Reinsurance under the LCR and ECOMOR Treatieswith Emphasis on Light-tailed Claims

Jun Jiang1, 2, Qihe Tang2, ∗1 Department of Statistics and Finance

University of Science and Technology of China

Hefei 230026, Anhui, P.R. China2 Department of Statistics and Actuarial Science

The University of Iowa

241 Schaeffer Hall, Iowa City, IA 52242, USA

August 16, 2008

Abstract

Suppose that, over a fixed time interval of interest, an insurance portfolio generatesa random number of independent and identically distributed claims. Under the LCRtreaty the reinsurance covers the first l largest claims, while under the ECOMOR treatyit covers the first l − 1 largest claims in excess of the lth largest one. Assuming thatthe claim sizes follow an exponential distribution or a distribution with a convolution-equivalent tail, we derive some precise asymptotic estimates for the tail probabilitiesof the reinsured amounts under both treaties.

Keywords: Asymptotics; Convolution-equivalence; Exponential distribution; LCRand ECOMOR treaties; Reinsurance; Tail probability

1 Introduction

Let {X1, X2, . . .} be a sequence of i.i.d. (independent and identically distributed) positive

random variables, representing successive claim sizes, with common continuous distribution

F on (0,∞). Assume that the claims arrive according to a counting process {N(t); t ≥ 0}independent of {X1, X2, . . .}; that is to say, the random variable N(t) counts the number of

claims up to time t ≥ 0. Let X∗1 < X∗

2 < · · · < X∗N(t) be the order statistics of the claim

sizes occurring in the time interval [0, t].

In this paper, we are interested in two large claims reinsurance treaties, LCR (largest

claims reinsurance) and ECOMOR (excedent du cout moyen relatif), which were introduced

∗Corresponding author. E-mails: junjiang@mail.ustc.edu.cn (J. Jiang), qtang@stat.uiowa.edu (Q. Tang);Tel.: 1-319-335–0730; Fax: 1-319-335-3017.

1

to the actuarial literature by Ammeter (1964) and Thepaut (1950), respectively. Under the

LCR treaty, the reinsurer pays the sum of the first l largest claims,

Ll(t) =l∑

i=1

X∗N(t)−l+i1{N(t)≥l}, l ≥ 1, (1.1)

while under the ECOMOR treaty, the reinsurer pays the sum of the parts of the first l − 1

largest claims in excess of the lth largest one,

El(t) =l∑

i=1

(X∗

N(t)−l+i −X∗N(t)−l+1

)1{N(t)≥l}, l ≥ 2, (1.2)

where 1A denotes the indicator function of a set A.

Through a series of papers since 1980’s, Kremer made numerous efforts on general formu-

las or upper bounds for the expectations of reinsured amounts, interpreted as net reinsurance

premiums, under some general reinsurance treaties including the present two; see Kremer

(1985, 1998) and references therein. Assuming that the number l of the order statistics in

(1.1) and (1.2) is fixed or increases in t at a certain rate and that the claim-size distribution

F belongs to the maximum domain of attraction of certain extremal value distributions,

Beirlant and Teugels (1992) as well as Ladoucette and Teugels (2006) obtained limiting

distributions for the quantities Ll(t) and El(t) as t → ∞. Hashorva (2007) extended the

scenario to the bivariate case. Embrechts et al. (1997) gave a short review of this study

using extreme value theory. A comprehensive review of the two reinsurance treaties LCR

and ECOMOR was made by Ladoucette and Teugels (2006). See also Teugels (2003) for an

extended review and a systematical treatment of these reinsurance treaties.

In recent years, several researchers started to investigate the asymptotic behavior of the

tail probabilities of Ll(t) and El(t). For the subexponential case, Ladoucette and Teugels

(2006) obtained a precise asymptotic estimate for the tail probability of El(t) with l and t

fixed; see the second relation of (2.7) below. Asimit and Jones (2008) used copulas belonging

to the maximum domain of attraction of an extreme value copula to describe the dependence

among the claim sizes and they derived some precise asymptotic estimates for the tail prob-

abilities of Ll(t) and El(t) with l fixed and N(t) ≥ l nonrandom and fixed. These works

initiate a new direction of the mainstream study of the two reinsurance treaties. A direct

application of such asymptotic results is to approximate risk measures of Ll(t) and El(t)

such as Value at Risk, Expected Shortfall, Conditional Tail Expectation, and so on.

The purpose of this paper is to establish precise asymptotic estimates for the tail proba-

bilities of Ll(t) and El(t), with l and t fixed. Our results show that, when F is an exponential

distribution, these tail probabilities are both asymptotic to a multiple of the tail of a gamma

distribution with suitable parameters, while when F has a convolution-equivalent tail, they

are both asymptotic to a multiple of the tail of F . The prefactors involved are completely ex-

plicit and transparent. Specifically, for the subexponential case, one of our results coincides

with a result first obtained by Ladoucette and Teugels (2006).

2

The rest of this paper is organized as follows: Section 2 presents the main results after

briefly introducing our conditions on the claim-size distribution, Section 3 prepares some

lemmas, and Sections 4 and 5 prove the main results.

2 Preliminaries and Main Results

Throughout the paper, all limit relationships are for x →∞ unless stated otherwise; for two

positive functions a (·) and b (·), we write a(x) ∼ b(x) if lim a(x)/b(x) = 1.

2.1 On the Claim-size Distribution

Since claim sizes are always nonnegative, we only consider distributions on [0,∞). Due to

its memoryless property, the exponential distribution with parameter γ > 0,

F (x) = 1− F (x) = e−γx, x > 0, (2.1)

is often an ideal candidate for claim-size distributions in the actuarial literature. It is well

known that, for each k = 1, 2, . . ., its k-fold convolution F ∗k is a gamma distribution with

probability density function

f (x; k, 1/γ) =γk

(k − 1)!xk−1e−γx, x > 0. (2.2)

Hence,

F ∗k(x) ∼ γk−1

(k − 1)!xk−1e−γx. (2.3)

As a natural generalization, a distribution F is said to belong to the class L(γ) for some

γ ≥ 0 if F (x) > 0 for all x ≥ 0 and the relation

limx→∞

F (x− y)

F (x)= eγy (2.4)

holds for all y. When γ > 0, we usually say that F has an exponential tail. In particular, if

a distribution F belonging to the class L(γ) is such that the limit

limx→∞

F ∗2(x)

F (x)= 2c (2.5)

exists and is finite, then we say that F has a convolution-equivalent tail, written as F ∈ S(γ).

As we go along we shall often suppress the phrase γ ≥ 0, but it remains in place.

Since it was introduced by Chistyakov (1964) and Chover et al. (1973a,b), the class S(γ)

has been extensively investigated by many researchers and applied to various fields. Recent

studies of this class can be found in Pakes (2004), Tang (2006), Foss and Korshunov (2007),

and Watanabe (2008), among many others. This class is often used to model claim-size

3

distributions; see, for example, Embrechts and Veraverbeke (1982), Kluppelberg (1989a),

and Tang and Tsitsiashvili (2004). It is well known that the constant c in relation (2.5) is

equal to the moment generating function of F at γ, defined to be

mF (γ) =

∫ ∞

0−eγxF (dx);

see Rogozin (2000) and references therein. Therefore, for F ∈ S(γ) it is necessary that

mF (γ) < ∞. This unfortunately excludes the exponential distribution as defined in (2.1).

Examples and criteria for membership of the class S(γ) for γ > 0 can be found in the

Theorem of Embrechts (1983) and Theorems 2-4 of Cline (1986). When γ = 0 it reduces

to the well-known subexponential class S (0), which contains Pareto, lognormal, and heavy-

tailed Weibull distributions. See Embrechts et al. (1997) for a review of applications of the

class S (0) to insurance and finance.

In this paper, we shall assume that F either is an exponential distribution or belongs to

the class S(γ).

2.2 Theorems

Denote by Qt(z) = EzN(t) the probability generating function of N(t). If it is analytic at

z > 0 then

Q(r)t (z) =

∞∑n=r

Pr (N(t) = n)n!

(n− r)!zn−r, r = 1, 2, . . . . (2.6)

Note that, when z = 1, the condition E (N(t))r < ∞ suffices for the series on the right-hand

side of equality (2.6) to converge. In the sequel, we shall borrow the notation Q(r)t (1) to

represent this series for z = 1, but we do not require that Qt(z) is analytic at z = 1.

It is clear that, for all t ≥ 0 for which 0 < EN(t) < ∞, the relation

Pr (L1(t) > x) ∼ EN(t)F (x)

holds as long as F (x) > 0 for all x; see Lemma 1 of Ladoucette and Teugels (2006). Hence

in this paper, we only deal with the case l ≥ 2 for both reinsurance treaties.

Theorem 2.1. Recall equality (1.1) with l ≥ 2 and t ≥ 0 fixed such that Pr (N(t) ≥ l) > 0.

Let the claim sizes X1, X2, . . . be i.i.d. with common continuous distribution F on (0,∞)

and independent of N(t). Assume E (N(t))l < ∞.

(i) If F is an exponential distribution with parameter γ as in (2.1), then

Pr (Ll(t) > x) ∼ γl−1xl−1e−γx

l! (l − 1)!Q

(l)t (1);

(ii) If F ∈ S(γ), then

Pr (Ll(t) > x) ∼ F (x)

(l − 2)!

∫ ∞

0

eγy

(∫ ∞

y

eγuF (du)

)l−2

Q(l−1)t (F (dy)) .

4

Theorem 2.2. Recall equality (1.2) with l ≥ 2 and t ≥ 0 fixed such that Pr (N(t) ≥ l) > 0.

Let the claim sizes X1, X2, . . . be i.i.d. with common continuous distribution F on (0,∞)

and independent of N(t). Assume E (N(t))l−1 < ∞.

(i) If F is an exponential distribution with parameter γ as in (2.1), then

Pr (El(t) > x) ∼ γl−2xl−2e−γx

(l − 2)!Pr (N(t) ≥ l) ;

(ii) If F ∈ S(γ), then

Pr (El(t) > x) ∼ F (x)

(l − 2)!

∫ ∞

0

e−(l−1)γy

(∫ ∞

y

eγuF (du)

)l−2

Q(l−1)t (F (dy)) .

Note that the probability generating function Qt(z) is automatically analytical at any

z ∈ (0, 1). Also recall our convention for Q(r)t (1). Therefore, both Q

(l)t (1) and Q

(l−1)t (F (y))

in Theorems 2.1 and 2.2 are well defined.

Theorems 2.1(i) and 2.2(i) show that the tail probabilities of Ll(t) and El(t) are both

asymptotically proportional to the tail of a gamma distribution with suitable parameters;

recall relation (2.3).

Note also that, when F ∈ S (0), Theorems 2.1(ii) and 2.2(ii) reduce to the relations

Pr (Ll(t) > x) ∼ Pr (El(t) > x) ∼ EN(t)1{N(t)≥l}F (x). (2.7)

Ladoucette and Teugels (2006) obtained the latter relation of (2.7) for El(t) under the

stronger condition that Qt(z) is analytic at z = 1.

Finally, we remark that all formulas obtained in Theorems 2.1 and 2.2 well capture the

impact of all stochastic factors including the claim-size distribution F , the total number of

claims N(t), as well as the number of claims l covered by reinsurance. Hence, these formulas

should work fine for a relatively large value of x when either N(t) has a non-degenerate

distribution or γ > 0. However, for the case where γ = 0 and N(t) is degenerate at some

constant they can poorly perform unless x is extremely large because, as seen from (2.7), for

this case they even fail to capture the impact of the number l. To keep the paper short, we

shall not pursue numerical studies of these formulas.

3 Lemmas

In this section we prepare several lemmas for later use.

Lemma 3.1. Assume F ∈ S(γ).

(i) For each function h(·) such that h(x) →∞ and x− h(x) →∞, it holds that

F (h(x)) F (x− h(x)) = o(F (x)

);

(ii) If, for each i = 1, . . . , n, the distribution Fi satisfies Fi(x) ∼ ciF (x) for some ci > 0,

then F1 ∗ · · · ∗ Fn ∈ S(γ) and F1 ∗ · · · ∗ Fn(x) = O(F (x)

).

5

Proof. The two items are direct consequences of Lemma 2 and Corollary 1 of Cline (1986),

respectively.

Lemma 3.2. Let X and Y be two independent positive random variables with distributions

F and G, respectively. Assume F ∈ S(γ) and G(x) ∼ cF (x) for some c > 0. Then,

Pr (X + Y > x,X > Y ) ∼ mG(γ)F (x). (3.1)

Proof. Since the class S(γ) is closed under tail equivalence (see page 260 of Kluppelberg

(1989b)), we have G ∈ S(γ). Moreover, as (x− y) ∨ y →∞ uniformly for all y > 0,

Pr (X + Y > x,X > Y ) =

∫ ∞

0

F ((x− y) ∨ y) G(dy) ∼ 1

c

∫ ∞

0

G ((x− y) ∨ y) G(dy),

which is further equal to c−1 Pr (Y1 + Y2 > x, Y1 > Y2) for Y1 and Y2 being i.i.d. copies of Y .

By Lemma 3.1(i), Pr (Y1 = Y2 > x/2) ≤ (G(x/2)

)2= o

(G(x)

). Hence,

Pr (X + Y > x,X > Y ) =1

2cPr (Y1 + Y2 > x) + o

(G(x)

) ∼ mG(γ)F (x).

This proves relation (3.1).

The following lemma will play a key role in proving Theorems 2.1(ii) and 2.2(ii):

Lemma 3.3. Let X1, X2, . . . be a sequence of independent positive random variables such

that X1 follows a distribution F∗ and X2, X3, . . . follow a common distribution F . Assume

F ∈ S(γ) and F∗(x) ∼ cF (x) for some c > 0. Write Sk =∑k

i=1 Xi for k = 1, 2, . . .. Then,

for each integer k ≥ 2 and each constant a ≥ 0,

Pr (Sk > x + ka, Xk > · · · > X1 > a)

∼ F (x)e−kγa

∫ ∞

0

eγy Pr (Sk−1 ∈ dy, Xk−1 > · · · > X1 > a) . (3.2)

Proof. For k = 1, 2, . . . and a ≥ 0 fixed, we write Ak = Pr (Xk > · · · > X1 > a) and write Gk

as the conditional distribution of Sk on (Xk > · · · > X1 > a). Thus by (2.4), relation (3.2)

amounts to the relation

Pr (Sk > x + ka, Xk > · · · > X1 > a) ∼ Ak−1mGk−1(γ)F (x + ka). (3.3)

We are to prove that relation (3.3) holds for each integer k ≥ 2 and each a ≥ 0. As a

by-product, our proof also shows that Gk−1 ∈ S(γ).

First, we claim that relation (3.3) holds for k = 2 with A1 = Pr (X1 > a) and G1 (y) =

Pr (X1 ≤ y|X1 > a), which obviously belongs to the class S(γ). Actually, by Lemma 3.2,

Pr (S2 > x + 2a,X2 > X1) ∼ mF∗(γ)F (x + 2a),

6

and by the local uniformity of the convergence of relation (2.4),

Pr (S2 > x + 2a,X2 > a ≥ X1) =

∫ a

0

F (x + 2a− y) F∗(dy) ∼ F (x + 2a)

∫ a

0

eγyF∗(dy).

Therefore,

Pr (S2 > x + 2a,X2 > X1 > a)

= Pr (S2 > x + 2a,X2 > X1)− Pr (S2 > x + 2a,X2 > a ≥ X1)

∼ F (x + 2a)

∫ ∞

a

eγyF∗(dy).

Now we inductively assume that relation (3.3) holds for k = j for some integer j ≥ 2

with Gj−1 ∈ S(γ). Straightforwardly, the distribution Gj also belongs to the class S(γ) since

Gj(x) = Pr (Sj > x|Xj > · · · > X1 > a) ∼ Aj−1

Aj

mGj−1(γ)F (x).

We need to prove that relation (3.3) still holds for k = j + 1.

For notational convenience, we write x = x + (j + 1) a, which tends to infinity uniformly

for all j = 2, 3, . . . and a ≥ 0. By comparing Xj+1 and Sj with jx/ (j + 1) and x/ (j + 1),

respectively, we split the probability of the left-hand side of relation (3.3) with k = j + 1

into four parts as

4∑s=1

Pr ((Sj+1 > x,Xj+1 > · · · > X1 > a) ∩Bs) ,4∑

s=1

Is(x), (3.4)

where the events B1-B4 are defined to be

B1 = {Xj+1 ≤ x/ (j + 1)},B2 = {x/ (j + 1) < Xj+1 ≤ jx/ (j + 1)},B3 = {Xj+1 > jx/ (j + 1) , Sj > x/ (j + 1)}, and

B4 = {Xj+1 > jx/ (j + 1) , Sj ≤ x/ (j + 1)}.Trivially, I1(x) = 0. By Lemma 3.1(ii),

I2(x) ≤∫ jex/(j+1)

ex/(j+1)

Pr (Sj > x− u) F (du) = O(1)

∫ jex/(j+1)

ex/(j+1)

F (x− u) F (du) = o(F (x)

),

where the last step follows from Lemma 5.5 of Pakes (2004). Moreover, by items (ii) and (i)

of Lemma 3.1, in turn,

I3(x) ≤ Pr

(Xj+1 >

jx

j + 1

)Pr

(Sj >

x

j + 1

)= O(1)F

(jx

j + 1

)F

(x

j + 1

)= o

(F (x)

).

Therefore, plugging these estimates into (3.4) yields that

Pr (Sj+1 > x,Xj+1 > · · · > X1 > a) = I4(x) + o(F (x)

). (3.5)

7

Now we focus on I4(x). Introduce a positive random variable Yj independent of Xj+1 and

distributed by Gj, so that

I4(x) = Pr

(Sj+1 > x, Sj ≤ x

j + 1, Xj > · · · > X1 > a

)

= Aj Pr

(Xj+1 + Yj > x, Yj ≤ x

j + 1

). (3.6)

On the one hand, recalling Gj ∈ S(γ), by Lemma 3.2 we have

I4(x) ≤ Aj Pr (Xj+1 + Yj > x,Xj+1 > Yj) ∼ AjmGj(γ)F (x). (3.7)

On the other hand, by the local uniformity of the convergence of relation (2.4), for arbitrarily

fixed M > 0 and all large x,

I4(x) ≥ Aj

∫ M

0

F (x− y) Gj(dy) ∼ AjF (x)

∫ M

0

eγyGj(dy). (3.8)

It follows from relations (3.7) and (3.8) and the arbitrariness of M > 0 that

I4(x) ∼ AjmGj(γ)F (x). (3.9)

We conclude from relations (3.5) and (3.9) that relation (3.3) holds for k = j + 1. This

completes the proof of Lemma 3.3.

4 Proof of Theorem 2.1

Hereafter, for notational convenience, we write pn(t) = Pr (N(t) = n) for t ≥ 0 and n =

0, 1, . . .. Since F is continuous on (0,∞), it is clear that the random variables X∗n−l+1, . . . , X

∗n

have a joint probability density function

Pr

(l⋂

i=1

(X∗

n−l+i ∈ dxi

))

=n!

(n− l)!F n−l (x1)

l∏i=1

F (dxi), xl > · · · > x1 > 0.

Hence,

Pr (Ll(t) > x) =∞∑

n=l

pn(t) Pr

(l∑

i=1

X∗n−l+i > x

)

=∞∑

n=l

pn(t)n!

(n− l)!

∫· · ·

∫

4L(x)

F n−l (x1)l∏

i=1

F (dxi),

where4L(x) ={

(x1, . . . , xl) :∑l

i=1 xi > x, xl > · · · > x1 > 0}

. Introduce a positive random

variable X ′1 independent of X2, . . . , Xl and distributed by F n−l+1, so that Pr (X ′

1 > x) ∼

8

(n− l + 1) F (x). Then,

Pr (Ll(t) > x) =∞∑

n=l

pn(t)n!

(n− l + 1)!Pr

(l∑

i=2

Xi + X ′1 > x, Xl > · · · > X2 > X ′

1

)

,∞∑

n=l

pn(t)n!

(n− l + 1)!PL(x, n). (4.1)

(i) Let F be an exponential distribution as given in (2.1). By comparing X ′1 with x/l,

we split PL(x, n) in (4.1) into two parts as

PL(x, n)

= Pr(Xl > · · · > X2 > X ′

1 >x

l

)+ Pr

(l∑

i=2

Xi + X ′1 > x, Xl > · · · > X2 > X ′

1, X′1 ≤

x

l

)

, PL,1(x, n) + PL,2(x, n). (4.2)

Clearly,

PL,1(x, n) ≤ Pr(Xl, . . . , X2, X

′1 >

x

l

)∼ (n− l + 1) e−γx. (4.3)

By the memoryless property of the exponential distribution,

PL,2(x, n) =1

(l − 1)!

∫ x/l

0

Pr

(l∑

i=2

(Xi − y) > x− ly, Xl, . . . , X2 > y

)F n−l+1(dy)

=1

(l − 1)!

∫ x/l

0

Pr

(l∑

i=2

Xi > x− ly

)(F (y)

)l−1F n−l+1(dy).

Recall that the sum∑l

i=2 Xi has the density function f (u; l − 1, 1/γ) as given in (2.2).

Plugging it into the above and using change of variables,

PL,2(x, n) =n− l + 1

(l − 1)! (l − 2)!γlxl−2e−γx

∫ x/l

0

∫ ∞

0

(u + x− ly)l−2

xl−2e−γu

(1− e−γy

)n−ldudy

, n− l + 1

(l − 1)! (l − 2)!γlxl−2e−γxJ(x). (4.4)

On the one hand,

J(x) ≤∫ ∞

0

∫ x/l

0

(u

x+ 1− ly

x

)l−2

e−γudydu

=x

l (l − 1)

∫ ∞

0

((u

x+ 1

)l−1

−(u

x

)l−1)

e−γudu

∼ x

l (l − 1) γ.

9

On the other hand, with M > 0 arbitrarily fixed,

J(x) ≥ 1

γ

∫ x/l

M

(1− ly

x

)l−2 (1− e−γy

)n−ldy

≥ 1

γ

(1− e−γM

)n−l∫ x/l

M

(1− ly

x

)l−2

dy

∼ (1− e−γM

)n−l x

l (l − 1) γ.

Therefore,

J(x) ∼ x

l (l − 1) γ. (4.5)

We conclude from relations (4.2)-(4.5) that

PL(x, n) ∼ n− l + 1

l!(l − 1)!γl−1xl−1e−γx. (4.6)

Now we return to equality (4.1). By relation (2.3) with k = l, there exists some constant

C1 > 0 such that, for all n = l, l + 1, . . . and all large x,

n!

(n− l + 1)!

PL(x, n)

xl−1e−γx≤ n!

(n− l)!

Pr(∑l

i=1 Xi > x)

xl−1e−γx≤ C1n

l.

Therefore, applying the dominated convergence theorem to (4.1) and using relations (4.6)

and (2.6),

Pr (Ll(t) > x) ∼ γl−1xl−1e−γx

l! (l − 1)!

∞∑

n=l

pn(t)n!

(n− l)!=

γl−1xl−1e−γx

l! (l − 1)!Q

(l)t (1).

(ii) For F ∈ S(γ), we begin with equality (4.1). By Lemma 3.1(ii), there exists some

constant C2 > 0 such that, for all n = l, l + 1, . . . and all large x,

n!

(n− l + 1)!

PL(x, n)

F (x)≤ n!

(n− l)!

Pr(∑l

i=1 Xi > x)

F (x)≤ C2n

l.

Hence, applying the dominated convergence theorem to (4.1) and using Lemma 3.3 with

k = l, a = 0, and F∗ = F n−l+1,

Pr (Ll(t) > x) ∼ F (x)∞∑

n=l

pn(t)n!

(n− l + 1)!KL(n), (4.7)

where

KL(n) =

∫ ∞

0

eγu Pr

(l−1∑i=2

Xi + X ′1 ∈ du,Xl−1 > · · · > X2 > X ′

1

)

=1

(l − 2)!

∫ ∞

0

eγy

l−1∏i=2

E(eγXi1{Xi>y}

)F n−l+1(dy)

=1

(l − 2)!

∫ ∞

0

eγy

(∫ ∞

y

eγuF (du)

)l−2

F n−l+1(dy). (4.8)

10

Therefore, by relations (4.7), (4.8), and (2.6) we have

Pr (Ll(t) > x) ∼ F (x)

(l − 2)!

∞∑

n=l

pn(t)n!

(n− l + 1)!

∫ ∞

0

eγy

(∫ ∞

y

eγuF (du)

)l−2

F n−l+1(dy)

=F (x)

(l − 2)!

∫ ∞

0

eγy

(∫ ∞

y

eγuF (du)

)l−2

Q(l−1)t (F (dy)) .

This completes the proof of Theorem 2.1.

5 Proof of Theorem 2.2

Similarly as in the proof of Theorem 2.1,

Pr (El(t) > x) =∞∑

n=l

pn(t) Pr

(l∑

i=1

(X∗

n−l+i −X∗n−l+1

)> x

)

=∞∑

n=l

pn(t)n!

(n− l)!

∫· · ·

∫

4E(x)

F n−l(x0)l−1∏i=0

F (dxi),

where in the integral we suitably changed the subscripts and

4E(x) =

{(x0, . . . , xl−1) :

l−1∑i=0

(xi − x0) > x, xl−1 > · · · > x0 > 0

}.

Introduce a positive random variable Y independent of X1, . . . , Xl−1 and distributed by

F n−l+1. Thus,

Pr (El(t) > x) ,∞∑

n=l

pn(t)n!

(n− l + 1)!PE(x, n), (5.1)

where

PE(x, n) = Pr

(l−1∑i=1

Xi − (l − 1) Y > x,Xl−1 > · · · > X1 > Y

)

=

∫ ∞

0

Pr

(l−1∑i=1

Xi > x + (l − 1) y, Xl−1 > · · · > X1 > y

)F n−l+1(dy).

(i) Let F be an exponential distribution as given in (2.1). Then by its memoryless

property,

PE(x, n) =1

(l − 1)!

∫ ∞

0

Pr

(l−1∑i=1

(Xi − y) > x, Xl−1, . . . , X1 > y

)F n−l+1(dy)

=n− l + 1

(l − 1)!γ Pr

(l−1∑i=1

Xi > x

)∫ ∞

0

e−lγy(1− e−γy

)n−ldy.

11

Hence, by relation (2.3) with k = l − 1,

PE(x, n) ∼ (n− l + 1)!

n! (l − 2)!γl−2xl−2e−γx, (5.2)

and there exists some constant C3 > 0 such that, for all n = l, l + 1, . . . and all large x,

n!

(n− l + 1)!

PE(x, n)

xl−2e−γx≤ n!

(n− l + 1)!

Pr(∑l−1

i=1 Xi > x)

xl−2e−γx≤ C3n

l−1.

Therefore, applying the dominated convergence theorem to (5.1) and using relation (5.2),

Pr (El(t) > x) ∼ γl−2xl−2e−γx

(l − 2)!Pr (N(t) ≥ l) .

(ii) For F ∈ S(γ), by Lemma 3.1(ii), there exists some constant C4 > 0 such that, for all

n = l, l + 1, . . . and all large x,

n!

(n− l + 1)!

PE(x, n)

F (x)≤ n!

(n− l + 1)!

Pr(∑l−1

i=1 Xi > x)

F (x)≤ C4n

l−1.

Hence, applying the dominated convergence theorem to (5.1) and using Lemma 3.3 with

k = l − 1, a = y, and F∗ = F ,

limx→∞

Pr (El(t) > x)

F (x)=

∞∑

n=l

pn(t)n!

(n− l + 1)!lim

x→∞PE(x, n)

F (x)

=∞∑

n=l

pn(t)n!

(n− l + 1)!

∫ ∞

0

KE (y) e−(l−1)γyF n−l+1(dy), (5.3)

where

KE (y) =

∫ ∞

0

eγu Pr

(l−2∑i=1

Xi ∈ du,Xl−2 > · · · > X1 > y

)

=1

(l − 2)!

l−2∏i=1

E(eγXi1{Xi>y}

)

=1

(l − 2)!

(∫ ∞

y

eγuF (du)

)l−2

. (5.4)

Therefore, by relations (5.3), (5.4), and (2.6) we have

Pr (El(t) > x)

∼ F (x)

(l − 2)!

∞∑

n=l

pn(t)n!

(n− l + 1)!

∫ ∞

0

e−(l−1)γy

(∫ ∞

y

eγuF (du)

)l−2

F n−l+1(dy)

=F (x)

(l − 2)!

∫ ∞

0

e−(l−1)γy

(∫ ∞

y

eγuF (du)

)l−2

Q(l−1)t (F (dy)) .

This completes the proof of Theorem 2.2.

Acknowledgments. The authors wish to thank an anonymous referee for his/her helpful

comments and suggestions.

12

References

[1] Ammeter, H. The rating of largest claim reinsurance covers. Quart. Algem. Reinsur.Comp. Jubilee. (1964), no. 2, 5–17.

[2] Asimit, A. V.; Jones, B. L. Asymptotic tail probabilities for large claims reinsurance ofa portfolio of dependent risks. Astin Bull. 38 (2008), no. 1, 147–159.

[3] Beirlant, J.; Teugels, J. L. Limit distributions for compounded sums of extreme orderstatistics. J. Appl. Probab. 29 (1992), no. 3, 557–574.

[4] Chistyakov, V. P. A theorem on sums of independent positive random variables and itsapplications to branching random processes. Theory Probab. Appl. 9 (1964), 640–648.

[5] Chover, J.; Ney, P.; Wainger, S. Functions of probability measures. J. Analyse Math.26 (1973a), 255–302.

[6] Chover, J.; Ney, P.; Wainger, S. Degeneracy properties of subcritical branching pro-cesses. Ann. Probability 1 (1973b), 663–673.

[7] Cline, D. B. H. Convolution tails, product tails and domains of attraction. Probab.Theory Relat. Fields 72 (1986), no. 4, 529–557.

[8] Embrechts, P. A property of the generalized inverse Gaussian distribution with someapplications. J. Appl. Probab. 20 (1983), no. 3, 537–544.

[9] Embrechts, P.; Kluppelberg, C; Mikosch, T. Modelling Extremal Events for Insuranceand Finance. Springer-Verlag, Berlin, 1997.

[10] Embrechts, P.; Veraverbeke, N. Estimates for the probability of ruin with special em-phasis on the possibility of large claims. Insurance Math. Econom. 1 (1982), no. 1,55–72.

[11] Foss, S.; Korshunov, D. Lower limits and equivalences for convolution tails. Ann.Probab. 35 (2007), no. 1, 366–383.

[12] Hashorva, E. On the asymptotic distribution of certain bivariate reinsurance treaties.Insurance Math. Econom. 40 (2007), no. 2, 200–208.

[13] Kluppelberg, C. Estimation of ruin probabilities by means of hazard rates. InsuranceMath. Econom. 8 (1989a), no. 4, 279–285.

[14] Kluppelberg, C. Subexponential distributions and characterizations of related classes.Probab. Theory Related Fields 82 (1989b), no. 2, 259–269.

[15] Kremer, E. Finite formulae for the premium of the general reinsurance treaty based onordered claims. Insurance Math. Econom. 4 (1985), no. 4, 233–238.

[16] Kremer, E. Largest claims reinsurance premiums under possible claims dependence.Astin Bull. 28 (1998), no. 2, 257–267.

[17] Ladoucette, S. A.; Teugels, J. L. Reinsurance of large claims. J. Comput. Appl. Math.186 (2006), no. 1, 163–190.

13

[18] Pakes, A. G. Convolution equivalence and infinite divisibility. J. Appl. Probab. 41(2004), no. 2, 407–424.

[19] Rogozin, B. A. On the constant in the definition of subexponential distributions. TheoryProbab. Appl. 44 (2000), no. 2, 409–412.

[20] Tang, Q. On convolution equivalence with applications. Bernoulli 12 (2006), no. 3, 535–549.

[21] Tang, Q.; Tsitsiashvili, G. Finite- and infinite-time ruin probabilities in the presence ofstochastic returns on investments. Adv. in Appl. Probab. 36 (2004), no. 4, 1278–1299.

[22] Teugels, J. L. Reinsurance Actuarial Aspects (Technical Report 2003–006). EURAN-DOM, Technical University of Eindhoven, 2003.

[23] Thepaut, A. Une nouvelle forme de reassurance. le traite d’excedent du cout moyenrelatif (ECOMOR). Bull. Trim. Inst. Actu. Francais 49 (1950), 273–343.

[24] Watanabe, T. Convolution equivalence and distributions of random sums. Probab. The-ory Relat. Fields. 142 (2008), no. 3-4, 367–397.

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