Asymptotic behavior of the Gerber Shiu discounted penalty function in the Erlang(2) risk process with subexponential claims

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1 Nonlinear Analysis: Modelling and Control, 2, Vol. 6, No. 3, Asymptoti behavior of the Gerber Shi disonted penalty fntion in the Erlang2 risk proess with sbexponential laims Jelena Kočetova, Jonas Šialys Department of Mathematis and Informatis, Vilnis University Nagardko str. 24, LT-3225 Vilnis, Lithania Reeived: Febrary 2 / Revised: 7 Jne 2 / Pblished online: 9 September 2 Abstrat. We investigate the asymptoti behavior of the Gerber Shi disonted penalty fntion φ = E e δt {T < } U =, where T denotes the time to rin in the Erlang2 risk proess. We obtain an asymptoti expression for the disonted penalty fntion when laim sizes are sbexponentially distribted. Keywords: Gerber Shi disonted penalty fntion, sbexponential laim sizes, defetive renewal eqation. Introdtion In this paper we onsider the insrer s srpls proess {Ut, t } whih is defined by the eqality Nt Ut = + t Y i, where is the insrer s initial srpls, > is the rate of premim inome per nit time, and {Nt} t is the renewal onting proess for the nmber of laims p to time t. As sal, Nt = {θ+θ 2+ +θ i t}, 2 i= where θ, θ 2,... is a seqene of independent and identially distribted random variables, whih represent the inter-arrival times, with θ being the time ntil the first laim. In The seond athor is spported by a grant No. MIP-55 from the Reasearh Conil of Lithania. i= Vilnis University, 2

2 36 J. Kočetova, J. Šialys addition, in this paper we sppose that θ has the Erlang2 density fntion with sale parameter λ > : ky = λ 2 ye λy, y. Individal laims Y, Y 2,... are non-negative, independent and identially distribted random variables with distribtion fntion Hy = PY y and finite mean EY = a. In addition, we sppose that the laim seqene is independent of the renewal proess Nt. Also we assme that the safety loading ondition holds, i.e. Sppose that ϱ := Eθ EY = 2 >. 3 λa T = inf { t > : Ut < U = } is the time to rin. Hene UT is the defiit at rin and UT is the srpls before rin. In [], Gerber and Shi onsidered a fntion assoiated with a given penalty fntion ω and the joint distribtion of T, UT, UT. The athors named this fntion by an expeted disonted penalty fntion and defined it by the following eqality φ ω = E e δt ω UT, UT {T < } U =, 4 where is an indiator fntion; ωx, y, x, y < is some non-negative fntion, whih an be interpreted as the penalty at the time to rin; and δ is a fore of interest. It is important to mention that, sine the onept of expeted disonted penalty fntion has been introded, many athors started to investigate the Gerber Shi disonted penalty fntion. A nmber of signifiant reslts abot this fntion was obtained, bt many problems haven t been stdied yet and the investigation of the Gerber Shi disonted penalty fntion is still atal and important. Cheng and Tang [2] investigated the disonted penalty fntion in Erlang2 risk proess. They demonstrated that if δ = and ωx, yhx + y dx dy <, 5 then the fntion φ ω satisfies some defetive renewal eqation. Here hy is the ontinos density fntion of individal laim sizes. In the ase where δ, sing a similar approah as in Cheng and Tang, Sn showed see [3] that nder ondition 5 the fntion φ ω satisfies the following defetive renewal eqation where B = λ2 + β 2 φ ω = + β e ρ2s e ρx s s φ ω y dgy + B, 6 + β x ωx, x 2 x dhx 2 dx ds, 7

3 Asymptoti behavior of the Gerber Shi disonted penalty fntion 37 2λδ + δ β = 2 ρ ρ 2 2λδ δ 2 if δ >, λ2 λa aλ 2 2λ + 2 if δ =, ρ 2 and ρ = ρ δ, ρ 2 = ρ 2 δ are two non-negative roots of Lndberg s fndamental eqation see, for example, [4] λ 2 provided ρ < λ + δ/ < ρ 2, ρ =. Frthermore, in 6, where and gx = λ2 2 D = x e sx dhx = s λ δ 2 8 Gy = D e ρ2v x y [v, gx dx, e ρz v dhz dv, gx dx = 2 ρ ρ 2 2λδ δ 2 2 ρ ρ 2 = + β. 9 In this paper we investigate the ase where ωx, y = for all x, y, namely the penalty at the moment T is aepted to be nit. In this ase, for δ let φ = E e δt {T < } U =. We obtain that the partial disonted penalty fntion φ satisfies a defetive renewal eqation withot the tehnial ondition 5 whih was reqired by Sn [3] and Cheng and Tang [2]. Below, we formlate this assertion. Theorem. Consider the Erlang2 risk model with the safety loading ondition 3. If δ >, then the penalty fntion φ satisfies the renewal eqation φ = φ y dgy + G + β + β [,] with distribtion fntion G = G for whih G =, and G = λ2 + β 2 e ρ2y y e ρx y Hx dx dy,. Here ρ, ρ 2 ρ < λ + δ/ < ρ 2 are two positive roots of eqation 8, and the positive onstant β is defined in 9. Nonlinear Anal. Model. Control, 2, Vol. 6, No. 3, 35 33

4 38 J. Kočetova, J. Šialys The defetive renewal eqation for the fntion φ ω defined in 4 has been investigated in many works. The papers of Gerber and Shi [5], Willmot [6] and Landrialt and Willmot [7] shold be mentioned where the general ase of inter-arrival times θ, θ 2,... was onsidered. Unfortnately, to obtain the renewal eqation for φ ω, the athors sppose that the joint distribtion of T, UT, UT has a density. This qestionable assmption greatly failitates the derivation of the renewal eqation. In Setion 2, we present the proof of eqation whih does not reqire the existene of sh a joint density. For this reason, or proof trns ot to be mh more ompliated. Before formlating the seond main reslt, we need to define the lass of sbexponential distribtion fntions. A distribtion fntion F with spport [, is alled sbexponential if F F x lim = 2, x F x where F F denotes the onvoltion of F with itself. As sal, the lass of all sbexponential distribtion fntions is denoted by S. As it was already mentioned, the Gerber Shi disonted penalty fntion was widely investigated. Mh less attention has been paid to the asymptoti behavior of this fntion. We mention only few works. Šialys and Asanavičiūtė [8] obtained an asymptoti formla for the Gerber Shi disonted penalty fntion in the lassial Poisson risk model with sbexponential laim sizes. They showed that the asymptoti formla φ µ H,, δ holds. Here H S, δ >, and µ is the intensity of the Poisson proess. Cheng and Tang [2] derived some asymptoti formlas for the moments of the srpls prior to rin and defiit at rin in the renewal risk model with onvoltion-eqivalent laim sizes and Erlang2 inter-arrival times. Tang and Wei [9] stdied the renewal risk model with absoltely ontinos laim sizes whose density fntion belongs to some lass of heavy-tailed distribtions and satisfies some additional onditions. They obtained asymptoti formlas for the Gerber Shi disonted penalty fntion whih involve the ladder heights and related qantities in the main terms. Or prpose is to find the simple asymptoti formla of the Gerber Shi disonted penalty fntion like in [8] for the Erlang2 risk proess with laims distribted aording to the lassial sbexponential law. The seond main reslt of this artile is the following theorem. Theorem 2. Consider the Erlang2 model with the safety loading ondition 3 with the laim distribtion fntion H S. Then, for δ >, φ H λ 2 2λδ + δ 2,.

5 Asymptoti behavior of the Gerber Shi disonted penalty fntion 39 If, in addition, then, for δ =, a φ ϱa Hy dy S, Hy dy. The seond part of Theorem δ = is well known see, for example, []. The proof in the ase δ > is presented in Setion 3. We an hek that the main formla of Theorem 2 oinides with the asymptoti formla 3.5 of Tang and Wei [9], whih was proved for absoltely ontinos laim distribtions. 2 Proof of Theorem In this setion we show that the Gerber Shi disonted penalty fntion φ satisfies the defetive renewal eqation as stated in Theorem. Proof of Theorem. Let δ be any fixed positive real nmber. Let the srpls proess Ut be defined by eqality with renewal onting proess Nt defined by eqality 2. Denote T = and T m = θ + θ θ m for m. It is evident that rin an or only at moments T m, m. For these moments, NT m = m and where NT m UT m = + T m Y n = S m, S =, S m = We have that for all nonnegative and for all m = 2, 3,... n= m Y n θ n, m =, 2,.... n= PT = T = PS >, PT = T m = PS m >, S m, S m 2,..., S. Therefore, a disonted penalty fntion φ = E e δt {T < } = E e δt = m= {T =Tm} m= E e δtm {S, S 2,..., S m, S m>} Nonlinear Anal. Model. Control, 2, Vol. 6, No. 3, 35 33

6 32 J. Kočetova, J. Šialys for nonnegative. Deomposing the last sm we obtain φ = Ee δt {S>} + Ee δθ e δtm θ m=2 {S, S 2 S S,..., S m S S, S m S > S }. Therefore, φ = Ee δt {S>} + Ee δt e δtm θ m=2 R R {y t, S2 S y+t,..., S m S y+t, S m S > y+t} dhyλ 2 te λt dt = E e δt {S>} + e δt Ee δtm θ m=2 [,+t] {S2 S y+t,..., S m S y+t, S m S > y+t} dhyλ 2 te λt dt = E e δt {S>} + e δt Ee δt l [,+t] l= {S y+t, S 2 y+t,..., S l y+t, S l > y+t} dhyλ 2 te λt dt = e δt dhyλ 2 t e λt dt + e δt φ + t y dhyλ 2 te λt dt = λ 2 y t> t e δ+λt [, +t] [,+t] φ + t y dhy + +t, dhy dt. Sbstitting the new variable z = + t into obtained eqality we get φ = λ2 z e δ+λz [,z] φz y dhy + z, dhy dz. 2 Sine the fntion φz y dhy + dhy = φz y dhy [,z] z, [,z] is dereasing and bonded by nity, the prodt z e δ+λz φz y dhy + [,z] z, dhy

7 Asymptoti behavior of the Gerber Shi disonted penalty fntion 32 is ontinos in z for all positive z with the possible exeption some finite or ontable sbset of R. In addition, this prodt of fntions is integrable over the interval [,. Hene, it follows from 2 that φ is absoltely ontinos dereasing fntion, and for almost all nonnegative φ = λ2 [ z e δ+λz [,z] φz y dhy + z, ] dhy dz. Simplifying the last eqation and sing 2, we get that for almost all nonnegative φ = δ + λ φ λ2 2 e δ+λz [,z] φz y dhy + z, dhy dz. Similarly to the above onsiderations from the last expression we obtain that φ is absoltely ontinos and for almost all nonnegative the seond derivative of φ satisfies φ 2λ + δ = φ + λ2 2 [,] λ + δ2 2 φ φ y dhy + λ2 H. 3 2 Now we se the Laplae transform for reonstrtion of eqation 3. It is well known that the Laplae transform of some fntion exists in the ase if this fntion has exponential order. Sine φ for all, we have that and φ δ + λ φ + λ2 2 e δ+λz φ 2λ + δ φ λ + δ φ + λ2 2 2λ + δ λ + δ λ2 2 dz δ + λ [,] + λ 2 δ + λ :=, φ y dhy + λ2 2 H for almost all. Ths the fntions in eqation 3 are bonded, and the Laplae transforms of these fntions exist. Taking the Laplae transform on the both sides of eqation 3 we obtain that for all omplex s with Res > s 2 φs sφ φ 2λ + δ λ + δ 2 λ = s φs φ φs φs Hs 2 + λ2 Ĥs, 4 2 Nonlinear Anal. Model. Control, 2, Vol. 6, No. 3, 35 33

8 322 J. Kočetova, J. Šialys where φs = bease e s φ d, Hs = [, φ s = s 2 φs sφ φ, and by Fbini s theorem e s φ y dhy d = = [, [,] e s dh, [, Hs = φ s = s φs φ, e sx+y φx dx dhy = φs Hs. y e s H d e s φ y d dhy After some simplifiations of 4 we get the following expression for the Laplae transform φs φs = 2 sφ 2λ + δφ + 2 φ + λ 2Ĥs s λ δ 2 λ 2 Hs. 5 For a latter expression we se the same transformations as in Sn s paper see [3, Thm. 2.2]. Let T ρ be an operator defined in Dikson and Hipp [4]. For a given integrable fntion f and any real ρ T ρ fx = Simple allations show that x e ρz x fz dz, x. T ρ fs = fs fρ ρ s for all ρ >, Res >, ρ s. Non negative real roots of eqation 8 ρ and ρ 2 are zeroes for the denominator of expression 5. Hene, they are also zeroes for the nmerator of this expression. In partilar, 2 ρ φ 2λ + δφ + 2 φ + λ 2Ĥρ =. Hene, from property 6 and expression 5 we get 6 φs = 2 sφ + λ 2 Hs 2 ρ φ λ 2Ĥρ s λ δ 2 λ 2 Hs Hs Ĥρ ρ s = s ρ 2 φ λ 2 s λ δ 2 λ 2 Hs = s ρ 2 φ λ 2 T ρ Hs s λ δ 2 λ 2 Hs 7

9 Asymptoti behavior of the Gerber Shi disonted penalty fntion 323 for all omplex s with positive real part. It was mentioned that ρ 2 is another zero for denominator of the last expression. Hene ρ 2 ρ 2 > ρ is a zero for nmerator. Therefore, 2 φ = λ 2 T ρ Hρ 2, and, sing the property 6 again, we obtain from 7 that for above s φs = λ2 s ρ s ρ 2 T ρ Hρ 2 Ls s ρ 2 T ρ Hs = λ2 s ρ s ρ 2 T ρ2 T ρ Hs, 8 Ls where Ls denotes the denominator of the fration in 7. denominator in more details. Evidently, Now we onsider this Ls = s λ δ 2 λ 2 Hs ρ λ δ 2 + λ 2 Hρ = s ρ 2 s + ρ 2λ + δ + λ Hρ 2 Hs. 9 s ρ Let τ ρ be a new operator defined by eqality τ ρ F x = e ρz x df z, x, [x, where F is a distribtion fntion of a nonnegative random variable and ρ is real positive nmber. Similarly to relation 6, we obtain that the Laplae transform of fntion τ ρ F has the following property τ ρ F s = F s F ρ, 2 ρ s where F s = [, e s df denotes the Laplae Stieltjes transform of the distribtion fntion F. Aording to this relation, it follows from 9 that Ls = s ρ 2 s + ρ 2λ + δ + λ 2 τ ρ Hs Sine ρ 2 is also a root of Ls =, we have Hene, sing the property 6 we obtain: 2 ρ 2 + ρ 2λ + δ + λ 2 τ ρ Hρ 2 =. Ls = s ρ 2 s ρ 2 + λ 2 τ ρ Hs λ 2 τ ρ Hρ 2 = s ρ s ρ 2 2 λ 2 T ρ2 τ ρ Hs. Nonlinear Anal. Model. Control, 2, Vol. 6, No. 3, 35 33

10 324 J. Kočetova, J. Šialys Sbstitting this expression into 8 we get that φs = λ2 T ρ2 T ρ Hs 2 λ 2 T ρ2 τ ρ Hs 2 for omplex s, Res >. For non negative, let η := T ρ2 T ρ H = γ := T ρ2 τ ρ H = e ρ2x e ρy x Hy dy dx, x e ρ2x [x, e ρy x dhy It follows from 2 that φs = λ2 φs γs + ηs 2 for any omplex s as above. Inverting the Laplae transform, we get the following form of the renewal eqation for φ φ = λ2 2 φ yγy dy + λ2 2 η. The desired renewal eqation follows immediately from the last eqality taking bease for details, see [] or [2] + β λ 2 2 γx dx = λ2 + β 2 G = λ2 + βη, 2 = λ2 + β 2 x x, = λ2 + β 2 ρ 2 ρ = λ2 + β 2 ρ 2 ρ x, e ρ2y x [y, z e ρ2x ρz x dx. e ρz y dhz dy dx e ρ2 ρy dy dhz dx e ρ z x e ρ2z x dhz dx e ρ x e ρ2x Hx dx

11 Asymptoti behavior of the Gerber Shi disonted penalty fntion 325 and G = λ2 + β 2 = λ2 + β 2 = λ2 + β 2 ρ 2 ρ e ρ2y y x Hxe ρx+ρ2 e ρx y Hx dx dy e ρ2 ρy dy dx e ρ x e ρ2x Hx dx. Finally, observe that the properties 6 and 2 imply identity 9. Namely, λ 2 2 Theorem is proved. 3 Proof of Theorem 2 γx dx = λ2 2 T ρ2 τ ρ H = λ2 τ ρ H τ ρ Hρ 2 2 = λ 2 λ 2 Hρ ρ + λ2 Hρ2 λ 2 Hρ ρ 2 ρ 2 ρ 2 ρ 2 λ 2 ρ λ δ 2 ρ = + ρ2 λ δ2 ρ λ δ 2 ρ 2 ρ 2 ρ 2 = 2 ρ ρ 2 2λδ δ 2 2. ρ ρ 2 In this setion, we present the proof of Theorem 2. For this we need the following two axiliary lemmas. The first lemma desribes the standard form of a soltion of eqation 6. The simple proof of lemma an be fond, for example, in [2] see Theorem 2.. Lemma. Assme that a fntion ψ satisfies the defetive renewal eqation ψv = ψv x dv x + W v, v, + b + b [,v] where b >, V x = V x is a distribtion fntion with V =, and W v is ontinos for all v. The soltion of this eqation an be expressed in the form ψv = b [,v] W v x dkx = b,v] W v x dkx + + b W v. Nonlinear Anal. Model. Control, 2, Vol. 6, No. 3, 35 33

12 326 J. Kočetova, J. Šialys Here Kx = Kx is the assoiated ompond geometri distribtion fntion, i.e. Kx = n= n b V + b + b n x, x, and V n denotes the tail of the n-fold onvoltion of V with itself. The following lemma is needed in order to ompare the tail of a onvoltion of distribtion fntions with a tail of some basi sbexponential distribtion. The proof of the lemma below an be fond in [3] see Corollary 3.9. Lemma 2. Let F be some distribtion fntion from S, and let G, G 2,..., G n be distribtion fntions sh that G i x lim x F x = i, i =, 2,... n, with some nonnegative onstants, 2,..., n. Then G G 2 G n x F x + + n as x. If, in addition n >, then G G 2 G n S. Proof of Theorem 2. In the ase δ >, aording to Theorem, we have that fntion φ satisfies the defetive renewal eqation with the distribtion tail G defined by. Hene, it follows from Lemma that, for any nonnegative, φ = G x dlx 22 β with [,] Lx = Lx = n= β + β n+ G n x. In order to obtain the asymptoti formla for φ onsider first the fntion G. Let for nonnegative y Then, aording to, ry := G = λ2 + β 2 y e ρx y Hx dx. e ρ2y ry dy.

13 Asymptoti behavior of the Gerber Shi disonted penalty fntion 327 We have that ry = ρ Hy + e ρz H dz + y, 23 and for every positive M Hy e ρz H dz + y [, = Hy [,M] Hy + M Hy [, e ρz H dz + y + Hy + e ρm. M, e ρz H dz + y The basi properties of sbexponential distribtion fntions see, for example, Lemma.3.5 in [4] imply that for every fixed M Hy + M lim =. y Hy Therefore, aording to the obtained estimates, lim e ρz H dz + y =. 24 y Hy This and 23 imply Hene for we get [, ry = ρ Hy + o, y. G = λ2 2 + β + o e ρ2y Hy dy ρ = λ2 2 + β + o H + e ρ2v H dv + ρ ρ 2 bease, similarly to 24, [, = λ2 2 ρ ρ 2 + βh + o 25 lim H [, e ρ2v H dv + =. Nonlinear Anal. Model. Control, 2, Vol. 6, No. 3, 35 33

14 328 J. Kočetova, J. Šialys Distribtion fntion H belongs to the lass S, so, aording to Lemma 2, we have that for every fixed n G lim n = λ2 H 2 + βn, ρ ρ 2 and distribtion fntion G also belongs to the lass S. Aording to the basi properties of sbexponential distribtion fntions see, for example, Lemma.3.5 in [4], there exists a finite onstant K β sh that for all n 2 G sp n K β + β n. G 2 Therefore, for every nonnegative G n + β n H = G n + β n G By the dominated onvergene theorem we obtain L lim H = β + β lim = β + β = βλ2 2 ρ ρ 2 n= n= G H K βλ β + β 2 n ρ ρ 2 + β n. n= + β n G n H + β n lim G n H n + β n = λ2 + β 2 ρ ρ 2 β and we get that L λ2 + β 2 H,. ρ ρ 2 β If is nonnegative, then G L = G y dly + L. [,] [,] Aording to Lemma 2 and 25 G y dly λ2 2 + βh, ρ ρ 2 Hene, from 22 we get that for φ λ2 2 ρ ρ 2 β + H. Theorem 2 is proved.

15 Asymptoti behavior of the Gerber Shi disonted penalty fntion Examples In this setion we present three simple examples whih illstrate the se of Theorem 2 for the evalation of the Gerber Shi disonted penalty fntion. All obtained formlas are appliable only for a large vales of initial apital. Example. Consider the Pareto laim distribtion fntion Hx = + x 3 {x }, the Erlang2 sale parameter λ = 2 and the serity loading oeffiient ϱ >. It is well known that distribtion fntion H is sbexponential see, for instane, [4]. In addition, a = EY = /2, and Hy dy/a is also sbexponential. Hene, aording to Theorem 2 we have that in the desribed model 4 δ φ 2 + 4δ + 3 if δ >, ϱ + 2 if δ =. Example 2. Consider the Weibll distribtion fntion Hx = e x {x }, for laims, the inter arrival times sale parameter λ, the fore of interest δ and the serity loading oeffiient ϱ >. Distribtion fntion H is sbexponential see, for instane, [4], a = EY = 2, and a x Hy dy = x + e x {x } S. In the onsidered ase Theorem 2 implies that λ 2 δ φ 2 + 2λδ e if δ >, + e if δ =. 2ϱ Example 3. Sppose now that laims of Erlang2 renewal risk model are distribted aording to the disrete law with distribtion fntion where A = Hx = A k= [x] k= e log2 k 2 log k, k 2 log k e log2 k k Nonlinear Anal. Model. Control, 2, Vol. 6, No. 3, 35 33

16 33 J. Kočetova, J. Šialys Let, in addition, λ = 2, δ =.3 and ϱ is positive. It is evident that Hx e log2 +x /A for large x. Distribtion fntion e log 2 +x {x } belongs to the lass S see, for instane, [5, p. 87]. Hene H S aording to the basi properties of sbexponential distribtion for details see [3, Corol. 3.3], and as, aording to Theorem 2. φ e log2 +, Aknowledgement The athors are gratefl for the omments and sggestions of anonymos referees. These have led to a mh improved paper. Referenes. H. Gerber, E. Shi, On the time vale of rin, N. Am. Atar. J., 2, pp , Y. Cheng, Q. Tang, Moments of the srpls before rin and the defiit at rin in the Erlang2 risk proess, N. Am. Atar. J., 7, pp. 2, L.J. Sn, On the disonted penalty at rin at the Erlang2 risk proess, Stat. Probab. Lett., 72, pp , D.C.M. Dikson, C. Hipp, On the time to rin for Erlang2 proess, Insr. Math. Eon., 29, pp , H. Gerber, E. Shi, The time vale of rin in a sparre Andersen model, N. Am. Atar. J., 92, pp , G.E. Willmot, On the disonted penalty fntion in the renewal risk model with general interlaims times, Insr. Math. Eon., 4, pp. 7 3, D. Landdrialt, G.E. Willmot, On the Gerber Shi disonted penalty fntion in the sparre Andersen model with an arbitrary inter-laim time distribtion, Insr. Math. Eon., 42, pp. 6 68, J. Šialys, R. Asanavičiūtė, On the Gerber Shi disonted penalty fntion for sbexponential laims, Lithanian Math. J., 46, pp , Q. Tang, L. Wei, Asymptoti aspets of the Gerber Shi fntion in the renewal risk model sing Wiener Hopf fatorization and onvoltion eqivalene, Insr. Math. Eon., 46, pp. 9 3, 2.. P. Embrehts, N. Veraverbeke, Estimates for the probability of rin with speial emphasis on the posibility of large laims Insr. Math. Eon.,, pp ,

17 Asymptoti behavior of the Gerber Shi disonted penalty fntion 33. C.C.L. Tsai, L.J. Sn, On the disonted distribtion fntions for Erlang2 proess, Insr. Math. Eon., 35, pp. 5 9, X.S. Lin, G.E. Willmot, Analysis of a defetive renewal eqation arising in rin theory, Insr. Math. Eon., 25, pp , S. Foss, D. Korshnov, S. Zahary, An Introdtion to Heavy-Tailed and Sbexponential Distribtions, Springer, P. Embrehts, C. Klüppelberg, T. Mikosh, Modelling Extremal Events, Springer, D.B.H. Cline, G. Samorodnitsky, Sbexponentiality of the prodt of independent random variables, Stoh. Pro. Appl., 49, pp , 994. Nonlinear Anal. Model. Control, 2, Vol. 6, No. 3, 35 33