) A+ (∞)µ−
µ(2) + 2c αµ− + − F (x) + o(F (x)), 2A+ (∞)µ2−
F (t )dt ≥
Hence F e ( x) ≤
Lemma 4.5. If α2 < ∞, F ∈ L and the distribution A is nonarithmetic, then L(x) = −
∞
x
By Lemma 4.3, the first integral on the right-hand side is o(A(x)). Then by (2.4), we obtain the desired result.
x
∞ x
A(x − t )dU− (t )
(4.9)
0
(4.8)
Proof. By integrating by parts, we have
By integrating by parts, we obtain
η(y)dy · F (x) 0
= E (cY1 )F (x) = c α F (x).
A− (x) ≤ A(x)/(1 − A+ (∞)),
µ(−2)
∞
φ ′ (0) = φ−′ (0)(1 − φ+ (0)),
(4.13)
J. Lin / Insurance: Mathematics and Economics 51 (2012) 422–429
i.e., c α − µ = −{1 − A+ (∞)}µ− .
(4.14)
Differentiating (4.13), then letting s = 0 and using the equation 1 − φ− (0) = 0, we have EX12
= {1 −
(2) A+ (+∞)}µ−
− 2µ+ µ− .
(4.15)
Substituting (4.14) and (4.15) into (4.12) and in view of EX12
2
Proof. For any v > 0 and x > 2v ,
which, combined with (4.18), gives F (x − y)F (y)dy F (x)
F (x) v
≥
v
+
x
x/2
F (x − y)F (y)dy =
yF (y)dy +
F (x)
0
yF (x − y)F (y)dy
x/ 2
v
x/ 2
yF (y)dy.
x/4
F (x)
x/ 2 x/4
yF (x−y)F (y)dy F (x)
v
v
lim
F (x − v) F ( x)
−
v 0
yF (y)dy
yF (y)dy
F ( x)
yF (x − y)F (y)dy ≥
=
.
(4.16)
[Fe (x/2) − Fe (x)] = 2
= 1,
yF (x − y)F (y)dy F (x)
≤
v
→
yF (y)dy
(4.17)
0
0
F (x − y)F (y)dy F (x)
v
→
F (y)dy.
(4.18)
1
yF (x − y)F (y)dy F (x)
≤
x → ∞.
1 4 1 4
x/2
xF (x/2)
x/ 4
(4.22)
F (x − y)dy
3x/4
xF (x/2)
x/ 2
1
F (y)dy.
(4.23)
µ2 1 2µ
x/2
F (y)dy
xF (x/2) 2
1
µ
2
x
xF (x/2) 2
x
x/2
F (y)dy
3x/4
x/ 2
Combining (4.22)–(4.24) gives (4.20).
Fe (x + ∆(t )) =
0
F (y)dy.
(4.24)
∞
yF (y)dy,
→
1
µ
x +t
F (y)dy ∼
x
t
µ
F (x),
x → ∞.
From this, it is easy to see that Fe ∈ L∆(t ) for all t ∈ (0, ∞). Since µ2 < ∞, then by Lemma 4.6, Lemma 4.8 and in view of
By (3.3) and (4.17), we obtain
x/2
(4.21)
Proof of Theorem 3.2(2). Since F ∈ H , then by Lemma 4.7, F ∈ L . Hence for every fixed t > 0,
and
v
(4.20)
= 0.
yF (x − y)F (y)dy = o(F (x)),
Proof of Theorem 3.2(1). Let F ∈ H . Then by Lemma 4.7, F ∈ L and thus for every fixed v > 0, as x → ∞, 0
x → ∞.
On the other hand, it is easy to see that
which completes the proof.
v
yF (x − y)F (y)dy
Let x → ∞ in (4.16) and by (3.3), we obtain
x→∞
F (x − y)F (y)dy,
Obviously,
x/2
x/2
it is easy to see that F ∈ S∗ .
F (x − v)
0
(4.19)
Hence
Hence we have
x/2
x → ∞.
0
v→∞ x→∞
F (x)
F (x − v)
F (y)dy, 0
From this and in view of
lim lim
yF (x − y)F (y)dy
0
∞
→
Proof. By (3.3) and (4.17), we have
F (x)
=
x → ∞,
1
x/ 2 x/2
F (y)dy,
→
F (x)
yF (x − y)F (y)dy
v
∞
[Fe (x/2) − Fe (x)]2 = o(F (x)),
Lemma 4.7. If F ∈ H , then F ∈ L .
≤
F (x − y)F (y)dy
Lemma 4.8. If F ∈ H , then
Now we prove the results in Section 3.2.
1 ≤
1
0
have L (x) = o(L(x, x + 1]). Hence Lemma 4.2(2) can be applied to (2.3) and thus, the proof of the first part can be reverted to obtain Fe ∈ S2 .
0
x/ 2
2
we obtain (3.1), as required. Next we prove that the condition Fe ∈ S2 is necessary for the relation (3.1). Since F ∈ L , then by (4.11), we know that L ∈ L∆(t ) for all t ∈ (0, ∞). Moreover, by (4.11), Lemmas 4.5 and 4.6, we
x/2
then in view of F ∈ L and by Pratt’s lemma (Pratt, 1960), which can be seen as an improved version of the dominated convergence theorem, we obtain
x/ 2
= E (Z1 − cY1 ) = µ2 + c α2 − 2c αµ, 2
427
1
x → ∞.
2
2
Fe (x/2) ≤ 2Fe (x) + 2[Fe (x/2) − Fe (x)]2 ,
Since for every fixed y ∈ (1, x/2),
we obtain
F (x − y)F (y) ≤ yF (x − y)F (y),
Fe (x/2) = o(F (x)),
2
x → ∞.
428
J. Lin / Insurance: Mathematics and Economics 51 (2012) 422–429
Note that
Thus, by (3.6) and the dominated convergence theorem, we obtain x/ 2
Fe∗2 (x) =
∞
2
Fe (x − y)dFe (y) + Fe (x/2)
v
yF (x − y)F (y)dy F (x)
0 2
= 2Fe (x)(1 − Fe (x/2)) + Fe (x/2) x/ 2 [Fe (x − y) − Fe (x)]dFe (y). +
Fe (x)dx = 0
µ2 2µ
x/ 2
x→∞
µ2 2µ2
F (x),
x → ∞.
(4.25)
Note that x/ 2
¯ Lemma 4.9 (Lemma 4(ii) of Baltrunas et al. (2006)). If r < 1, then F ∈ L and s(x) is nonincreasing for x large enough. Proof of Proposition 3.4. By Lemma 4.9, there exists v > 0 such that s(x) is nonincreasing on [v, ∞) and for x > v , xq(x)
[Fe (x − y) − Fe (x)]dFe (y) x/2 x 1 F (t )dt F (y)dy = 2 µ 0 x−y
≤ r (ε),
Q (x)
0
and thus, by the mean value theorem, we have for x − y > y > v , Q (x) − Q (x − y) = yq(z ) ≤ r (ε)ys(z ) ≤ r (ε)ys(y) = r (ε)Q (y),
and
where z ∈ (x − y, x). It follows that for x > 2v ,
x
F (t )dt ≤ yF (x − y);
x/2
x −y
v
yF (x − y)F (y)dy F (x)
hence 1
µ2
F ( x)
x/ 2
yF (y)dy ≤
0
x/2
≤
[Fe (x − y) − Fe (x)]dFe (y)
µ
x/ 2
1 2
= v
yF (x − y)F (y)dy.
v
From this and in view of (3.3) and (3.4), we obtain (4.25), as required.
Proof of Proposition 3.2. Note that for all y ∈ (0, x/2), F (x − y) ≤ F (x/2), then in view of F ∈ D ∩ L and by the dominated convergence theorem, we obtain (3.3), i.e., F ∈ H .
x/ 2
≤
0
Proof of Proposition 3.1. Assume F ∈ H , then F ∈ L and the relation (4.21) holds. Since G and F are weakly tail-equivalent, it is easy to see that G also satisfies (4.21) with F replaced by G. Since G ∈ L , G also satisfies (4.17) with F replaced by G, and thus, G∈H. If G and F are tail-equivalent, then the result follows from the facts that H ⊂ L and F ∈ L ⇔ G ∈ L .
x/ 2
0
y exp{Q (x) − Q (x − y) − Q (y)}dy y exp{(r (ε) − 1)Q (y)}dy
x/ 2
=
x
v
Proof of Corollary 3.4. By Proposition 3.4, we only need to verify the condition (3.7). Since xq(x) → ∞, then for every fixed ε ∈ (0, 1 − r ), there exists v > 0 such that for x > v , q(x) ≥
3
yF (x − y)F (y)dy F ( x)
Q (x) − Q (v) ≥
∞
x/2
=
x x −y
x/ 2
≤ v
q(t )dt F (y)dy
y exp v
.
By integration we obtain
yF
1
1 − r (ε) x
3 1 − r (ε)
ln(x/v).
Hence
q(t )dt .
Then F ∈ L since q(x) converges to 0 as x tends to ∞, and thus the relation (4.17) holds. Since q(x) is decreasing eventually, that is to say, there exists a constant v > 0 such that q(x) is decreasing on [v, ∞), then for x > 2v , v
(y)dy.
Then by (3.7) and in view of F ∈ L (by Lemma 4.9), the dominated convergence theorem can be applied to obtain (4.26), which, combined with (4.17), gives (3.3), as required.
0
x/ 2
1−r (ε)
yF
Proof of Proposition 3.3. It is easy to see that F (x) = exp −
r := lim sup xq(x)/Q (x).
[Fe (x − y) − Fe (x)]dFe (y) ∼
yF (x) ≤
(4.26)
Recall that
,
0
yF (y)dy,
s(x) := Q (x)/x.
we only need to prove
v
∞
Define
Thus, in order to prove Fe ∈ S2 , in view of ∞
→
which, combined with (4.17), gives (3.3), as required.
0
y exp{yq(y)}F (y)dy.
1−r (ε)
v
∞
= v ∞ ≤ v
(y)dy
y exp{−(1 − r (ε))Q (y)}dy y exp{−(1 − r (ε))Q (v) − 3 ln(y/v)}dy
= exp{−(1 − r (ε))Q (v) + 3 ln(v)}
v
which is equivalent to (3.7), as required.
∞
y−2 dy < ∞,
J. Lin / Insurance: Mathematics and Economics 51 (2012) 422–429
5. Extensions to the case of random walk Notice that Theorem 3.1 is stated in the context of insurance, that is to say, the relation (1.1) is assumed. However, such a result can be easily extended to the case of general random walk without assuming (1.1). In the following discussion, let {Xn }n≥1 be a sequence of i.i.d. rv’s with finite mean EX1 < 0 and common distribution A. The related random walk is defined as in (1.2). Then by examining the proof of Theorem 3.1 and using Lemma 4.4 in place of Lemma 4.5, it is easy to obtain the following result. Theorem 5.1. Assume E (X12 ) < ∞, A ∈ L and A is non-arithmetic. Then the relation
P
sup Sn > x
+
n ≥0
∞
1 EX1
A(t )dt ∼ Ka · A(x),
x → ∞,
x
where Ka :=
E (X12 ) − 2µ+ µ− 2(EX1 )2
,
holds if and only if Ae ∈ S2 , where the distribution Ae has support on [0, ∞) and is defined as Ae (x) = ∞ 0
1 A(t )dt
x
A(t )dt ,
x ≥ 0.
0
Remark 5.1. The first order result was established by Veraverbeke (1977) and Korshunov (1997). Our result above improves Theorem 5.1 of Borovkov (2002). Acknowledgment The author is thankful to the anonymous referee for his/her helpful comments. References Albrecher, H., Hipp, C., Kortschak, D., 2010. Higher-order expansions for compound distributions and ruin probabilities with subexponential claims. Scandinavian Actuarial Journal 2, 105–135. ˘ ˙ A., Leipus, R., Siaulys, Ale˘skevi˘ciene, J., 2009. Second-order asymptotics of ruin probabilities for semiexponential claims. Lithuanian Mathematical Journal 49, 364–371. Asmussen, S., Foss, S., Korshunov, D., 2003. Asymptotics for sums of random variables with local subexponential behaviour. Journal of Theoretical Probability 16, 489–518. ¯ Baltrunas, A., 1999. Second order behaviour of ruin probabilities. Scandinavian Actuarial Journal 2, 120–133. ¯ Baltrunas, A., 2005. Second order behaviour of ruin probabilities in the case of large claims. Insurance: Mathematics & Economics 36, 485–498. ¯ Baltrunas, A., Omey, E., 1998. The rate of convergence for subexponential distributions. Liet. Matem. Rink. 38 (1), 1–18. ¯ Baltrunas, A., Omey, E., 2002. The rate of convergence for subexponential distributions and densities. Lithuanian Mathematical Journal 42, 1–14.
429
¯ Baltrunas, A., Omey, E., Van Gulck, S., 2006. Hazard rates and subexponential distributions. In: Nouvelle série, vol. 80. Publications de L’ Institut Mathématique, Tome, pp. 29–46. Barbe, P., McCormick, W.P., 2009. Asymptotic Expansions for Infinite Weighted Convolutions of Heavy Tail Distributions and Applications. Memoirs of the American Mathematical Society, 197. Barbe, P., McCormick, W.P., Zhang, C., 2007. Asymptotic expansions for distributions of compound sums of random variables with rapidly varying subexponential distribution. Journal of Applied Probability 44, 670–684. Barbe, P., McCormick, W.P., Zhang, C., 2007. Tail expansions for the distribution of the maximum of a random walk with negative drift and regularly varying increments. Stochastic Processes and their Applications 117, 1835–1847. Borovkov, A.A., 2002. On subexponential distributions and asymptotics of the distribution of the maximum of sequential sums. Siberian Mathematical Journal 43, 995–1022. Chistyakov, V.P., 1964. A theorem on sums of independent positive random variables and its application to branching random processes. Theory of Probability and its Applications 9, 640–648. Chover, J., Ney, P., Wainger, S., 1973. Functions of probability measures. Journal of D’Analyse. 26, 255–302. Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling Extremal Events for Insurance and Finance. Springer, Berlin. Embrechts, P., Veraverbeke, N., 1982. Estimates for probability of ruin with special emphasis on the possibility of large claims. Insurance: Mathematics & Economics 1, 55–72. Feller, W., 1971. An Introduction to Probability Theory and its Applications, vol. 2. Wiley, New York. Geluk, J.L., 1992. Second order tail behaviour of a subordinated probability distribution. Stochastic Processes and their Applications 40, 325–337. Geluk, J.L., 1996. Tails of subordinated laws: the regularly varying case. Stochastic Processes and their Applications 61, 147–161. Geluk, J.L., Pakes, A.G., 1991. Second order subexponential distributions. Journal of the Australian Mathematical Society (Series A) 51, 73–87. Grübel, R., 1987. On subordinated distributions and generalized renewal measures. Annals of Probability 15 (1), 394–415. Klüppelberg, C., 1988. Subexponential distributions and integrated tails. Journal of Applied Probability 25, 132–141. Klüppelberg, C., 1989. Estimation af ruin probabilities by means of hazard rates. Insurance: Mathematics & Economics 8, 279–285. Korshunov, D., 1997. On distribution tail of the maximum of a random walk. Stochastic Processes and their Applications 72, 97–103. Lin, J., 2010. Second order subexponential distributions with finite mean and their applications to subordinated distributions. Journal of Theoretical Probability Online first. Lin, J., 2008. Studies of some probability problems associated with heavy-tailed risk. Ph.D. Thesis. Xiamen University (in Chinese). Lukacs, E., 1970. Characteristic Functions. Charles Griffin, London, Hafner Publ., New York. Mikosch, T., Nagaev, A., 2001. Rates in approximations to ruin probabilities for heavy-tailed distributions. Extremes 4 (1), 67–78. Omey, E., 1994. On the difference between the product and the convolution product of distribution functions. In: Nouvelle série, vol. 55. Publications de L’ Institut Mathématique, Tome, pp. 111–145. (69). Omey, E., Willekens, E., 1986. Second order behaviour of the tail of a subordinated probability distribution. Stochastic Processes and their Applications 21, 339–351. Omey, E., Willekens, E., 1987. Second-order behaviour of distributions subordinate to distribution with finite mean. Communications in Statistics Stochastic Models 3, 311–342. Pratt, J.W., 1960. On interchanging limits and integrals. Annals of Mathematics Statistics 31, 74–77. Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.L., 1998. Stochastic Processes for Insurance and Finance. Wiley, Chichester. Sgibnev, M.S., 2007. Homogeneous conservative Wiener–Hopf equation. Sbornik: Mathematics 198 (9), 1341–1350. Veraverbeke, N., 1977. Asymptotic behavior of Wiener–Hopf factors of a random walk. Stochastic Processes and their Applications 5, 27–37. Wolff, R.W., 1984. Conditions for finite ladder height and delay moments. Operations Research 32 (4), 909–916.