Insurance: Mathematics and Economics 49 (2011) 487–495

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Insurance: Mathematics and Economics

journal homepage: www.elsevier.com/locate/ime

Archimedean copulas in finite and infinite dimensions—with applicationto ruin problemsCorina Constantinescu a, Enkelejd Hashorva a, Lanpeng Ji b,a,∗a Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, Bâtiment Extranef, UNIL-Dorigny, 1015 Lausanne, Switzerlandb School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

a r t i c l e i n f o

Article history:Received March 2011Received in revised formAugust 2011Accepted 22 August 2011

Keywords:Dirichlet distributionArchimedean copulaRuin probabilityPerturbed risk modelRandom scalingMixingk-monotone functionsCompletely monotone functionsMax-domain of attractionGumbel distributionDavis–Resnick tail propertyWeibull distribution

a b s t r a c t

In this paper we discuss the link between Archimedean copulas and L1 Dirichlet distributions for bothfinite and infinite dimensions. With motivation from the recent papers Weng et al. (2009) and Albrecheret al. (2011) we apply our results to certain ruin problems.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Let ψ be a positive, strictly decreasing, continuous functionwith ψ(0) = 1, lims→∞ ψ(s) = 0, and let (U1,U2) be a bivariaterandom vector with distribution function (df) Cψ defined by

Cψ (u1, u2) = ψ(ψ−1(u1)+ ψ−1(u2)), u1, u2 ∈ [0, 1], (1)

where ψ−1(x) := inf{t : ψ(t) ≤ x}. Since U1,U2 are uniformlydistributed on (0, 1), the df Cψ is a copula which is commonlyreferred to as the Archimedean copula, whereas ψ denotes theso-called Archimedean generator. Distributional and asymptoticproperties of the Archimedean copulas have been investigated bynumerous researchers (see e.g., the recent contributions of Denuitet al. (2006), Charpentier and Segers (2007, 2008, 2009), Embrechtset al. (2009) and the references therein).

∗ Corresponding author at: Department of Actuarial Science, Faculty of Businessand Economics, University of Lausanne, Bâtiment Extranef, UNIL-Dorigny, 1015Lausanne, Switzerland.

E-mail addresses: Corina.Constantinescu@unil.ch (C. Constantinescu),Enkelejd.Hashorva@unil.ch (E. Hashorva), Lanpeng.Ji@unil.ch (L. Ji).

0167-6687/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.insmatheco.2011.08.006

The impetus for writing this paper came while readingProposition 4.8.2 in Denuit et al. (2006) (see also Genest andRivest, 1993), which shows that Cψ (U1,U2) is independent ofψ−1(U1)/(ψ

−1(U1) + ψ−1(U2)). Consequently, the bivariaterandom vector (X1, X2), with stochastic representation

X1d= RO1, X2

d= R(1 − O1), (2)

is an L1 Dirichlet random vector, where

O1d=

ψ−1(U1)

ψ−1(U1)+ ψ−1(U2)and R d

= ψ−1(U1)+ ψ−1(U2) (3)

are independent. (Here d= stands for equality of distribution

functions.) It is not difficult to prove the converse result, thusestablishing the link between the Archimedean copula and theL1 Dirichlet distributions. Recently, McNeil and Nešlehová (2009)proved the general result in the k-dimensional setup; specifically,as shown therein, the survival copula of a k-dimensional L1Dirichlet random vector is Archimedean, and conversely, anArchimedean copula can be retrieved as the survival copula of ak-dimensional L1 Dirichlet random vector.

In this paper, a direct proof of the link between Archimedeancopulas and L1 Dirichlet distributions is established, providing that

488 C. Constantinescu et al. / Insurance: Mathematics and Economics 49 (2011) 487–495

the Archimedean generator is sufficiently smooth. We also discusssome properties of the Archimedean copulas and L1 Dirichletdistributions in infinite dimensions, i.e., the extension of the k-dimensional framework (Xi, i ≤ k) to the random sequence Xi,i ≥ 1. Further, we exploit the radial representation of L1 Dirichletdistributions to derive explicit or asymptotic results for ruinprobabilities in risk models with Archimedean dependent risks.In an insurance framework, Xi, i ≥ 1, may represent claim sizes;when Xi, i ≥ 1, are identically distributed with df F , and the jointdf of (Xi, i ≤ n) is, for any n > 1, an L1 Dirichlet distribution, orequivalently the survival copula of (Xi, i ≤ n) is Archimedean, thenthe claim sizes form an exchangeable random sequence. Therefore,in view of the well-known de Finetti representation, the claimsizes are simply a mixing of i.i.d. random variables. As shown inthis paper, the claim sizes are simply generated by measurabletransforms of randomly scaled, independent exponential randomvariables with mean 1. Note that one of the strengths of thisstochastic representation approach is that it does not requireLaplace transformation for finding the mixing distribution, asillustrated in Example 2.

With motivation from Weng et al. (2009) and Albrecher et al.(2011), in Sections 4 and 5 we discuss asymptotics and exactexpressions for ruin probabilities, in discrete and continuous timeframe, for losses with survival copula being Archimedean. Ournovel asymptotic results are supported by recent findings ofHashorva et al. (2010), where extreme value theory is employedfor the investigation of certain ruin models.

An additional novel result of this contribution is the explicitclosed-form expression of the ruin probability in the classicalperturbed riskmodel (see Dufresne and Gerber, 1991)where claimsizes have a certain L1 Dirichlet distribution.

Organisation of the paper: In the next section we brieflydiscuss the L1 Dirichlet distributions. Section 3 is dedicated toArchimedean survival copulas in finite and infinite dimensions,highlighting their link with L1 Dirichlet distributions and randomsequences. In Section 4 we consider the discrete time ruinprobability, with net losses specified as the components of aweighted L1 Dirichlet randomvector,whereas in Section 5we focuson the continuous time ruin probability in finite or infinite time.The proofs of all the results are relegated to Section 6.

2. L1 Dirichlet distributions

Throughout this paper Yi, i ≥ 1, denotes a sequence of inde-pendent exponential random variables with mean 1, referred toas unit exponentials. Define the random vectors Y = (Y1, . . . , Yk),k ≥ 2, and set

O = (O1, . . . ,Ok)d=

Y1∑1≤i≤k

Yi, . . . ,

Yk∑1≤i≤k

Yi

. (4)

By the independence of Yi, 1 ≤ i ≤ k,

Oid= B1,k−1, i = 1, . . . , k,

where Ba,b is a beta distributed random variable, with probabilitydensity function (pdf) Γ (a + b)/(Γ (a)Γ (b))xa−1(1 − x)b−1, x ∈

(0, 1), a, b > 0. Here Γ (·) is the Euler gamma function. Further-more, the random vector O is uniformly distributed on {x ∈

(0,∞)k :∑k

i=1 xi = 1}, and Y has the radial representation

Y d=RO := (RO1, . . . ,ROk), R d

= Y1 + Y2 + · · · + Yk, (5)

withR andO independent. Ifwedrop thedistributional assumptionon the associated random radiusR, we can define a larger classof random vectors consisting of L1 Dirichlet random vectors. More

specifically, if R > 0 with df F and further, R is independent of Odefined in (4), then X with stochastic representation

X d= RO (6)

is referred to as an L1 Dirichlet random vector. When F possesses apositive pdf f , then X possesses a pdf h, given by

h(x1, . . . , xk) = Γ (k)f

k−

i=1

xi

k−

i=1

xi

1−k

,

xi > 0, i = 1, . . . , k. (7)Conversely, if h is the pdf of X, then R possesses a pdf f related toh as above. One canonical example is when R is unit exponential,implying that the pdf h is given by

h(x1, . . . , xk) = Γ (k) exp

−

k−i=1

xi

k−

i=1

xi

1−k

,

xi > 0, i = 1, . . . , k.See Fang and Fang (1990) or Balakrishnan and Hashorva (in press)for the basic properties of L1 Dirichlet distributions. Indeed, theL1 Dirichlet distributions are very tractable due to the radial rep-resentation (6). It is of interest to work with such simple ran-dom structures also in infinite dimensions, i.e., considering Xi, i ≥

1, a random sequence with common marginal df F , such that(X1, . . . , Xk), k ≥ 1, are L1 Dirichlet random vectors. This assump-tion implies immediately thatXi, i ≥ 1, is an exchangeable randomsequence, which restricts the dependence structure significantly,since in view of the well-known de Finetti representation, therandom sequence is then a mixing of i.i.d. random variables. i.e.,Xi|Θ = θ, i ≥ 1, are i.i.d. for some given realization θ of the ran-dom parameterΘ . Clearly, the stochastic representation (6) can bewritten alternatively as

X d= (SkY1, . . . , SkYk), Sk

d=

RY1 + · · · + Yk

,

but note that Sk and Yi, i ≤ k, are in general dependent.

Theorem 1. Let Xi, i ≥ 1 be a random sequence such that for anygiven integer k the random vector (X1, . . . , Xk) is an L1 Dirichletrandom vector. Then we have

Xid= SYi, i ≥ 1, (8)

with S independent of Yi, i ≥ 1.

Remarks. (i) The corresponding result of Theorem 1 for spher-ically symmetric random sequences is well-known (see e.g.,Schoenberg (1938) or Bryc (1995)). In that framework, any spher-ically symmetric random sequence is a simple random scaling of aGaussian random sequence Yi, i ≥ 1, where Yi, i ≥ 1, are inde-pendent, standard Gaussian random variables.

(ii) Weighted L1 Dirichlet random vectors are easily introducedby using indicator random variables. More specifically if R > 0is independent of O with representation (4), and both R,O areindependent of the Bernoulli random variables Ii, i ≤ k, withP{Ii = 1} = qi = 1 − P{Ii = −1}, qi ∈ (0, 1], i ≤ k, thenthe random vector X with stochastic representation

X d= (RI1O1, . . . , RIkOk) (9)

is referred to as a weighted L1 Dirichlet random vector.(iii) If Xi, i ≥ 1, is such that, for any fixed k ∈ N, (Xi, i ≤ k) is

a weighted L1 Dirichlet random vector with Ii, i ≤ k, i.i.d., then byTheorem 1

Xid= SIiYi, i ≥ 1, (10)

with S some non-negative random variable independent of Yi, i ≥

1, and Ii, i ≥ 1.

C. Constantinescu et al. / Insurance: Mathematics and Economics 49 (2011) 487–495 489

3. Archimedean copula and L1 Dirichlet distributions

A k-dimensional Archimedean randomvectorU = (U1, . . . ,Uk)with components uniformly distributed on (0, 1) is defined simi-larly to (1) by specifying its df Cψ as

Cψ (u1, . . . , uk) = ψ

−1≤i≤k

ψ−1(ui)

, (11)

with ψ an Archimedean generator. It follows from McNeil andNešlehová (2009) that Cψ possesses a positive pdf (i.e., Cψ doesnot have a singular part) if the (k − 1)th derivative ψ (k−1) isabsolutely continuous. It is well-known that the k-monotonefunctions play an important role in the characterization of L1Dirichlet distributions, see e.g., Fang and Fang (1988). By definition,a function f on (0,∞) is called k-monotone, if for any i =

1, . . . , k−2 the function (−1)if (i) is non-negative, non-increasing,and convex on (0,∞). In view of the known link between theArchimedean copula and the L1 Dirichlet random vectors in thebivariate setup (recall (2)), it is natural to conjecture that thisrelation holds also for higher dimensions (k ≥ 3), i.e.,

X d= (ψ−1(U1), . . . , ψ

−1(Uk)) (12)

is an L1 Dirichlet random vector. Recently, this has been shown inMcNeil and Nešlehová (2009). Our next results establishes this linkutilising a known result of Fang and Fang (1988) and the structureof L1 Dirichlet random vectors.

Theorem 2. The random vector X ∈ Rk, k ≥ 2, with stochasticrepresentation (12) is an L1 Dirichlet random vector, if ψ , with ψ (k)

continuous, is an Archimedean generator, defining the Archimedeancopula Cψ in (11), and (−1)kψ (k)

≥ 0. Furthermore, such a generatorψ is a k-monotone function.

Conversely, if X is an L1 Dirichlet random vector with P{X1 =

0, . . . , Xk = 0} = 0, then there exists a k-monotone functionψ , suchthat X has representation (12), and furthermore,ψ is the generator ofthe Archimedean copula defined in (11).

In light of McNeil and Nešlehová (2009), the restriction that theArchimedean copula possesses a continuous pdf is not necessary.Therefore, in the next corollary we drop this restriction.

Corollary 3. Let Ui, i ≥ 1, be uniformly distributed random vari-ables, such that, for any k ≥ 2, the random vector (U1, . . . ,Uk) has dfgiven by an Archimedean copula, with Archimedean generator ψ .Then the df of (U1, . . . ,Uk) agrees with the copula of w(SYi), i ≤ k,with Yi, i ≥ 1, independent of the positive random variable S whichsatisfies

U1d= ψ(SY1), (13)

wherew is some positive strictly monotone decreasing function.

It is well-known (see e.g. Denuit et al., 2006) that anArchimedean copula is a product df if and only if its generatorψ isgiven by

ψ(x) = exp(−cx), x > 0, for some c > 0. (14)

An alternative proof of this fact is the following: By Theorem 1of Balakrishnan and Hashorva (in press) an L1 Dirichlet randomvector X d

= RO has independent components if and only if RO1 isexponentially distributed. In this case, S is almost surely a constant,and in view of (13) we find that ψ satisfies (14).

4. Discrete time risk model with dependent net losses

Consider a discrete stochastic model, starting at some initialpoint u ≥ 0, and described by

U0 = u, Ui = Ui−1(1 + η̄)− Xi, i ∈ N, (15)with η̄ some positive constant and Xi, i ≥ 1, a sequence of randomvariables.

In the insurance modeling framework of Weng et al. (2009),the initial capital at time 0 corresponds to U0 = u,Ui denotesthe surplus of the portfolio at the end of the ith period, whereasXi stands for the net loss, i.e., the total claim amount minus thetotal premium income during the period i. In this setting, the ruinprobability in the finite time interval [0, n], n ∈ N, is defined by(set X0 := 0 and η = 1 + η̄)

Ψ̆ (η, u; n) := Pmin0≤j≤n

Uj < 0|U0 = u

= P

max0≤i≤n

i−j=0

Xjη−j > u

. (16)

Taking n = ∞, the infinite time ruin probability is defined as

Ψ̆ (η, u) := Pminj≥0

Uj < 0|U0 = u

= P

maxi≥0

i−j=0

Xjη−j > u

. (17)

Numerous contributions consider the tail asymptotics of Ψ̆ (η,u; n), see e.g., Tang and Tsitsiashvili (2003, 2004), Tang (2007,2008), Denuit et al. (2006), Weng et al. (2009), Tang et al. (2011),Yang et al. (2011) and the references therein.

Next, we assume that the random vector (X1, . . . , Xn) of the netlosses has stochastic representation (9), with k replaced by n. Withthe working asymptotic assumption that |X1| is regularly varyingat infinity, with some index γ > 0, i.e.,

limu→∞

P{|X1| > xu}P{|X1| > u}

= x−γ , x > 0, (18)

we derive below the asymptotic behavior of Ψ̆ (η, u; n) andΨ̆ (η, u), as u → ∞; see Embrechts et al. (1997) for details on reg-ular variation.

Theorem 4. In the framework of the model (15), assume that the netloss vector (X1, . . . , Xn) is a weighted L1 Dirichlet random vector. IfX1 satisfies (18), then we have

limu→∞

Ψ̆ (η, u; n)P{|X1| > u}

= Cγ ,η,n, (19)

with Cγ ,η,n given by (defined also for n = ∞)

Cγ ,η,n :=1

Γ (γ + 1)E

max

max1≤i≤n

i−j=1

IjYjη−j, 0

γ∈ (0,∞).

Furthermore, if for any n ≥ 1 the net loss vector (X1, . . . , Xn) is aweighted L1 Dirichlet random vector, then

limu→∞

Ψ̆ (η, u)P{|X1| > u}

= Cγ ,η,∞. (20)

Remarks. (i) In view of Hashorva (2006), condition (18) isequivalent to R being regularly varying with index γ , or theweighted L1 Dirichlet random vector X = (X1, . . . , Xn) beingregularly varying.

(ii) The result of Theorem 4 does not follow from Theorem3.1 of Weng et al. (2009). The assumption that |X1| is regularlyvarying at infinity, with some index γ > 0, implies that Xi, Xj areasymptotically dependent, for any two distinct indices i, j ≤ k.For the definition of asymptotic (in-)dependence see for instanceAsimit et al. (2011).

490 C. Constantinescu et al. / Insurance: Mathematics and Economics 49 (2011) 487–495

(iii) Instead of η−j above we can consider ηj, j ≤ n, some posi-tive bounded random variables; for amore general setup the resultin (19) still holds.

The recent paper of Hashorva et al. (2010) is the first contri-bution which derived an asymptotic result for the ruin probabilitywhere claim sizes have df F in the Gumbel max-domain of attrac-tion. A df F is said to be in the Gumbel max-domain of attraction if(write xF for the upper endpoint of F )

limx↑xF

1 − F(x + sa(x))1 − F(x)

= exp(−s), for all s ∈ R, (21)

where a(x) is some positive scaling function satisfying

limx↑xF

a(x)x

= 0, and limx↑xF

a(x)xF − x

= 0 if xF < ∞.

It iswell-known (see e.g., Mikosch, 2009, Eq. (9.2.26)) that for somex0 < xF

1 − F(x) = c(x) exp

−

∫ x

x0

1a(t)

dt, x0 < x < xF , (22)

where c(x) is a measurable function such that limx→xF c(x) = c >0. Here the scaling function a(x) has a positive Lebesgue density a′

and limx→xF a′(x) = 0. The df F is said to be in the Weibull max-

domain of attraction with some parameter α > 0 if (set xF = 1, byconvention)

lims→0

1 − F(1 − ts)1 − F(1 − s)

= tα, ∀t > 0. (23)

For more details on the max-domains of attraction, see e.g. Em-brechts et al. (1997).

In the last result of this section we obtain an approximation ofΨ̆ (η, u; n) as u → xF/η, under Gumbel or Weibull max-domain ofattraction assumptions on the net loss variable X1.

Theorem 5. Suppose that the net loss vector (X1, . . . , Xn) isa weighted L1 Dirichlet random vector having the stochasticrepresentation (9), with k replaced by n. If X1 has df F satisfying (21),with some scaling function a(·), or (23), with someα = ν+n−1, ν ≥

0, then we have

limu→xF /η

Ψ̆ (η, u; n)P{X1 > ηu}

= Cη,nDα,n, (24)

where (V1, . . . , Vn−1) are uniformly distributed on the simplex ofRn−1,

Cη,n =

∫ 1

0Pmax

I2V1

η, . . . , I2

V1

η+ · · · + In

Vn−1

ηn−1

> y

×

n − 1(1 − y)n

dy + 1 ∈ (0,∞)

and, Dα,n is 1 for the Gumbel case and is Γ (ν + 1)/Γ (ν + n)if (23) holds.

Remarks. (a) If the df F of X1 satisfies (21) with xF = ∞, then

limu→∞

P{X1 > ηu}P{X1 > u}

= 0.

In fact, a stronger result, which is referred to as the Davis–Resnicktail property, holds. Namely, for any c > 1 and µ ≥ 0,

limu→∞

P{X1 > cu}P{X1 > u}

u

a(u)

µ= 0. (25)

(b) In the above problems, we can consider a general randomvector X with radial representation RO, where R is independent ofO. Since our results are asymptotical in nature, under assumptions

on R and with no specific distributional assumptions on O, it ispossible to derive the above results. Note that we do not imposeassumptions on R, but on the marginal distributions Xi, i ≤ d.

5. Continuous time ruin with dependent risks

As in the classical setting of ruin theory, we consider next thesurplus process R(c, u; t), t ≥ 0, given by

R(c, u; t) = u + ct −

N(t)−i=1

Xi, c > 0, t ≥ 0, (26)

with initial capital u, counting process N(t), t ≥ 0, modeling thenumber of claims up to t , and random claim sizes Xi, i ≥ 1. As inthe previous model, the quantity of central interest is the ruinprobability

Ψ (c, u; T ) = P{R(c, u; t) < 0, for some finite t ∈ (0, T ]},

T ∈ (0,∞].

In the more special case of independent, unit exponential claimsizes, we denote, for distinction, the ruin probability by Ψ (c, u; T ).Further, we suppress T if it is equal to ∞, i.e., for the infinite timeruin probability.

This section is motivated by Albrecher et al. (2011), whichattempts to remove the common independence assumption onboth claim sizes and inter-arrival times (whenever N(t), t ≥ 1 is arenewal counting process). For the former, as discussed therein, atractable instance is that of a common mixing variable Θ , with dfH , such that Xi|Θ = θ, i ≥ 1 are independent with common df Fθ .Clearly, for any T ∈ (0,∞],

Ψ (c, u; T ) =

∫RΨθ (c, u; T ) dH(θ), (27)

where Ψθ (c, u; T ) is the ruin probability corresponding to i.i.d.claim sizes, with df Fθ .

The special case considered in Proposition 2.1 of Albrecher et al.(2011) rests on the assumption that the claim sizes are generatedby an L1 Dirichlet sequence. A slightly more general framework isto assume that the claim sizes have the stochastic representation

Xid= g(SYi), i ≥ 1, (28)

with g some positive, measurable function and S with df H beingindependent of Yi, i ≥ 1. In the special case that g is the identityfunction, and SYi, i ≥ 1 has an Archimedean survival copula, withgenerator ψ , then

Ψ (c, u; T ) =

∫∞

0

Ψ (c/s, u/s; T ) dH(s) (29)

holds for any T ∈ (0,∞], hence, for this case, the mixing randomvariableΘ is related to S, by

S d=

1Θ. (30)

The novelty of this section lies in the discovery of the stochasticrepresentation (30), which allows us to derive explicit formulas,for various ruin probabilities, and connect them to results fromAlbrecher et al. (2011).

Example 1. Consider L1 Dirichlet claim sizes Xid= SYi, i ≥ 1, with

S > 0 independent of Yi, i ≥ 1,which are unit exponential randomvariables. We specify S to be such that 1/S has df Gamma(q, 1), forsome q > 0, where Gamma(α, β) denotes the Gamma df, withpdf xα−1 exp(−βx)βα/Γ (α), for x > 0 and with α, β positive con-stants. Consequently, SYi has the Pareto df with parameter q. Thiscompares to Example 2.3. of Albrecher et al. (2011), with α =

q, β = 1.

C. Constantinescu et al. / Insurance: Mathematics and Economics 49 (2011) 487–495 491

Example 2. Again, for L1 Dirichlet claim sizes Xi ∼ Gamma(α, 1),i ≥ 1, we consider the scaling random variable S to be bounded. Itfollows that S is beta distributed with pdf h, given by

h(x) =xα−1(1 − x)−α

Γ (α)Γ (1 − α), 0 < x < 1,

and necessarily α ∈ (0, 1). When N(t), t ≥ 0, is a homogeneousPoisson process with intensity λ > 0, we have

Ψ (c, u) =

∫∞

0

Ψ (c/s, u/s) dH(s)= 1 − H(c/λ)+

∫ (c/λ)∧1

0

λsc

× exp

−

1s

−λ

c

u

sα−1(1 − s)−α

Γ (α)Γ (1 − α)ds

= 1 − H(c/λ)+1

Γ (α)Γ (1 − α)

λ

ceλuc

×

∫ (c/λ)∧1

0e−

us

s

1 − s

αds.

Here the bound c/λ of the integral comes from the net profitcondition 1/s > λ/c. By Euler’s celebrated reflection formula,

Γ (α)Γ (1 − α) =π

sin(απ), α ∈ (0, 1),

thusΘ d= 1/S has pdf f , given by

f (x) =sin(απ)π

x−1(x − 1)−α, x ∈ (1,∞),

which is mentioned in Example 2.5 of Albrecher et al. (2011).

Example 3. Let g(x) = exp(x), x > 0, and let claim sizes be anL1 Dirichlet random sequence with ln Xi ∼ Gamma(α, 1), i ≥ 1.Utilising Example 2, we obtain that

Ψ (c, u) =

∫ 1

0Ψs(c, u) h(s)ds,

where Ψs(c, u) is the ruin probability with claim sizes beingPareto(1/s, 1) distributed. When N(t), t ≥ 0, is a homogeneousPoisson process, with intensity λ, satisfying 0 < λ < c , it follows(see e.g. Rolski et al., 1999) that

Ψs(c, u) ∼

s

(c/λ)(1 − s)− 1u−(1/s−1), 0 < s < 1 − λ/c,

1, 1 − λ/c ≤ s(31)

as u → ∞. Consequently, we have that, as u → ∞,

Ψ (c, u) ∼1

Γ (α)Γ (1 − α)

u∫ 1−λ/c

0

sα(1 − s)−α

(c/λ)(1 − s)− 1u−1/sds

+

∫ 1

1−λ/csα−1(1 − s)−αds

.

Example 4. Consider a classical risk model with double-exponential jumps, defined as

R(1, u; t) = u + t −

N(t)−i=1

Xi, t ≥ 0, (32)

where N(t), t ≥ 0, is a Poisson process, with parameter λ. Thecommon pdf of Xi is given by

f (x) = qθe−θxI(x>0) + (1 − q)θeθxI(x<0),

with q assumed to be greater than 1/2. From Zhang and Yang(2010), it follows that the ruin probability of model (32) can beexpressed as

Ψθ (1, u) =

1, θ ≤ λ(2q − 1),

λ(ξθ + θ)

θ(λ+ θ)eξθ u, θ > λ(2q − 1),

where ξθ =λ−

√λ2+4(θ2+θλ−2qθλ)

2 is the negative root of theequation

λ

λ− x

[qθθ + x

+(1 − q)θθ − x

]= 1.

Assume that Xi, i ≥ 1, from the classical risk model (26) has thestochastic representation (10), i.e.,

Xid= SIiYi, i ≥ 1,

where S is a non-negative random variable with df H and P{Ii =

1} = q = 1 − P{Ii = −1}. Consequently, the corresponding ruinprobability can be given by

Ψ (c, u) =

∫∞

0P

u/c + t −

N(t)−i=1

s/cIiYi < 0,

for some t ≥ 0

dH(s)

=

∫∞

0Ψc/s(1, u/c) dH(s)

= 1 − H

cλ(2q − 1)

+

∫ cλ(2q−1)

0

λ(ξ̃ + c/s)(c/s)(λ+ c/s)

eξ̃u/c dH(s),

where ξ̃ =λ−

√λ2+4((c/s)2+(c/s)λ−2q(c/s)λ)

2 .

Example 5. Since the seminal paper of Dufresne and Gerber(1991), exact expressions for the ruin probabilities in the perturbedrisk model

R(c, u; t) = u + ct −

N(t)−i=1

Xi + σB(t), c > 0, u ≥ 0, t ≥ 0, (33)

were well sought out. Here B denotes the standard Brownianmotion (with volatility σ > 0), independent of the claim sizes andthe homogeneous Poisson process N(t), t ≥ 0 (with intensity λ >0). Consider an L1 Dirichlet random sequence of claim sizes Xi

d=

SYi, where S has df H and is independent of the unit exponentialsYi, i ≥ 1. This means that Xi, i ≥ 1, are conditionally independent.Employing the algebraic operator approach of Albrecher et al.(2010) we obtain the explicit form of the probability of ruin in theperturbed risk model with independent exponential claims to be

Ψν(c, u) =γ2eγ1u − γ1eγ2u

γ2 − γ1Ψν(c, 0)

+eγ1u − eγ2u

γ2 − γ1

dduΨν(c, u)

u=0,

with Ψν(c, 0) = 1, dduΨν(c, u)|u=0 =

2σ 2 (λ/ν − c), ν the mean of

the claim sizes and γ1, γ2 the negative roots of the correspondingLundberg equation (see e.g., Eq. (7.2) in Dufresne and Gerber,1991), given by

492 C. Constantinescu et al. / Insurance: Mathematics and Economics 49 (2011) 487–495

γ1 =12ν

+cσ 2

+

12ν

+cσ 2

2

+2 (λ− c/ν)

σ 2,

γ2 =12ν

+cσ 2

−

12ν

+cσ 2

2

+2 (λ− c/ν)

σ 2.

Using the notation α =ddu Ψ̃ (c/s, 0)|u=0 =

2σ 2 (λ − c/s) and a =

12 +

csσ 2 , the solutions of Lundberg’s equation in a model with

independent, exponential claims havingmean 1, and premium ratec/s, can be written as γ̃1 = a +

√a2 + α, γ̃2 = a −

√a2 + α.

Furthermore, the corresponding explicit ruin probability is givenby

Ψ̃ (c/s, u/s)

=(a −

√a2 + α)e(a+

√a2+α)u/s

− (a +√a2 + α)e(a−

√a2+α)u/s

−2√a2 + α

+e(a+

√a2+α)u/s

− e(a−√

a2+α)u/s

−2√a2 + α

α

= −a + α

2√a2 + α

e(a+

√a2+α)u/s

− e(a−√

a2+α)u/s

+12

e(a+

√a2+α)u/s

+ e(a−√

a2+α)u/s.

Note that a and α are functions of s. As before, one can calculateΨ (c, u) by integrating Ψ̃ (c/s, u/s) with respect to the df H ofS, leading to an explicit expression for the ruin probability in aperturbed risk model, with conditionally independent claim sizes,

Ψ (c, u) =

∫∞

0Ψ̃ (c/s, u/s) dH(s)

= 1 − H(c/λ)+

∫ c/λ

0

a + α

2√a2 + α

+12

e(a−

√a2+α)u/s

−

a + α

2√a2 + α

−12

e(a+

√a2+α)u/s

dH(s). (34)

Example 6. Consider the perturbed risk model, with conditionallyindependent heavy-tail claim sizes Xi, i ≥ 1. More precisely, ln Xi,

i ≥ 1, is an L1 Dirichlet random sequence, i.e., Xid= exp(SYi), where

S has df H and is independent of the unit exponentials Yi, i ≥ 1.This means that, given a value of S = s, the claim sizes are inde-pendent Pareto(1/s, 1) distributed. Schlegel (1998, Cor. 4.1) showsthat in the case of subexponential claim sizes, the probability ofruin in a model with ‘‘light perturbation’’ has the same asymptoticbehavior as the probability of ruin in the non-perturbedmodel. TheBrownian motion B satisfies the condition of ‘‘light perturbation’’,namely that, for a fixed ϵ > 0, the distribution function given byP{supt≥0(σB(t)− ϵt) > u} = exp

−

2ϵσ 2 u

is light tailed. In view

of (31), for 0 < λ < c , one may conclude that, under the net profitcondition

E{X1} < c/λ (35)

(for whichwe assume thatH(1−λ/c) = 1), the probability of ruinhas a power-like decay

Ψ (c, u) ∼

∫ 1−λ/c

0

s(c/λ)(1 − s)− 1

u1−1/s dH(s), u → ∞. (36)

Remark. In Examples 5 and 6, we illustrate the fact that the ruinprobability in a perturbed risk model, with conditionally indepen-dent claims, has the same asymptotic behavior as the one in a

perturbed risk model, with independent claims. More precisely, forconditionally independent light tail claims, under (35) (for whichassume that H(λ/c) = 1) one perceives exponential-like decays,

Ψ (c, u) ∼

∫ c/λ

0

a + α

2√a2 + α

+12

e(a−

√a2+α)u/s

−

a + α

2√a2 + α

−12

e(a+

√a2+α)u/s

dH(s),

whereas, for conditionally independent heavy tail (i.e., Pareto type)claims, the probability of ruin exhibits a power-like decay (36), asin a perturbed risk model with independent claims.

6. Proofs

Proof of Theorem 1. The claim follows from Fang and Fang (1988,Thm. 4.2), McNeil and Nešlehová (2010) or the references therein.We give next an alternative proof.

In view of Theorem 4.3.4 of Bryc (1995) the proof follows if weshow that, for any given integer k, and any given positive constantsa1, . . . , ak, we have

min1≤j≤k

Xj

ajd=

X1∑1≤j≤k

aj. (37)

Since (X1, . . . , Xn) is an L1 Dirichlet random vector, then for anyn ≥ 1, we have Rn

d=∑n

j=1 Xj is independent of (O1, . . . ,On) withstochastic representation (4). Thus, for any n ≥ k and Yi, i ≤ nindependent of Rn,

min1≤j≤k

Xj

aj

d=

min1≤j≤k

RnYj

ajn∑

j=1Yj

d=

Rn

nmin1≤j≤k

Yj

aj

1n∑

j=1Yj/n

.We write → and

d→, for the almost sure convergence and conver-

gence in distribution, respectively (as n → ∞). Since Yi, i ≥ 1,are i.i.d. unit exponential random variables, we obtain

∑nj=1 Yj/n

→ 1. Consequently,

Rn

nmin1≤j≤k

Yj

ajd

→ min1≤j≤k

Xj

aj,

implying Rn/nd

→ S, with S independent of Yi, i ≤ k. Hence, by thefact that min1≤j≤k

Yjaj

d= Y1/

∑kj=1 aj

,

Rn

nmin1≤j≤k

Yj

ajd=

Rn

nY1k∑

i=1ai

d→

1k∑

i=1ai

SY1d=

1k∑

i=1ai

X1,

where the last equality in distribution follows from the validity ofthe previous step at k = 1. Hence, the proof is complete. �

Proof of Theorem 2. Let X be a random vector with stochasticrepresentation (12). From the characterization of L1 Dirichletdistributionswith positive pdf, the proof of the direct claim followsif we show that the pdf of X has the form given by (7). LetH denotethe df of X. It is well-known that, if h = ∂kH/(∂x1 · · · ∂xk) iscontinuous and positive (in the support of H), then h is a pdf. Bydefinition, for any xi ∈ (0,∞), i ≤ k, we have

P{X1 ≥ x1, . . . , Xk ≥ xk}= P{ψ−1(U1) ≥ x1, . . . , ψ−1(Uk) ≥ xk}= P{U1 ≤ ψ(x1), . . . ,Uk ≤ ψ(xk)}= ψ(x1 + · · · + xk).

C. Constantinescu et al. / Insurance: Mathematics and Economics 49 (2011) 487–495 493

Since we assume that ψ has a continuous kth derivative and(−1)kψ (k)

≥ 0, h defined by

h(x1, . . . , xk) = (−1)kψ (k)(x1 + · · · + xk)

is indeed a pdf. Noting that

f (y) =1

Γ (k)(−1)kψ (k)(y)yk−1

is a pdf, by (7) this implies that X is an L1 Dirichlet random vector.Since any component subvector of X is again a Dirichlet randomvector with a positive pdf, it follows that ψ is a k-monotonefunction.

We show next the converse. Suppose that X is a L1 Dirichletrandom vector. Then by Theorem 3.2 of Fang and Fang (1988),there exists a functionψ which is k-monotone on (0,∞), such thatψ(0) = 1, and further

P{X1 > x1, . . . , Xk > xk} = ψ

k−

i=1

xi

, ∀xi ∈ (0,∞), i ≤ k,

which implies that X has stochastic representation (12), and thusthe claim follows. �

Proof of Theorem 4. Recall that Xid= RIiOi, with Ii a Bernoulli

random variable and Oi ∼ B1,k−1. By the independence of∑n

j=1 Yj

and Y1/∑n

j=1 Yj, . . . , Yn/∑n

j=1 Yj, we have

Γ (γ + 1) = E{Y γ1 } = E

n−

j=1

Yi

γE{Oγ1 } = Γ (γ + n)E{Oγ1 }.

Since further (I1, . . . , In) are independent of (O1, . . . ,On)

E

max

max1≤i≤n

i−j=1

IjYjη−j, 0

γ

= E

n−

j=1

Yi

γE

max

max1≤i≤n

i−j=1

IjOjη−j, 0

γ.

In view of Hashorva (2006) |X1|d= RO1 is regularly varying at

infinity, with index γ ≥ 0, if and only if R is regularly varying atinfinity, with the same index. Since further, E{Oα1 } < ∞ for anyα > 0, using Breiman’s Lemma (see Breiman (1965) or Lemma 2.3in Davis and Mikosch (2008)) we obtain

limu→∞

Ψ̆ (η, u; n)P{|X1| > u}

= limu→∞

P{R > u}P{|X1| > u}

limu→∞

P

R max

1≤i≤n

i∑j=1

IjOjη−j > u

P{R > u}

=

E

n∑

j=1Yj

γΓ (γ + 1)

limu→∞

P

Rmax

max1≤i≤n

i∑j=1

IjOjη−j, 0

> u

P{R > u}

=

E

n∑

j=1Yj

γΓ (γ + 1)

E

max

max1≤i≤n

i−j=1

IjOjη−j, 0

γ

=1

Γ (γ + 1)E

max

max1≤i≤n

i−j=1

IjYjη−j, 0

γ.

Clearly, Cγ ,η,n is a finite positive constant.

We assume next that (X1, . . . , Xn) is L1 weighted Dirichlet ran-dom vector, for any n ≥ 1. By (10) we have Xi

d= SIiY1 with S > 0,

Ii, Yi, i ≥ 1, independent random variables. In the light of Jacobsenet al. (2009), |X1| is a regularly varying survivor function, implyingthat S also has a regularly varying survivor function, with the sameindex γ as |X1|. In order to apply again Breiman’s Lemma we needto show that, for some α > γ ,

E

max

maxi≥1

i−j=1

IjYjη−j, 0

α=: Cα,η ∈ (0,∞).

Here Cα,η > 0 is obvious. The finiteness of Cα,η follows if we showthat E{Zα} < ∞, with Z =

∑∞

j=0 Yjη−j. This will be established if

we prove that

E{eθZ } < ∞, (38)

for some θ > 0. It is easy to see that, for 0 < θ < η,

E{eθZ } =

∞∏j=1

11 − θη−j

.

Clearly,∏

∞

j=11

1−θη−j < ∞, since ln 11−θη−j ∼ θη−j, for j → ∞.

Therefore, (38) is valid for any 0 < θ < η. Consequently, as in theproof above, Breiman’s Lemma implies

limu→∞

Ψ̆ (η, u)P{|X1| > u}

= limu→∞

P{S > u}P{SY1 > u}

× limu→∞

P

S max

i≥1

i∑j=1

IjYjη−j > u

P{S > u}

=1

Γ (γ + 1)E

max

maxi≥1

i−j=1

IjYjη−j, 0

γ=: Cγ ,η,∞ ∈ (0,∞),

and thus the proof is complete. �

Proof of Theorem 5. We present next the proof of the Gumbelcase, that for the Weibull case follows with similar arguments andis therefore omitted. Set next I0 = O0 = 0 and write

η max0≤i≤n

i−j=0

Xj

ηjd= R max

0≤i≤n

i−j=0

IjOj

ηj−1=: RZη,n.

Since∑n

i=1 Oi = 1 almost surely, and η > 1, we have that Zη,n hasdf with upper endpoint 1. Let G denote the df of O1, and recallthat O1 is Beta distributed, with parameters 1 and n − 1. By theassumption that X1 has df in the Gumbelmax-domain of attraction,then by Theorem 4.1 of Hashorva and Pakes (2010), we have thatalsoRhas df in theGumbelmax-domain of attraction. SinceRhas dfin the Gumbel max-domain of attraction, then, if Zη,n has df in theWeibull max-domain of attraction, with some parameter γ > 0,i.e.,

lims→0

P{Zη,n > 1 − ts}P{Zη,n > 1 − s}

= tγ , ∀t > 0, (39)

by Theorem 3.1 of Hashorva et al. (2010), we obtain

P

max0≤i≤n

i−j=0

Xjη−j > u

= P{RZη,n > ηu}

= (1 + o(1))Γ (γ + 1)PZη,n > 1 −

a(ηu)ηu

P{R > ηu},

494 C. Constantinescu et al. / Insurance: Mathematics and Economics 49 (2011) 487–495

as u → xF/η. Since

1 − G(1 − s) = (n − 1)∫ 1

1−s(1 − x)n−2 dx = sn−1, s ∈ (0, 1)

applying again Theorem 3.1 of Hashorva et al. (2010), we obtain

P{X1 > ηu}= q1P{RO1 > ηu}

= (1 + o(1))q1Γ (n)PO1 > 1 −

a(ηu)ηu

P{R > ηu}

= (1 + o(1))q1Γ (n)[a(ηu)ηu

]n−1

P{R > ηu}, u → xF/η.

The proof follows if we show that (39) holds, with γ = n − 1, andfurther find the tail asymptotic behaviour of P{Zη,n > 1 − ts}, ass → 0 and t > 0. We may further write

P{Zη,n > 1 − ts}

= P

max

0, I1O1, I1O1 + I2

O2

η, . . . , I1O1 + I2

O2

η

+ · · · + InOn

ηn−1

> 1 − ts

= q1P

max

O1,O1 + I2

O2

η, . . . ,O1 + I2

O2

η

+ · · · + InOn

ηn−1

> 1 − ts

+ (1 − q1)P

max

−O1,−O1

+ I2O2

η, . . . ,−O1 + I2

O2

η+ · · · + In

On

ηn−1

> 1 − ts

:= q1J1(η, n, t, s)+ (1 − q1)J2(η, n, t, s).

By the properties of (O1, . . . ,On), we have the stochasticrepresentation

(O1, . . . ,On)d= (O1, (1 − O1)V1, . . . , (1 − O1)Vn−1),

with (V1, . . . , Vn−1) uniformly distributed on the simplex of Rn−1,independent of O1. Hence

J1(η, n, t, s)

=

∫ 1−ts

0P

max

O1 + I2(1 − O1)

V1

η, . . . ,O1

+ I2(1 − O1)V1

η+ · · · + In(1 − O1)

Vn−1

ηn−1

> 1 − ts|O1 = x

dG(x)+

∫ 1

1−tsdG(x)

=

∫ 1−ts

0P

max

I2V1

η, . . . , I2

V1

η+ · · · + In

Vn−1

ηn−1

> 1 −ts

1 − x

(n − 1)(1 − x)n−2 dx + (ts)n−1

=

∫ 1−ts

0Pmax

I2V1

η, . . . , I2

V1

η+ · · · + In

Vn−1

ηn−1

> y

×n − 1(1 − y)n

dy + 1

(ts)n−1.

Furthermore, for s < 1t −

1tη we obtain

J2(η, n, t, s) ≤ P−O1 +

O2

η+ · · · +

On

ηn−1> 1 − ts

≤ P

n−

i=2

Oi

η> 1 − ts

= 0,

hence the proof is complete. �

Acknowledgments

The authors would like to thank the referees for their carefulreading and helpful comments. We would also like to thankHansjörg Albrecher, Bikramjit Das, Francois Dufresne, Rajat Hazra,Jinzhu Li, Zuoxiang Peng, Qihe Tang, Yuebao Wang, MarioWüthrich and Yang Yang for various discussions related to thetopic of this paper. Corina Constantinescu has been partiallysupported by the Swiss National Science Foundation Project200021-124635/1. Enkelejd Hashorva gratefully acknowledges thesupport from Swiss National Science Foundation Project 200021-134785. Lanpeng Ji has been partially supported while staying atNankai University from the Natural Science Foundation of China(No. 11171164).

References

Albrecher, H., Constantinescu, C., Loisel, S., 2011. Explicit ruin formulas for modelswith dependence among risks. Insurance: Mathematics and Economics 48 (2),265–270.

Albrecher, H., Constantinescu, C., Pirsic, G., Regensburger, G., Rosenkranz, M.,2010. An algebraic operator approach to the analysis of Gerber–Shiu functions.Insurance: Mathematics and Economics 46 (1), 42–51.

Asimit, A.V., Furman, E., Tang, Q., Vernic, R., 2011. Asymptotics for risk capitalallocations based on conditional tail expectation. Insurance: Mathematics andEconomics 49 (3), 310–324.

Balakrishnan, N., Hashorva, E., 2011. Scale mixtures of Kotz-Dirichlet distributions.Journal of Multivariate Analysis (in press).

Breiman, L., 1965. On some limit theorems similar to the arc-sin law. Theory ofProbability and its Applications 10 (2), 323–331.

Bryc, W., 1995. The Normal Distribution: Characterizations with Applications.In: Lecture Notes in Statistics, vol. 100. Springer-Verlag, New York.

Charpentier, A., Segers, J., 2007. Lower tail dependence for Archimedean copulas:characterizations and pitfalls. Insurance: Mathematics and Economics 40 (3),525–532.

Charpentier, A., Segers, J., 2008. Convergence of Archimedean copulas. Statistics &Probability Letters 78 (4), 412–419.

Charpentier, A., Segers, J., 2009. Tails of multivariate Archimedean copulas. Journalof Multivariate Analysis 100 (7), 1521–1537.

Davis, R.A., Mikosch, T., 2008. Extreme value theory for space-time processes withheavy-tailed distributions. Stochastic Processes and their Applications 118 (4),560–584.

Denuit, M., Dhaene, J., Goovaerts, M., Kass, R., 2006. Actuarial Theory for DependentRisks: Measures, Orders and Models. Wiley.

Dufresne, F., Gerber, H.U., 1991. Risk theory for the compound Poisson processthat is perturbed by diffusion. Insurance: Mathematics and Economics 10 (1),51–59.

Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling Extremal Events: ForInsurance and Finance. In: Applications of Mathematics (New York), vol. 33.Springer-Verlag, Berlin.

Embrechts, P., Nešlehová, J., Wüthrich, M.V., 2009. Additivity properties forvalue-at-risk under Archimedean dependence and heavy-tailedness. Insurance:Mathematics and Economics 44 (2), 164–169.

Fang, K.T., Fang, B.Q., 1988. Some families of multivariate symmetric distributionsrelated to exponential distribution. Journal of Multivariate Analysis 24 (1),109–122.

Fang, K.T., Fang, B.Q., 1990. Generalized symmetrized Dirichlet distributions.In: Statistical Inference in Elliptically Contoured and Related Distributions.Allerton, New York, pp. 127–136.

Genest, C., Rivest, L.-P., 1993. Statistical inference procedures for bivariateArchimedean copulas. Journal of the American Statistical Association 88 (423),1034–1043.

Hashorva, E., 2006. On the regular variation of elliptical random vectors. Statistics& Probability Letters 76 (14), 1427–1434.

Hashorva, E., Pakes, A.G., 2010. Tail asymptotics under beta random scaling. Journalof Mathematical Analysis and Applications 372 (2), 496–514.

Hashorva, E., Pakes, A.G., Tang, Q., 2010. Asymptotics of random contractions.Insurance: Mathematics and Economics 47 (3), 405–414.

C. Constantinescu et al. / Insurance: Mathematics and Economics 49 (2011) 487–495 495

Jacobsen, M., Mikosch, T., Rosiński, J., Samorodnitsky, G., 2009. Inverse problems forregular variation of linear filters, a cancellation property for σ -finite measuresand identification of stable laws. The Annals of Applied Probability 19 (1),210–242.

McNeil, A.J., Nešlehová, J., 2009. Multivariate Archimedean copulas, d-monotonefunctions and l1-norm symmetric distributions. Annals of Statistics 37 (5B),3059–3097.

McNeil, A.J., Nešlehová, J., 2010. From Archimedean to Liouville copulas. Journal ofMultivariate Analysis 101 (8), 1772–1790.

Mikosch, T., 2009. Non-Life Insurance Mathematics: An Introduction with thePoisson Process, second ed. In: Universitext, Springer-Verlag, Berlin.

Rolski, T., Schmidli, H., Schmidt, V., Teugels, J., 1999. Stochastic Processes forInsurance and Finance. In: Wiley Series in Probability and Statistics, JohnWiley& Sons Ltd., Chichester.

Schlegel, S., 1998. Ruin probabilities in perturbed risk models. Insurance:Mathematics and Economics 22 (1), 93–104.

Schoenberg, I.J., 1938. Metric spaces and completely monotone functions. Annals ofMathematics. Second Series 39 (4), 811–841.

Tang, Q., 2007. The subexponentiality of products revisited. Extremes 9 (3–4),231–241. (2006).

Tang, Q., 2008. From light tails to heavy tails through multiplier. Extremes 11 (4),379–391.

Tang, Q., Tsitsiashvili, G., 2003. Precise estimates for the ruin probability in finitehorizon in a discrete-time model with heavy-tailed insurance and financialrisks. Stochastic Processes and their Applications 108 (2), 299–325.

Tang, Q., Tsitsiashvili, G., 2004. Finite- and infinite-time ruin probabilities in thepresence of stochastic returns on investments. Advances in Applied Probability36 (4), 1278–1299.

Tang, Q., Vernic, R., Yuan, Z., 2011. The finite-time ruin probability in the presenceof dependent extremal insurance and financial risks. Preprint.

Weng, C., Zhang, Y., Tan, Ken S., 2009. Ruin probabilities in a discrete time riskmodelwith dependent risks of heavy tail. Scandinavian Actuarial Journal (3), 205–218.

Yang, Y., Leipus, R., Siaulys, J., 2011. On the ruin probability in a dependent discretetime risk model with insurance and financial risks. Preprint.

Zhang, Z., Yang, H., 2010. A generalized penalty function in the Sparre–Andersen riskmodel with two-sided jumps. Statistics & Probability Letters 80 (7–8), 597–607.