TAIL PROBABILITIES OF RANDOMLY WEIGHTED SUMS OF RANDOM VARIABLES WITH DOMINATED VARIATION

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1 Stochastc Models, :53 7, 006 Copyrght Taylor & Francs Group, LLC ISSN: prnt/ onlne DOI: / TAIL PROBABILITIES OF RANDOMLY WEIGHTED SUMS OF RANDOM VARIABLES WITH DOMINATED VARIATION Dngcheng Wang School of Appled Mathematcs and School of Management, Unversty of Electronc Scence and Technology of Chna, Chengdu, P.R. Chna Qhe Tang Department of Statstcs and Actuaral Scence, The Unversty of Iowa, Iowa Cty, Iowa, USA Ths paper nvestgates the asymptotc behavor of tal probabltes of randomly weghted sums of ndependent heavy-taled random varables, where the weghts form another sequence of nonnegatve and arbtrarly dependent random varables. The results obtaned are further appled to derve asymptotc estmates for the run probabltes n a dscrete tme rsk model wth dependent stochastc returns. Keywords Heavy tal; Maxmum of sums; Randomly weghted sums; Run probablty; Stochastc return; Tal probablty. Mathematcs Subject Classfcaton Prmary 6E0, 60G70; Secondary 91B INTRODUCTION Let X ; = 1,, be a sequence of ndependent, dentcally dstrbuted..d., and real-valued random varables wth generc random varable X and common dstrbuton F = 1 F, whch satsfes F x >0 for all real numbers x, and let ; = 1,, be another sequence of nonnegatve random varables ndependent of the sequence X ; = 1,,. We wrte S n = X, n = 1,,, 1.1 Receved March 004; Accepted August 005 Address correspondence to Dngcheng Wang, School of Appled Mathematcs and School of Management, Unversty of Electronc Scence and Technology of Chna, Chengdu , P.R. Chna; E-mal: wangdc@uestc.edu.cn

2 54 Wang and Tang whch are called randomly weghted sums. To avod trvalty, we slently assume that each weght s not degenerate at 0. Note that we wll not make any specfc assumptons on the dependence structure of the sequence ; = 1,,. The goal of ths work s to establsh for the randomly weghted sums 1.1 the asymptotc relaton Pr S n > x Pr X > x 1. for each fxed n = 1,, or unformly for n = 1,,. Hereafter, all lmtng relatonshps are for x unless stated otherwse; by usng the symbol we mean that the rato of the two sdes tends to 1. Admttedly, ths s an nterestng queston and has many applcatons. For example, we may assgn the sums n 1.1 an economc explanaton. Consder a dscrete tme rsk model, n whch the surplus of an nsurance company s nvested nto a rsky asset. We understand each X as the net loss the total clam amount mnus total ncomng premum wthn the tme perod and understand each as the dscount factor from tme to tme 0 the present. Then, Sn n 1.1 denotes the dscounted amount of the total net loss of the company at tme n. If we wrte the maxmum of the frst n = 1,, weghted sums and the ultmate maxmum as M n = max 1 k n S k and M = max 1 k< S k, then the probabltes of run by tme n = 1,, and of ultmate run can be defned by x; n = Pr M n > x and x = Pr M > x, 1.3 respectvely, where x 0 s the ntal surplus. Recently, the run probabltes n 1.3 have been extensvely nvestgated by Nyrhnen [6,7] and Tang and Tstsashvl [9,11], among others. The standng assumpton of ths work s that the common dstrbuton F s heavy taled. The most mportant class of heavy-taled dstrbutons s the subexponental class. By defnton, F concentrated on [0, s sad to be subexponental, denoted by, f the relaton F lm n x x F x = n 1.4 holds for any or, equvalently, for some n, where F n denotes the n-fold convoluton of F. More generally, F on, s stll sad to

3 Tal Probablty of Randomly Weghted Sums 55 be subexponental f F + x = F x1 0 x< s subexponental, where 1 A denotes the ndcator functon of A. Closely related are the class of dstrbutons wth domnatedly varyng tals, the class L of dstrbutons wth long tals, and the class of dstrbutons wth consstently varyng tals. By defnton, F f the relaton lm sup x F xy F x < holds for some or, equvalently, for all 0 < y < 1; F L f the relaton F x + y lm = 1 x F x holds for some or, equvalently, for all y > 0; F f F lx lm lm sup l 1 x F x F lx = 1, or, equvalently, lm lm nf = l 1 x F x The ntersecton L s a useful class of heavy-taled dstrbutons. Specfcally, t contans the famous class R of dstrbutons wth regularly varyng tals. By defnton, F R f there exsts some 0 such that the relaton F xy lm x F x = y 1.6 holds for any y > 0. Denote by F R the regularty property n 1.6. A slghtly larger one s the class of dstrbutons wth extended regularly varyng ERV tals. By defnton, F belongs to the class ERV f there exst some 0 < such that the relaton y lm nf x F xy F x lm sup x F xy F x y 1.7 holds for any y > 1. Denote by F ERV, the regularty property n 1.7. It s well known that R ERV L L For more detals of heavy-taled dstrbutons, we refer the reader to Bngham et al. [1] and Embrechts et al. [5]. Let us revew some results regardng the asymptotc tal probabltes of the weghted sums n 1.1. When the generc random varable X s

4 56 Wang and Tang nonnegatve wth F R for some >0, Resnck and Wllekens [8] proved that the relaton Pr j X j > x Pr X > x 1.8 holds as long as the weghts ; = 1,, fulfll sutable summablty condtons. If the random varables X ; = 1,, are real valued, as those n the above-mentoned probablstc model, the result of Resnck and Wllekens [8] can not be appled mmedately. For ths case, under the assumptons F L and = Y j, = 1,,, 1.9 wth Y ; = 1,, beng a sequence of..d. nonnegatve random varables, Tang and Tstsashvl [9] establshed the asymptotc relaton Pr M n > x Pr X > x 1.10 for each fxed n = 1,,. Later, under the restrcton F ERV, Tang and Tstsashvl [11] further verfed the unformty of the asymptotc relaton 1.10 wth respect to n = 1,, hence, t apples to the case of n =. For the case of general weghts, Tang and Tstsashvl [10] proved that relatons 1. and 1.10 holds smultaneously under the assumptons F and Pra b = for some 0 < a b < and for all = 1,,. That s, the relatons Pr M n > x Pr S n > x Pr X > x 1.1 hold for each fxed n = 1,,. However, 1.11 s apparently too restrctve for applcatons. A more recent dscusson n the context of actuaral scence s Goovaerts et al. [13] In the present paper, we reconsder the queston wthout restrcton Frst we prove that relatons 1.1 hold for each fxed n = 1,,. f F L and ; = 1,, fulfll sutable moment condtons. Then, we dscuss the unformty of relatons 1.1 for the case F ERV. The remanng part of ths paper conssts of three sectons. Secton shows the man results, Secton 3 apples the man results to approxmatng

5 Tal Probablty of Randomly Weghted Sums 57 the fnte and nfnte tme run probabltes wth dependent return rates, and Secton 4 proves the man results.. MAIN RESULTS For a dstrbuton F, we defne J + F where for any y > 0, = lm y log F y log y F y = lm nf x F xy F x and and J F log F y = lm,.1 y log y F y = lm sup x F xy F x In the termnology of Tang and Tstsashvl [9], we call the quanttes J + F and J F the upper and lower Matuszewska ndces of the dstrbuton F. For more detals of the Matuszewska ndces, see Bngham et al. [1] Ch..1, Clne and Samorodntsky [4], as well as Tang and Tstsashvl [9]. Clearly, f F then J + F <, ff R then J + F = J F =, and f F ERV,, then J F J + F. Now we are ready to state the man results of ths paper. Theorem.1. Consder the weghted sums n 1 1. If F L and E p < for some p > J + F and all = 1,,, then for each n = 1,,, Pr S n > x Pr X > x. We leave the proof of Theorem.1 to Secton 4. The followng s an mmedate consequence of Theorem.1. Theorem.. Under the condtons of Theorem 1, t holds for each n = 1,, that Pr M n > x Pr X > x.3 If, n partcular, F R for some 0, then for each n = 1,,, Pr S n > x Pr M n > x F x E.4

6 58 Wang and Tang Proof. Clearly, Pr S n > x Pr M n > x Pr X + > x, where x + = max x,0 for any real number x. By Theorem.1 we know that both sdes of the above are asymptotc to n Pr X > x. Hence, relaton.3 holds. Relaton.4 s a consequence of relatons. and.3 and t can be verfed by a classcal result of Breman [], who proved that for two ndependent random varables and wth nonnegatve, f the dstrbuton of belongs to the class R for some 0 < and E p < for some p >, then Pr > x lm x Pr > x = E Ths ends the proof of Theorem.. For two bvarate functons a, and b, : 0, 1,, 0,, wesayax, n bx, n unformly for n = 1,, f lm sup ax, n x bx, n 1 = 0 n 1 Recently, for a specal case that the weghts are correspondng products of a sequence of..d. nonnegatve random varables see 1.9, Tang and Tstsashvl [11] proved n ther Theorem 3.1 that the asymptotc relaton 1.10 s unform wth respect to n = 1,,. The followng theorem extends ths result to the case of general random weghts and dscusses the unformty of asymptotc relatons 1.1 under an addtonal condton. Theorem.3. Consder the weghted sums n 1 1. Suppose F ERV, for some 0 < < and p E <.5 for some 0 < <mn 1, and p > /. Then, relaton 3 holds unformly for n = 1,,. If, n partcular, = >0 that s, F R, then t holds unformly for n = 1,, that Pr M n > x F x E.6

7 Tal Probablty of Randomly Weghted Sums 59 Furthermore, f X = X 1 X <0 has a fnte mean, relatons 1 1 hold unformly for n = 1,,. If, n partcular, = >0, relatons 4 hold unformly for n = 1,,. We also leave the proof of Theorem.3 to Secton 4. As for the sgnfcance of the unformty, see Tang and Tstsashvl [11]. 3. APPLICATION TO RUIN THEORY Recently, a vast amount of lterature has been devoted to the run probabltes n a stochastc economc envronment. Nyrhnen [6] and Tang and Tstsashvl [9,11] proposed the followng dscrete tme rsk model for such a stuaton: P 1. The net losses durng the referenced perods, X, = 1,,, consttute a sequence of..d. real-valued random varables wth common dstrbuton F ; P. The surplus s currently nvested nto a rsky asset, whch may lead to a negatve and stochastc return rate R wthn perod, and R, = 1,,, also consttute a sequence of..d. random varables takng values from 1, ; P 3. The two sequences X ; = 1,, and R ; = 1,, are ndependent. For = 1,,, we wrte Y = R, = Y j, whch represent dscount factors from tme to tme 1 and to the present, respectvely. Recall the defntons of the fnte and nfnte tme run probabltes n 1.3. Under the condtons F L and EY p < or, equvalently, E p < for some p > J + F and all = 1,,, Tang and Tstsashvl [9] proved n ther Theorem 5.1 that for each n = 1,,, x; n Pr X > x 3.1 From Theorem. we mmedately see that the..d. assumpton n P s not necessary for ths result. In partcular, f F R for some >0 and the sequence ; = 1,, satsfes the moment condton descrbed n Theorem.3, then the relaton x; n F x E 3.

8 60 Wang and Tang holds unformly for n = 1,,. Hence, x F x E 3.3 Below we apply relatons 3. and 3.3 to a model proposed by Ca [3]. Example 3.1. Instead of the standard assumpton P above, Ca [3] assumed that the return rates follow an autoregressve structure of order 1, that s, R j = R j 1 + W j, j = 1,,, wth 0 < and R 0 = r 0 0 beng constants, and W j ; j = 1,, beng a sequence of..d. nonnegatve random varables whch are nondegenerate at zero. Clearly, for each q > 0 and = 1,,, E q = E 1 + R j q E 1 + W j q = [ E1 + W 1 q] Thus, f we assume F R for some >0, then E u p < for some 0 < <mn 1,, p > /, and u > p 1. Therefore,.5 s satsfed snce, by Hölder nequalty, p p E = E u p u p p 1 p 1 u u E p < Then, by Theorem.3, relatons 3. and 3.3 hold. In order to approxmate the fnte and nfnte tme run probabltes, t remans to calculate the moments E for = 1,,. For ths purpose, we denote g, r 0 = E 1 + R j = E wth g 0, 1. Let W j ; j = 1,, be an ndependent copy of the sequence W j ; j = 1,,. Then, we use the technque developed by Ca [3] to derve a recurrence equaton for E. Clearly, gven W 1 = w the return rate R j s equally dstrbuted as R j 1, where the sequence R j ; j = 1,, has the same autoregressve structure as that of R j ; j = 1,,, namely, R j = R j 1 + W j, j = 1,,,

9 Tal Probablty of Randomly Weghted Sums 61 but t has a dfferent ntal value R 0 = r 0 = r 0 + w. Denote by G the dstrbuton of W 1. Hence, [ g, r 0 = E E 1 + R j ] W1 [ = E 1 + r 0 + W 1 E = R ] j W1 1 + r 0 + w g 1, r 0 + wgdw Usng ths equaton we can recursvely calculate E, = 1,,. 4. PROOFS In the sequel, C = Cn always represents an absolute postve constant, whch s ndependent of x and may vary from dfferent places. For two functons a and b wth b postve satsfyng l 1 lm nf x ax bx lm sup x ax bx l for 0 l 1 l, we wrte ax = Obx f l <, ax = obx f l = 0; we also wrte ax bx f l = 1, ax bx f l 1 = 1, and ax bx f 0 < l 1 l < Some Lemmas Let F be a dstrbuton wth upper and lower Matuszewska ndces 0 J F J + F < defned by.1. From Proposton..1 n Bngham et al. [1] we know that, for any p 1 < J F and p > J + F, there exst postve constants C and D, = 1,, such that the nequalty holds for all x y D 1, and that the nequalty F y F x C 1x/y p F y F x C x/y p 4. holds for all x y D. By fxng the varable y n 4., we obtan the followng lemma; see also Lemma 3.5 of Tang and Tstsashvl [9].

10 6 Wang and Tang Lemma For F, the relaton x p = of x holds for any p > J + F. The second lemma below s a drect consequence of Lemmas 3.8 and 3.10 of Tang and Tstsashvl [9]. Lemma Let X and be two ndependent random varables, where X s dstrbuted by F L and s nonnegatve and nondegenerate at 0 satsfyng E p < for some p > J + F. Then, the dstrbuton of the product X belongs to the ntersecton L and Pr X > x F x. Consder the weghted sums n 1.1. For any fxed n = 1,,,0< <1, and 0 < L < x, we splt the tal probablty Pr S n > x nto three parts as Pr S n > x = Pr A 1 x, L, n + Pr A x, L,, n + Pr A 3 x, L,, n, 4.3 where n A 1 x, L, n = S n > x, X > x L, n n A x, L,, n = S n > x, X x L, j X j x, and n A 3 x, L,, n = S n > x, X x L, n j X j > x The followng lemma shows that, as x, the contrbuton of PrA x, L,, n to Pr S n > x s asymptotcally neglgble n comparson to F x. Lemma Under the condtons of Theorem 1, t holds for each fxed n = 1,, and 0 < <1 that n Pr S n > x, X x = of x 4.4 Proof. For x > 0, denote N = nf j 1 : j X + > 1 x,

11 Tal Probablty of Randomly Weghted Sums 63 where nf Ø = by conventon. In the dervaton below, we wll also use the conventons 0 = n = 0. Wth =n+1 n = 1,, n, we have n Pr S n > x, X x n = Pr X + > x, Pr =j+1 Pr =j+1 X + X + > > n X + x, N = j n 1 x, N = j n 1 x n Pr N = j n, where the last step s due to the condtonal ndependence between the event N = j and the random varables X ; = j + 1,, n. We contnue the dervaton as follows. For x > 0, n Pr S n > x, X x n Pr X + > [ Pr X + 1 x > 1 x n Pr N = j n ] n Snce for x > 0, X + > 1 x = n n X + > X > 1 x n, 1 x n for any fxed 0 < <1, whch wll be specfed later, we have n Pr S n > x, X x [ 1 x ] 1+ n Pr X > n n 4.5

12 64 Wang and Tang Next we deal wth each probablty on the rght-hand sde of 4.5. Recall nequalty 4. wth some p J + F, p. For each fxed nteger = 1,, n and all large x, accordng to the events B 1 = 1 < 1 x, B = nd 1 x, B 3 = < 1, nd we derve 1 x Pr X > n n = 3 k=1 Pr X > 1 x, B k n n Usng nequalty 4., we have for all large x > 0, 1 x Pr X > n n 1 x Pr X >, B 1 n + 1 B + Pr X > n 1 x n CF x p + Cx p p + CF x C [ F x p + F x ], 4.6 where we used Lemma Choose >0 suffcently small such that 1 + p < p. Substtutng 4.6 nto the rght-hand sde of 4.5 leads to n Pr S n > x, X x p CE[ F x + F x ] 1+ CF x 1+ E 1+ p + 1 Snce n s fxed and E 1+ p < for all = 1,,, the desred result 4.4 follows mmedately. Ths ends the proof of Lemma Recall equalty 4.3. The lemma below wll be used to prove that, as frst x then L, the contrbuton of PrA 3 x, L,, n to PrSn > x s also asymptotcally neglgble n comparson to F x. Lemma Under the condtons of Theorem 1, t holds for each fxed n = 1,, and >0 that n lm lm sup Pr j X j > x, : 1 n & =j X > L = L x F x

13 Tal Probablty of Randomly Weghted Sums 65 Proof. For a fxed n = 1,,, defne [ ] f L = log Pr X > L It s easy to see that as L, f L and f p L Pr X > L 0, 4.8 where p J + F, p. We derve Pr j X j > x, We rewrte I 1 as : 1 n & =j Pr j X j > x, j > f L + Pr X j > x f L, X > L : 1 n & =j X > L = I 1 + I 4.9 I 1 = E [ 1 j >f L Pr ] j X j > x j 4.10 Smlarly to the treatment n 4.6, by 4. we have for all large x > 0, Pr j X j > x j Pr j X j > x,1 j < xd 1 j + 1 j xd + Pr 1 j X j > x,1> j j CF x p j + Cx p p j + F x CF x p j + F x Substtutng ths nto 4.10 and notcng F x F x and E p j each j = 1,,, n, we obtan that < for I 1 lm lm sup = L x F x

14 66 Wang and Tang For I, agan by 4. we have for all large x > 0, I = Pr X j > x Pr X > L f L : 1 n & =j CnF xf p L Pr X > L Therefore by 4.8, I lm lm sup = L x F x Substtutng 4.11 and 4.1 nto 4.9 yelds 4.7. The followng two lemmas wll be used n the proof of Theorem.3. Lemma Let X and be two ndependent random varables, where X s dstrbuted by F ERV, for some 0 < < and s nonnegatve and nondegenerate at 0. Then for any fxed p 1 and p wth 0 < p 1 < < p <, there exsts some C = Cp 1, p >0rrespectve to such that for all large x say x D = Dp 1, p >0, Pr X > x CF x max p 1, p 4.13 Proof. Recall nequaltes 4.1 and 4.. Accordng to the events V 1 = x/d, V = x/d > 1, and V 3 = < 1, we dvde the probablty Pr X > x nto three parts as Pr X > x = 3 Pr X > x, V k k=1 Lemma gves that for all x 1, Pr X > x, V 1 1 x/d D p x p p CF x p Applyng nequalty 4. we can derve that for all x > D, Pr X > x, V C F x p Applyng nequalty 4.1 we also obtan that for all x > D 1, Ths proves Pr X > x, V 3 C 1 1 F x p 1

15 Tal Probablty of Randomly Weghted Sums 67 Lemma Consder the randomly weghted sums 1 1. If 1 F wth ts Matuszewska ndces 0 < J F J + F <. E p < for some 0 < <mn 1, J F and p > J + F /, then t holds unformly for all n = 1,, that Pr M n > x Pr X > x Proof. See Theorem 1.1 n Wang et al. [1]. 4.. The Proof of Theorem.1 We formulate the proof of Theorem.1 nto two parts, whch provde PrSn > x wth asymptotc upper and lower bounds, respectvely The Proof of an Upper Bound In ths part we use equalty 4.3 to prove the correspondng asymptotc upper bound. Note that, by Lemma 4.1., the product X s long taled and Pr X > x F x for each = 1,,. It holds for any fxed L > 0 that Pr A 1 x, L, n Pr X > x L Pr X > x F x 4.14 Furthermore, Lemma tells us that for fxed 0 < <1 and n, PrA x, L,, n = of x 4.15 As for PrA 3 x, L,, n, we have n PrA 3 x, L,, n Pr j X j > x, S n > x, X x L Hence by Lemma 4.1.4, Pr j X j > x, Pr j X j > x, : 1 n & =j : 1 n & =j X > L X > L PrA 3 x, L,, n lm lm sup = L x F x

16 68 Wang and Tang Smply combnng 4.14, 4.15, 4.16, and 4.3 we conclude that Pr S n > x Pr X > x 4... The Proof of a Lower Bound For L > 0 and = 1,,, n, we wrte Clearly, A = X > x + L, B = Pr n S n > x Pr S n > x, A Pr S n > x, A PrA,j : 1 <j n j : 1 j n&j =,j : 1 <j n PrA A j j X j L PrA A j PrA B c 4.17 Recall that X s long taled and Pr X > x F x for each = 1,, It holds that PrA Pr X > x F x 4.18 As for the thrd term on the rght-hand sde of 4.17, we have Pr A B c Pr X > x + L, Pr X > x, whch, together wth Lemma 4.1.4, gves that j : 1 j n & j = j : 1 j n & j = j X j > L j X j > L, n lm lm sup PrA B c = L x F x

17 Tal Probablty of Randomly Weghted Sums 69 We then consder the second term on the rght-hand sde of Clearly, for 1 < j n, PrA A j Pr > xl 1 + Pr X > x + L, j X j > x + L, xl 1 Pr > xl 1 + F L Pr j X j > x + L x p L p E p + F L Pr j X j > x, where n the last step we used Chebyshev s nequalty. Hence by Lemma 4.1.1, t holds for each fxed n that lm lm sup PrA,j : 1 <j n A j = L x F x Smply substtutng 4.18, 4.19, and 4.0 nto 4.17 gves the desred lower bound Pr S n > x 4.3. The Proof of Theorem.3 Pr X > x We only prove the results under the condton F ERV snce the proof of the other results under the condton F R can be gven smlarly. Accordng to Lemma we know that relaton.3 holds unformly for n = 1,,. Hence, t suffces to prove the unformty of relatons 1.1 under the addtonal condton that X has a fnte mean. For any gven >0, we choose some 0 < l 0 < 1 such that 1 <1 + l 0 1 <1 l 0 1 < Choose constants r > 0, p 1 > 0 and p > 0 such that <p 1 <1 + r p 1 < <p <1 + r p < p. Clearly, the nequalty y q j yq j holds for any fxed q > 1 and y j 0, j = 1,,. Usng ths, Jensen nequalty, and condton.5, we obtan that for = 1,, E p j 1+r E Hence, { E max { p 1 j, p } } 1+r j j 1+r p E { E j p } 1+r p p max { p 1 j, p } 1+r j < < 4.

18 70 Wang and Tang From Lemma we obtan that for all = 0, 1,,, Pr { p X > x CF x max 1, p } 4.3 By 4., 4.3, and Lemma 4.1., we fnd that for the >0 fxed n 4.1, there exst a postve nteger m 0 and a postve number x 0 such that for all x x 0, =m 0 +1 Pr X > x Pr 1 X 1 > x 4.4 We start to prove the unformty of relatons 1.1. Recallng the unformty of relaton.3, t suffces to prove that there exsts a postve functon L 1 rrespectve to n wth lm 0 L 1 = 1 such that L 1 Pr X > x Pr S n > x 4.5 holds for all n = 1,, and all large x > 0. By Theorem.1 and nequaltes 4.1, there exsts some x 1 > x0 such that for all 1 n m 0 and all x x Pr X > x Pr S n > x Pr S n >1 + l 0x 1 Pr X >1 + l 0 x l Pr X > x Pr X > x 4.6 For the case of n > m 0, notce that for all n > m 0 and x > 0, S n > x S n > x, X > l 0 x = m 0 +1 S m 0 >1 + l 0 x, = m 0 +1 X > l 0 x

19 Therefore, Tal Probablty of Randomly Weghted Sums 71 Pr S n > x Pr S m 0 >1 + l 0 x Pr S m 0 >1 + l 0 x, Pr S m 0 >1 + l 0 x Pr S m 0 >1 + l 0 x, =m 0 +1 = m 0 +1 X l 0 x X l 0 x = K 1 l 0, x K l 0, x 4.7 For K 1 l 0, x, we see from 4.6 and 4.4 that there exsts some x x 1 such that for all n > m 0 and x x, K 1 l 0, x 1 = m 0 +1 Pr X > x 1 3 Pr X > x = Now we deal wth K l 0, x. Usng Chebyshev s nequalty, Lemma and Theorem.1, we see that there exsts some x 3 x such that for all n > m 0 and all x x 3, K l 0, x Pr =m 0 +1 [ > x + E Pr S m 0 > x, =m 0 +1 [ 1 ] p x E p =m 0 +1 [ + 1 l0x E 1 Sm >x, 0 =m0 +1 x =m 0 +1 X l 0 x, x X1,, X m0 ; 1,, =m 0 +1 o1f x + EX Pr S m 0 > x l 0 E X 1,, ] o1 Pr 1 X 1 > x EX m 0 Pr X > x, 4.9 where n the last step we have used Lemma Substtutng 4.8 and 4.9 nto 4.7 and then takng nto account 4.6, we obtan the announced result n 4.5. Ths ends the proof of Theorem.3. l 0 ]

20 7 Wang and Tang ACKNOWLEDGMENTS We wsh to thank two anonymous referees, whose constructve comments have greatly extended the scope of ths paper. Wang s work was supported by Chna Postdoctoral Scence Foundaton Project No: and the Youth Scence and Technology Foundaton of UESTC Project No: JX REFERENCES 1. Bngham, N.H.; Golde, C.M.; Teugels, J.L. Regular Varaton; Cambrdge Unversty Press: Cambrdge, Breman, L. On some lmt theorems smlar to the arc-sn law. Teor. Verojatnost. Prmenen. 1965, 10, ; translaton n Theor. Probablty Appl. 1965, 10, Ca, J. Run probabltes wth dependent rates of nterest. J. Appl. Probab. 00, 39, Clne, D.B.H.; Samorodntsky, G. Subexponentalty of the product of ndependent random varables. Stochastc Process. Appl. 1994, 49 1, Embrechts, P.; Klüppelberg, C.; Mkosch, T. Modellng Extremal Events for Insurance and Fnance; Sprnger-Verlag: Berln, Nyrhnen, H. On the run probabltes n a general economc envronment. Stochastc Process. Appl. 1999, 83, Nyrhnen, H. Fnte and nfnte tme run probabltes n a stochastc economc envronment. Stochastc Process. Appl. 001, 9, Resnck, S.I.; Wllekens, E. Movng averages wth random coeffcents and random coeffcent autoregressve models. Comm. Statst. Stochastc Models 1991, 7 4, Tang, Q.; Tstsashvl, G. Precse estmates for the run probablty n fnte horzon n a dscrete-tme model wth heavy-taled nsurance and fnancal rsks. Stochastc Process. Appl. 003, 108, Tang, Q.; Tstsashvl, G. Randomly weghted sums of subexponental random varables wth applcaton to run theory. Extremes 003, 6 3, Tang, Q.; Tstsashvl, G. Fnte- and nfnte-tme run probabltes n the presence of stochastc returns on nvestments. Adv. n Appl. Probab. 004, 36 4, Wang, D.C.; Su, C.; Zeng, Y. Unform estmate for maxmum of randomly weghted sums wth applcatons to nsurance rsk theory. Sc. Chna Ser. A 005, 48 10, Goovaerts, M.J.; Kaas, R.; Laeven, R.J.A.; Tang, Q.; Vernc, R. The tal probablty of dscounted sums of Pareto-lke losses n nsurance. Scand. Actuar. J. 005, 6,