fi K Z 0 23A:4(2002), ρ [ fif;μ=*%r9tμ?ffi!5 hed* c j* mgi* lkf** J W.O(^ jaz:ud=`ψ`j ψ(x), p: x *fl:lffi' =Λ " k E» N /,Xß=χο6Πh)C7 x!1~πψ(x)

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1 fi K Z 23A:4(22), ρ [ fif;μ=*%r9tμ?ffi!5 hed* c j* mgi* lkf** J W.O(^ jaz:ud=`ψ`j ψ(), p: *fl:lffi' =Λ " k E» N /,Xß=χο6Πh)C7!1~Πψ() =,? s3πμ;χwc&c= Cramer-Lundberg UD6=χL&pUy ', +WοΞfiffbHΞ_χ_IΞY9TCΞi@% MR (2) bkψ1 62P5, 62E2, 6K3, 6K5 alffiψ1 O211.3, O213.2 NPΩD8 A N^Ψ( (22) Z Λ V 5ΩΠTC!]`RΩfi9yS </fiξνξ^ Grandell [7] 71H<± [M ID Z i ;i 1 ΞTfG<!Wο (i.i.d) P F _<b(ξ [ e} i, i 1 T n = n n 1 ffi#)9[}4idξj} I [;t] Q <9[-ff) N(t) = supfn 1;T n» tg, t, νn$tfi@%ξjb fz i ;i 1g G<±`Ξi@Y9%DY) N(t) X S(t) = i=1 i=1 i, Z i ct; (1.1) d!ff c ( <c<1)»λfi9i±vv i, i 1 Ξ+_kmWοzff G b_</u i, i 2 Ξ+ _kmwοzff G 1 G<ΞO 1 [. G < ] WοΞΨ Z G e (t) = 1 t G(u)du; E 2 l#zωtc)] i@y9tc±vv 2 [.vffwο (!P 1 R[.vffWο), Z, N(t) Ξe- Poisson %Ξ= <TC#) Cramer-Lundberg TC±+fl<4fl3fi μ [7, 8]. ffl-π w}id ρ = CE 2 EZ2 EZ 2 ψ() DY) ψ() =P. 2 Z 8 f 4 t±9π 21 Z 7 f 2 t±9f^c ΛΩΛ3KfiK3Π}V ΛΛ u2kfiψ3k"ffi"u3π}v >, #ν)=jψo]hffiν±_χ_i sup S(t) > ; J` : (1.2) t

2 532 fi K Z 23 - A ν»λffldfi9kfl& ffl) }1<l z<_χ_i± v!hl%b<-ntpξpabm<ξ [M[. X Ξνj [; 1] z<wοn_s <_<vff*ξ (Ψ*Xzffνj) l#)ν <Wο) +Wο±5QD(ßQ]8< +Wο<T? QΞ8ΞiL<4fl3fiμ [4] >± F ( y) (1) L 8Φ F M± lim =1Js y>(ξ>` y =1). F () (2) D 8Φ F M± lim sup F (3) S 8Φ F M± lim nλ () F () Λ F < n,±ξ F nλ () =1 F nλ (). (4) ERV 8Φ F M± y fi» lim inf F (y) F () < 1, Js <y<1, (Ξ>` y = 1 2 ). = n, J`s n 2(Ξ>` n =2),d F nλ» F (y) F () F (y)» lim sup» F () y ff, J`s y>1, d 1 <ff» fi<1 ΞsXqD< ff± r`_χ_i ψ() <+>r2rpv%g(b5ωffiv [4;5;11 14] : ο2 A 1H=JΨo]hffiν ρ> $<< Cramer-Lundberg TCΞvv [M Wο F 2 D l_ ψ() ο ρ 1 F e (); 6!1}: (1.3) οg~b + [9] ßzΩffiv#t8aψi@Y9TCΦ ο2 B j=jψo]hffiν ρ> $<<aψi@y9tc ΞfflD81} e 2.J=JΞvv [MWο F 2 D, l ψ() <+>r2 (1.3) R$<± ffl-$tρßzωffiv#t8] i@y9tcznξμjssj`o~ke ψ() +_!P<+>r2± 2. ]Φ`E 2.1 C 1 )BffΦz<eZΞ[$ +Wο<Tf@<Ξ8 C οx 2.1 (C 8) F Ξ (; 1] z<tfwοzffξvv F M± lim lim sup F (l) F () =1, l# F 2 C. r`ξ8 C, _5Ω»4ffivΞo?ffiv»SΞ8 C ffyt?_]<wοξv ERV 8 A 3 fl ßQ]8<Wο8± Y2 2.1 ERV ρ C ρ D L. _ J` 1 <ff» fi<1, ERV(ff; fi) <DY>` l ff» lim inf F (l) F ()» lim sup F (l) F ()» l fi J` <l<1; F l! 1 tff3: ERV ρ C, 8q C ρ D, ZO#GsS C ρ L. J` F 2 C < l < 1, 6 ±o2}ξ 1» F ( 1) F () F ( 1) lim =1,ZO F 2 L. F () Y2 2.2 F 2 D gyb F e 2 C. _ J` <l<1, _ R F e (l) F (t)dt l R 1 F (t)dt ;» F e () =1+ R» F (l) F () : `Ξ7F! 1 if l! 1, :8 F (t)dt l F R 1» R (l)(1 l) F (t)dt 2 F (t)dt» F (l)(1 l) : F (2)

3 4 B χ fl 4 M.3 HC) ff»flg<ν>+&s:uν@ffl"6 533 ^` F 2 D gy F (l) F (2) ΞT_!<Ξp_ lim lim sup R R 1 F (t)dt l =; F (t)dt ZO F e 2 C I$ß ψff ;j[$^ (1.1) DY<i@Y9% S(t) <?f Q<ρqΦ sup t>fin 1 (a) Mdmb<fiff}4 fi n =infft >;S(t) >S(fi n 1 )g; S(t) <S(fi n 1 ) DY fi n = 1 j S(fi )=S() =. (b) Mdmb<fiffbH L n = S(fi n ) S(fi n 1 ); n 1: w fi n = 1 DY L n = 1. ]8ffl<TRΞDY bfiff}4 bfiffρhv5 n 1: )BTflΞw fi n = infft n : S(T n )» S(fi n 1 )g; n 1; L n = S(fi n ) S(fi n 1 ); n 1: )Tfl S(fi )=S()=,fi =. 8qΞ f(fi i ;l i );i 1g f(fi i ;L i );i 1g ΞG<<PdΠIDΞ5Ξ^`j] i@tc 1 b 2 ν!wοξ*u 2 f(fi i ;l i );i 2g f(fi i ;L i );i 2g Ξ i.i.d <PdΠID±,$Ξvj [2, 6, 9] >- μξof L i;i 1 FΞ_7 L i, i 1 H 1 H W»ΛATfiff L 1 APfiff L 2 <Wο (W +_7@ q 1 = P (L 1 = 1) > q = P (L 2 = 1) > ). ^`J;j<Y9TCΞ_ lim t!1 N(t) t a.s. = 1 ; EN(1) = 1 : (2.1) E 2 E 2 p- [2,6] r` H 1 <+Wο 7@ q 1 g<ffivv!j5qd9 C Λ<YP3W g(e$<»λ± (;v3μ Ross [8] r`> (2.1) <ss) ο2 C j=jψo]hffiν ρ> $<<] i@y9tc Ξ_ q 1 = CE 2 EZ 2 CE 2 ; H 1 () = EZ 2 CE 2 F e (); (2.2) d H 1 () =H 1 (1) H 1 () =1 q 1 H 1 (). οg~ + [1] $TρN'i@Y9TCΞ:8Br` H <+Wο zω7@ q <U!8ffl<ffiv± ο2 D j=jψo]hffiν ρ> $<<] i@y9tc ΞfflD 2 Ξ.J= J<Ξvv F 2 D, l q = CE 2 EZ 2 EjL 2 j ; H() ο EZ 2 EjL 2 jf e(); (2.3) d H <+Wο3W8ffl@9ψ)D9 C < H 1 <+Wο± )»4fμΞDY L 1 L 2 <Wοzff< E v5 H 1s () =P (L 1» jfi 1 < 1) = H 1() 1 q 1 ; H s () =P (L 2» jfi 2 < 1) = H() 1 q : (2.4) jzω?fd9<ffiν5ξvv F 2 D, H 1s H s FΞLvffWοΞ+fl4fl3fiμ [4].

4 534 fi K Z 23 - A 3. A #7 ο2 3.1 j=jψo]hffiν ρ> $<<] i@y9tc ΞfflD 2 Ξ.J =J<Ξvv F 2 D, l_ (1.3) $<± _ DY v = inffk 1;fi h = 1g, 3μ v Ξ[.Ω P (v =1)=q 1 ; P (v = k) =q(1 q) k 2 (1 q 1 ); k 2: ZOj (1.1) DY<i@Y9% S(t) < 2u3W ) M =sup t L =,`Ξ ψ() =P (M >)= = n 1 X P 1 > L k >;v= n k=1 n 1 X P 1 > L k >;L n = 1) =qh 1 ()+q = qh 1 ()+q = H 1 ()+q k=1 n=2 Z (1 q) n 1 + Z 1 n=2 P 1 > H s (n 1)Λ ( t)dh 1 (t) P S(t) = v 1 L k d nx h=1 k= L k > Z (1 q) n H nλ ( t)dh 1 (t) (3.1) J`s qd< <l<1, zqap>3wwψ)v5?ffw (I) + (II) q Z l (1 q) n H nλ ( t)dh 1 (t)+q Z (1 q) n H nλ ( t)dh 1 (t): 71ffi (I), v!a 2 fl<v+ KΩ<YP_ H s 2 S, `ΞJOTf "> W Tf k(") >, JTi n 1, Ti WΦTi l t, T@_ H nλ s ( t) H s ( t)» k(") (1 + ")n : (3.2) o3fiμ [1,3,4]. j (3.2) F "> M± (1 q)(1 + ") < 1, `ΞJTi >, T@_ (I) R l H s( t)dh 1 (t)» qk(") Z,^6zΠ>D9#: ZO ^` lim R l (I) H s( t)dh 1 (t) = q Z l = q (I) ο 1 q H s ( t)dh 1 (t) = 1 q q (III) (IV): hz» (IV) (III)» H 1(l) H 1 () ο H 1 ()H() l [(1 q)(1 + ")] n C<1: (1 q) n lim R l H s R nλ l n(1 q) n = 1 q : q H( t)dh 1 (t) ( t)dh 1 (t) H s( t)dh 1 (t) Z l EjL j 2 Fe (l) ce 2 H 1 () F e () 1 : i H( t)dh 1 (t)

5 4 c 5S" fl G '"A ßh Ka3N» =^ jaud=`ψ`j 535 p ^[9 2.2, _ F e 2 C,,$ (III) = 1 q Z (IV) lim lim sup (III) =: Z H( t)dh 1 (t) ο (EZ 2) 2 qce 2 EjL j F e ( t)df e (t) 2 Z = ρ EZ 1 2 F e ( t)df e (t): ce 2 J` (II) _ (II)» q (1 q) n H1 (l) (H 1 (l) H 1 ()) = (1 q) H 1 () 1 H 1 () Fe (l) =(1 q) F e () 1 H 1 (): ZO p ^` F e 2 C ρ S, $O3W:8 (II) lim lim sup H 1 () =: ψ() =H 1 () + (I) + (II) ο H 1 () + (III) ο EZ Z 2 F e ()+ρ 1 F e ( t)df e (t) : ce 2 R lim F e( t)df e (t) F e () = lim F 2Λ e () F e () =1 F e () ψ() ο EZ 2 ce 2 (1 + ρ 1 )F e () =ρ 1 F e (): op)%$bd9 3.1 <ss± fi Λ / Λ O Λ Q [ 1] Athreya, K. B. & Ney, P. E, Branching processes [M], Springer, Berlin, [ 2] Bjök, O. & Grandell, J., An insensitivity property of the ruin probability [J], Scand. Act. J., (1985), [ 3]»Cistyakov, V. P., A theorem on the sums of independent positive random variables and its applications to branching random processes [J], Theory Prob. Appl., 9(1964), [ 4] Embrechts, P, Goldie, C. & Mikosch, T., Modelling etremal events for insurance and finance [M], Springer, Berlin, [ 5] Embrechts, P. & Veraverbeke, N., Estimates for the probability of ruin with special emphasis on the possibility of large claims [J], Insurance. Math. Econom., 1(1982),

6 536 fi K Z 23 - A [ 6] Frenz, M. & Schmidt, V., An insensitivity property of ladder height distributions [J], J. Appl. Prob., 29(1992), [ 7] Grandell, J., Aspects of risk theory [M], Springer-Verlag, Berlin, [ 8] Ross, S. M., Stochastic process [M], Wiley, New York, [ 9] Tang Qihe, Ruin probabilities for large claims in renewal risk model [C], Proceedings of Third Symposium of Post-Graduates of USTC, (2), [1] Su Chun, Tang Qihe & Jiang Tao, A Contribution to large deviations for heavy-tailed random sums [J], Chinese Science, 44:4(21), [11] Thorin, O., On the asymptotic behavior of the ruin probability for an infinite period when the epochs of claims form a renewal process [J], Scand. Actuar. J., (1974), [12] Thorin, O., Probability of ruin [J], Scand. Actuar. J., (1982), [13] Thorin, O. & Wikstad, N., Calculation of ruin probabilities when the claim distribution is longormal [J], ASTIN Bull., 9(1977), [14] Von Bahr, B., Asymptotic ruin probability when eponential moments do not eist [J], Scand. Actuar. J., (1975), 6 1. RUIN PROBABILITIES FOR LARGE CLAIMS IN EQUILIBRIUM RENEWAL MODEL KONG Fanchao* CAO Long* WANG Jinliang* TANG Qihe** ΛDepartment of Mathematics, Anhui University, Hefei 2339, China. ΛΛDepartment of Statistics and Finance, University of Science and Technology of China, Hefei 2326, China. Abstract This paper investigates ruin probabilities ψ() in the equilibrium renewal risk model, where is the initial capital of an insurance company. Under the assumption that the claim size is heavy-tailed, we aim at a tail equivalence relationship of ψ() as!1and obtain the desired result in the paper, which is surprisingly the same as that in the Cramer-Lundberg model. Keywords Heavy-tailed distributions, Ladder height, Ruin probabilities, Risk model, Renewal process 2 MR Subject Classification 62P5, 62E2, 6K3, 6K5 Chinese Library Classification O211.3, O213.2 Article ID (22) The English translation of this paper will be published in Chinese Journal of Contemporary Mathematics, Vol.23 No.3, 22 by ALLERTON PRESS, INC. NEW YORK, USA