ASYMPTOTIC BEHAVIOR OF FINITE-TIME RUIN PROBABILITY IN A BY-CLAIM RISK MODEL WITH CONSTANT INTEREST RATE

Transcription

1 Joural of Mahmacs ad Sascs 3: ISSN: Scc Publcaos do:.3844/mssp Publshd Ol 3 4 hp:// ASYMPTOTIC BEHAVIOR OF FINITE-TIME RUIN PROBABILITY IN A BY-CLAIM RISK MODEL WITH CONSTANT INTEREST RATE L Wag Dparm of Mahmacs Th Uvrsy of Souhr Msssspp USA Rcvd 4-5-7; Rvsd 4-6-9; Accpd ABSTRACT Ths sudy vsgas h ru probably of a rwal rs modl wh cosa rs ra ad by-clam pars. W assum ha h clam sz ad h r-arrval m sasfy a cra dpd srucur wh som addoal assumpos o hr dsrbuo fucos. I parcular w sudy h asympoc bhavor of PR δ > whch holds uformly a f rval. I hs way w sgfcaly d h L s rsul rgardg h parws srog quas-asympocally dpd radom varabls. Kywords: Rwal Rs Modl Subpoal Dsrbuo Uform Asympoc Parws Srog Quas-Asympocally Scc Publcaos. INTRODUCTION Rs hory plays a mpora rol facal mahmacs ad acuaral scc. I has b sudd by may domsc ad forg scholars. A vary of rs modls wh may spcal faurs also hav b vsgad by may paprs h lraur. Thy provd us dffr prspcvs o udrsad h rs modl ad rlad hors. Abou a cury ago Ludbrg 93 lad h foudao of acuaral rs modl hs Uppsala hss. Wars ad Paparadafylou 983 roducd dlay clams slms a dscr-m rs modl ad appld margal chqus o drv uppr bouds for ru probabls. I addo Yu al. 5 appld h probably-grag fucos o oba ru probabls for h compoud bomal modl wh dscr dlay m for by-clam. I addo Tag 5 vsgad a smpl asympoc formula for h ru probably of h rwal rs modl wh cosa rs forc ad rgularly varyg ald clams. Rcly Wg al. was rsd h al probably of h Posso sho os procss ad sablshd som asympoc formulas for h f 339 ad f ru probabls of a couous m rs modl. Corrspodg rsuls ca also b foud Ch ad Ng 7; Wag 8; L al. 9; Yag ad Wag ad som ohrs. Cosdr rwal rs modl wh h oal capal rsrv up o m dod by R δ as prssd by h followg qulbrum Equao.: δ = δ δ s δ c ds X { } = δ T Z { } T = R. whr dos h al capal of h surac compay δ> s h cosa rs ra ad c s h cosa gross prmum ra δ dos h oal capal afr m. W hav alrady ow h gral formula of h compoud-rs: δ m P = m

2 L Wag / Joural of Mahmacs ad Sascs 3: P s h amou of moy accumulad afr yars s h prcpal amou s h umbr of yars h amou s dposd or borrowd for m s h umbr of ms h rs s compoudd pr yar ad δ s h aual ra of rs. Th w sar h compoudg mor ad mor h frqucy of compoudg clud yarly half-yarly quarrly mohly daly v f gos o fy ad h w drv ha: Scc Publcaos δ m P= lm m = δ m I hs rwal rs modl h drmsc lar fuco c dos h oal amou of prmums accumulad up o m. Th δ s c δ cɶ = c ds = dos oal capal δ grad by h prmums by h m. Cosdr h rs modl whch h clam szs ad h arrval ms of succssv clams fulfll h followg rqurms: Th clams szs: X cosu a squc of o cssarly dpd ad ogav radom varabls wh commo dsrbuo H: H = H = P{ } > for all > Th arrval ms of succssv clams ar: N = Y =. Th r-arrval m {Y ; } form a squc of radom varabls wh commo dsrbuo fuco V bu ar o cssarly dpd. Th arrval ms of succssv clams ca gra a rwal coug procss Equao.: = = { }. whr A s h dcaor fuco of a v A. Th N also dscrbs h oal umbr of clams f rval [ ]. Do h rwal fuco by λ = EN ad assum ha λ< for all << ha: [ ] P Y P Y Λ = > ] = = W assum ha {X ; } {Y ; } ad {c; } ar muually dpd 34 I our rs modl hr ar wo pars of muually dpd clams amly ma clams ad byclams Equao.3: δ δ = X { } = δ T Z { } T = R.3 W rfr Z as by-clams or dlayd clams h rwal rs modl. Thy ar dcally dsrbud wh commo dsrbuo F. Thy ar usually ducd by h ma clam wh som probably ad h occurrc of a by-clam may b dlayd dpdg o assocad ma clams amou. If h ma clam occurs a h h h by-clam occurs a h T. L T b h corrspodg dlay ms of h by-clam ad hy ar dcally dsrbud wh commo dsrbuo fuco G ad form a squc radom varabls whch ar ogav bu possbly grad a. I hs sudy w assum ha h {X Z ; } ad {T ; } ar muually dpd. Th clams ca produc h dpd fluc o ach ohr ad som addoal damags ad coss such as a orado hurrca ad havy ra-sorm ad so o. W df hm as by-clam modl. Hc our rwal modl wh by-clam pars ca br rflc h ruh. Th ulma ru probably h f m s dfd as Equao.4: Φ = P Rδ s < for soms.4 Ad h ru probably h f m rval [ ] s gv by Equao.5: Φ = P Rδ s < for som s.5 Th ru probably h f m mas h surr s capal falls blow zro h f m rval [] ha s h oal clam cds h al capal plus prmum com. W also vsga h asympoc bhavor of h ru probably h f m. Oc h capal s lss ha zro h ru occurs ad h compay wll barup.. PRELIMINARIES W frs roduc som oaos. Throughou h papr all lm rlaoshps ar for dg o f ulss ohrws sad. Df:

3 L Wag / Joural of Mahmacs ad Sascs 3: Scc Publcaos = f lm a o b a / b = Furhrmor for wo posv bvara fucos a ad b Equao.: a a = b b f lm sup. Clarly h asympoc rlao a~b holds uformly for : Ad: a lm sup sup b a lm f f b As s h cas for may rc rfrcs h flds of rs hory w ar rsd ru probabls f rval udr h assumpo ha H s havyald. I parcular h grad-al dsrbuo of H s subpoal. For dsrbuo H o h - H = H s h al dsrbuo of h fuco H. W do h uppr ad lowr Mauszwsa d of H: log H J lm L H H L log H y = lm f for y > H log H J lm U H HU log H y = lm sup for y > H A d.f. H wh suppor s subpoal f for all : H lm = H whr h H dos h -fold covoluo of H. Tag Th class of subpoal dsrbuo fucos wll b dod by S. Th class of subpoal dsrbuo plays a crucal rol havy-ald dsrbuo. I h surac dusry pracors usually choos havy-ald radom varabls o modl larg clams so w hav cludd for a mor dald aalyss of hr proprs. A suffc codo for subpoaly f hr s a gr such ha: H lm H Th H S: Thrfor udr h hypohss of h formula w oba: H lm H Th class of domad varyg dsrbuo s dfd as: H y D = Hdf o : lm sup < for ay y > H y If w suppos H D h for ay > J H hr s wo posv cosas c ad d such ha wh y d: H y c H η y η W hav alrady courd mmbrs of h followg hr famls: Th dfo of h class of log-ald dsrbuo: H y L = Hdf o : lm sup = for ay y > H y W ca s ha a dsrbuo H L ad oly f hr ss a fuco l : [ [ such ha l l = o ad: H y H Holds uformly for all y l.

4 L Wag / Joural of Mahmacs ad Sascs 3: Accordg o h dfo w may df a ll bggr dsrbuo H ERV o [ f hr ar som <α β< such ha: β H s H s α s lmf lmsup s forall s H H Whch dod by H ERV -α -β. If α = β w say ha H blogs o h rgular varao class ad wr H R- α. I s wll ow ha h followg propr cluso rlaoshp should hold for h dsrbuo of havy-al: Scc Publcaos a L D D R ERV α β Embrchs al. 997; Klupplbrg ad Sadmullr 998; Tag ad Tssashvl 3; L al. 9; Hao ad Tag 8 From h sudy of may cos ad lraurs w asly foud ha h rwal rs modl wh cosa rs ra maly volvs h dpd srucur bw h clam szs ad arrval ms of succssv clams; hs lms h usfulss of h obad rsuls o som. Howvr h roduco of dpd srucur o rs modls has capurd mor ad mor rsarchrs ao rc yars ad provds a spcal prspcv for h ru probabls hory. May prvous paprs hav alrady word o hs w opc for ampl Yag ad Wag ; Lu al. ; Wag al. 3 ad ohrs. Th dpd srucur also allows h udrlyg radom varabls o b posv or gav. Hc w smply summarz h curr corrspodg rsuls ad ma clar h rlaoshp bw hm. I s also cssary for our proof. W furhr d h sudy o h dpd cas ad g svral smlar rsuls abou h dpd radom varabls. W may df radom varabls {ξ } as Lowr Ngavly Dpd LND ad Uppr Ngavly Dpd UND f for ach ad all : = { ξ } ξ P P = = { ξ > } ξ > P P = If h squc ca sasfy boh h LND ad UND w ca am Ngavly Dpd ND srucur. Wh = h LND UND ad ND srucurs ar quval Lhma W say ha wo radom varabls {ξ } ar parws Ngavly Quadra Dpd NQD f for all posv grs ξ ad ξ ar NQD: P ξ ξ P ξ P ξ Or quvally: P ξ > ξ > P ξ > P ξ > Addoally w also amd h LND as h NLOD L al. 9 wh dffr oaos ad dffr formulas. W ca df ha {ξ } ar WUOD wdly uppr orha dpd. If hr ss a f ral squc {g U } sasfyg for ach ad for all : = { ξ > } U ξ > P g u P = W ca also df ha {ξ } ar WLOD wdly lowr orha dpd. If hr ss a f ral squc {g L } sasfyg for ach ad for all : = { ξ } L ξ > P g P = Accordgly w would l rmar ha f h {ξ } ca hold.9 ad. s also sad o b WOD wdly orha dpd Wag al. 3: Ad h w should b famlar wh som mpora proprs of h WUOD ad WLOD Frsly w suppos {ξ } sasfy h WLOD or WUOD If {f } ar o-dcrasg h {f ξ } ar sll WLOD or WUOD Ivrsly f {f } ar o-crasg h {f ξ } ar WUOD or WLOD Scodly f {ξ } ar ogav ad WUOD h for ach : ξ E g Eξ U = = I parcular f h {ξ } ar WUOD h for ach ad ay v>:

5 L Wag / Joural of Mahmacs ad Sascs 3: Scc Publcaos { ξ } = E p v g Epvξ u I h followg w wll us h assumpos ha for ay ε> Equao. ad.3: g =. lm U ε Ad g =.3 lm L ε W df h squc of h ral valu radom varabls {ξ } as Parws Quas-Asympocally Idpd PQAI for ay : ξ ξ ξ ξ lm z P > z > z = W also df h squc of h ral valu radom varabls {ξ } as parws Srog Quas- Asympocally Idpd psqai for ay : ξ ξ lm z z P > z > z = W ls svral corrspodg rsuls ad rmar h mhods usd som paprs mod abov havly rly o h..d assumpo o h clam sz ad h arrval ms of succssv clams. By obsrvao ad aalyss w compar h advaags ad dsadvaags of abov dpd srucurs. Th dpd cas of WUOD ad WLOD ca allow som gavly ad posvly radom varabls. Th psqai srucur ca clud h WUOD ad PQAI. I addo wh h radom varabls ar ogav h wo srucurs of psqai ad PQAI radom varabls ar quval ad h psqai srucur s a mor gral dpd cas ha h WUOD. Th ND srucur s a rlavly wa codo h asympoc bhavor of h ru probably s o ssv o h ND srucur. Rsul Thorm of Ch ad Ng 7. Cosdr h rwal rs modl sco f h clam szs {X ; } ar parws ND wh commo dsrbuo H ERV h r-arrval ms Y ar..d radom varabls ad h {c } s a drmsc lar fuco ad h h asympoc for h ulma ru probably Φ: 343 Rsul Φ H δ dλ Thorm of L al. 9. Cosdr h rwal rs modl Sco f h clam szs {X ; } ar parws NQD wh commo dsrbuo H D h rarrval ms {Y ; } ar NLOD ad h{c } s a drmsc lar fuco. I parcular f H L ad J > w oba h Φ: H Rsul 3 Φ H δ dλ Thorm of Sh ad L 8. Cosdr h rwal rs modl f h clam szs {X ; } ar NOD radom varabls wh commo dsrbuo H L D h r-arrval ms h r-arrval ms {Y ; } ar..d wh commo poal dsrbuo {N } s a homogous Posso procss: Rsul 4 Φ δ H dλ Thorm. of Wag al. 3. Cosdr h rwal rs modl Sco. If h clam szs {X ; } ar WUOD wh commo dsrbuo H L D h r-arrval ms {Y ; } ar WLOD. Also holds h rlaos. ad.. Th for ay f T Λ h rlao.5 holds uformly for Λ [T] ad h w oba h quval form for h ΦT: Rsul 5 Φ H δ dλ Thorm. of Lu al.. Cosdr h rwal rs modl Sco. If h clam szs {X ; } ar UTAI wh commo dsrbuo H L D h r-arrval ms {Y ; } ar WLOD such ha h rlao. holds. Th for ay fd T Λ h: Φ H δ dλ

6 L Wag / Joural of Mahmacs ad Sascs 3: Rsul 6 Thorm 3. of L 3. Cosdr h by-clam modl Sco assumg ha {X X ; } {θ ; } ad {T ; } ar muually dpd X Y X Y ar PQAI ad radom pars X Y X Y... ar dcally dsrbud. L dsrbud. L H ERV ad F also ERV h w oba: Φ Scc Publcaos δ H λ δ s F d G s dλ For h rs modls ad rsuls w may dscuss hm varous aspcs accordg o h movao of rsarch such as h gral rs modl or rwal rs modl dpd srucur or dpd srucur som commo havy-ald dsrbuo classs h cosa rs ra or o ad so o. By aalyss w foud ha h clam szs ad h r-arrval ms rsuls of L al. 9; Wag al. 3; L 3 sasfd h dffr dpd srucurs s a srogr rsrco ha h..d codo rsul of Ch ad Ng 7. Bu amog h dffr dpd srucurs w may hav dffr chocs dffr rs modls ad h lad o dffr rsuls such as Lu al. h rqurd boh h commo dsrbuo of clam szs ad r-arrval ms follow h rsco class bu may cass h auhor chos a mor mld codo ERV. Furhrmor rms of commo dsrbuo som paprs volvs a mor complcad cas L al. 9 papr ad Yag ad Wag h rmard h uppr ad lowr Mauszwsa d w also cosdr h uppr ad lowr Mauszwsa d h rwal rs modl. Bu Wag al. 3 papr h cacld h codo J H I parcular bacgroud sco w roducd h rlao. ad. Wag al. 3 cosdrd hm [Rsul 4] ad w wll dscuss hm our rwal rs modl. I addo h [Rsul ] o [Rsul 5] maly vsga h asympoc bhavor of ru probably f m ad h h [Rsul ] ad [Rsul 6] word o h formula of ulma ru probably rs modl. Grally spag h prmum fuco c s a gral sochasc procss bu som papr s assumd ha h c s a drmsc lar fuco such as L al. 9 ad Ch ad Ng 7. Furhrmor w do o always rqur δ> ad h r-arrval ms may o hav a poal dsrbuo bu mos cass w 344 df ha h δ s cosa rs ra somms δ yld ad h r-arrval ms may follow a commo poal dsrbuo. Fally w cosdr h N facor rs modl sco w df N o cosu a rwal coug procss bu h rsul of Sh ad L 8 h N s a homogous Posso procss whch follow h Posso dsrbuo wh assocad paramr λ. 3. MAIN RESULTS I hs sudy w sll vsga h rwal rs modl ad rqur h clam szs ad h r-arrval ms sasfy h psqai ad WLOD srucur. W ca g a srogr rsul udr mld assumpo whch h H ad F blog o L D. I addo h rs modl volvg by-clam pars ca also lad o a dffr rsul. So w hav h followg rsuls: Lmma 3. If P s a probably fuco of A ad A ar ay s h Equao 3.: = m m P A P A P A A = m 3. Proof W us mahmacal duco o prov h rlao 3.. Wh m = w ca asly draw h cocluso ha Equao 3.: P A P A 3. Assum ha s ru ha wh m =.. Equao 3.3: = P A P A P A A = 3.3 Wh m = w ca us h basc probably formula PA B = PAPB-PA B Equao 3.4: = = = A A P = A P A P = A A P A P = 3.4 Cosquly by duco assumpo Equao 3.5: = A = P A P = A A P P A P A A 3.5

7 L Wag / Joural of Mahmacs ad Sascs 3: Thrfor combg h rlao from 3. o 3.5 w fally g ha Equao 3.6: = P A P A P A A 3.6 = Scc Publcaos Ths ds h proof of h Lmma 3.. Accordg o h Thorm Chapr of Kabaov al. 986 w hav: Lmma 3. Igrao by pars Suppos f ad g ar rgh couous o-dcrasg ad wh lf-had lm fucos o [a b] whr a < b <R. Th Equao 3.7: b g df = g b f b g a f a f dg 3.7 a Lmma 3.3 Cosdrg h rwal coug procss {N } dfd.. Suppos ha {Y } sasfy h WLOD srucur ad h also holds h. rlao. For ay T Λ ad ay γ> w oba ha Equao 3.8: γ lm sup Λ[T] λ EN {N > } 3.8 Proof S h proof of Lmma. Wag al. 3. Lmma 3.4 L {X ; } ad {Z ; } b h muual dpd squcs wh commo dsrbuo fucos H ad F blog o h class L D uformly for Λ[ T] = or. Th Equao 3.9 ad : P X δ { < } δ { < } P X > ± l / N P > / N = b a = / N < > = δ T { T } < > ± N = δ T Z { T } P Z l / Lmma Udr h assumpo of Lmma 3.4 X ad Z b h muual dpd squcs wh commo dsrbuo fucos H ad F blog o h L D ad 345 boh sasfy h psqai srucur. For ay holds uformly Λ[T]: δ T P X T δ { < } Z { < } > l N = lm f f Λ[T] δ P X N Proof { < } > = δ T { T < } P Z > N = By h Lmma 3. ad h formula PA B PAPB w ca asly foud: δ δ T { < } { T < } P X Z > l N = δ { < } δ T { T < } P X > l N = Z > N = δ { < } = P X > l N = δ T { T < } P Z > N = P X δ { < } δ T { T < } δ { < } δ T { T < } δ { < } δ T { T < } > l N = Z > N = P X > l N = P Z > N = P X > l Z > N = From h las sp bcaus of h fac ha h dpd rlaoshp amog X Z ad w coclud: δ { < } P X > l Z > N = δ { < } = P X > l N = P Z > N = Bcaus h basc propry of probably ad h l s larg ough: Lmma 3.6 lm P Z > N = = L X ad Z b h muual dpd squcs wh commo dsrbuo fucos H ad F blog o h L D ad boh sa ad boh sasfy h psqai srucur. For ay holds uformly Λ [T] Equao 3.:

8 L Wag / Joural of Mahmacs ad Sascs 3: lm δ δ T P X { } < Z { T } < > l N = δ T > = > = sup sup P X N P Z N δ Λ[T] { < } { T < } 3. For h covc of proof w df ha: J = P X Z T > l N = δ δ T { < } { < } δ δ T J = P X N P Z N A = B = { } < > = { T } < > = δ δ T { X { } < Z { T } < > l } δ δ { X { < } > l N = } { Z T { T } < > l N = } Proof Tha s o say: Scc Publcaos J lm sup sup Λ[T] J Frsly f s larg ough w cosdr h dfo of l ad h formula PA = PAB PAB c s obvously ru: δ J P X { < } Z { T < } > l δ { < } δ T { T < } δ P X { < } Z { T < } > l δ { < } δ T { δ T X > l N = Z > l N = X l Z δ T T < } l N = Followd by h abov qualy w apply h smpl formula ha P A = B P = B P B = ad Bool s qualy w fd: δ J P X { < } > l N = δ T { T < } P Z > l N = δ l P X { < } > δ T { T < } Z > l N = δ T l P Z { T < } > δ { < } X > l N = 346 Cosdrg h fac ha -δt ad -δ blog o h rval [ -δ ] w may coclud ha h s δ l δ T { } X < > Z { T } < > l δ l X { < } > Z > l ad h s δ T l δ Z { T } { } < > X < > l δ T l Z { T } < > X > l So w ca apply h commo probably formula f A B h PA<PB o g h followg prsso: δ J P X { < } > l N = δ T { T < } P Z > l N = δ l P X { < } > Z > l N = δ T l P Z { T < } > > = X l N By h dpd rlaoshp amog h X Z ad { T } w hav:??? < δ T { T < } J P X > l N = P Z > l N = δ l P X { < } > N = P Z > l δ T l P Z { T < } > = > N P X l By h propry of class D ad Lmma 3.4 w ca g:

9 L Wag / Joural of Mahmacs ad Sascs 3: lm lm δ l P X { } < > N = Scc Publcaos δ { } < > = P X l N δ { < } P X > N = δ T l P Z { T } < > N = δ T { T < } δ T { T < } P Z > l N = P Z > N = Thus f s larg ough: l P X > N = P Z > l = δ { < } δ T P Z { T < } l > N = P X > l = Hc w may fd: J lm sup sup Λ[T] J Cosquly w hold h rlao 3.. Ths ds h proof of Lmma 3.6. Lmma 3.7 For h rwal rs modl roducd Rs Thory sco w hav: δ P X { } < > N = δ T { T } < P Z > N δ = H dλ F dg s d Proof δ s λ Frsly w should df h commo dsrbuo of as V Equao 3.: δ { } < > δ T P Z { T < } N P X N > = H F δ dv δ s dg s dv Accordg o h o-dcrasg codo Lmma 3. w rorgaz h rlao 3.3: δ { } < > N δ T P Z { T < } N P X > δ dv = H F δ s dg s dv δ V V H d V = V G G δ s V F d G s d δ V V H d V = V V G G H δ s F d G s dv δ s F G dv δ V V H V H V = δ H G G V dh V V δ F G dv δ s F G d V δ s Gs d F d V δ s F G d V V d H H V δ s s G d F dv δ F G d V δ δ W should ma clar ha V P = < so h s quval o h N ad < s quval o h N> w ca drv:

10 L Wag / Joural of Mahmacs ad Sascs 3: Ad: = = V = P N = EN < - = = V = P N = EN < I h rs modl sco w hav alrady show ha EN s a o-dcrasg ad rgh couous fuco. Accordg o h Lmma 3. w ca asly oba ha: V = δ δ { } < > T P Z T < N = δ { } = d H H P X N > δ = V δ δ s Gs d F d V = δ δ F G V = = EN d H H EN G δ s s d F den δ F G EN EN δ { } < > T P Z { } T < N P X N = δ > δ δ s = H d λ F d G s d λ I ds h proof of Lmma 3.7. Our ma rsul s for h appromao of f ru probably of h rwal rs modl wh cosa rs ra. I h followg sco w wll gv proofs of som rlad Lmmas mod h followg sco. Lmma 3.8 Cosdr h rs modl Sco. Assum ha h {X Y ; }{ ; } ad {T ; } ar muually dpd. L clam sz {X ; } ad by-clam pars {Z ; } b psqai dpd srucur wh commo dsrbuo fucos H ad F blog o h L D ad h r-arrval m {Y ; } b WLOD wh commo dsrbuo V ad also hold h rlao. ad.. Th for ay fd T Λ holds uformly Λ[T]:..: Φ δ H d λ δ s F d G s dλ W should apply h grao by pars aga. W g: Scc Publcaos δ { } < > T P Z { T < } N P X N = δ > δ δ s λ d H Gs d F d λ δ δ H EN H d λ = = δ F G F Thrfor: EN EN δ s d G s d λ 348 Φ lm sup Λ[T] = δ H dλ δ s F d G s dλ Lmma PROOFS Cosdr h rs modl roducd Sco. L X b h squc of psqai srucur radom varabls wh a commo dsrbuo fuco of H ad Z b h squc of psqai srucur radom varabls wh a commo dsrbuo fuco of F. Th X ad Z ar H L D ad F L D for ay fd holds uformly Λ[T] Equao 4.:

11 L Wag / Joural of Mahmacs ad Sascs 3: δ δ T P { } X { } < Z T < > N = = δ T P X { } { } < > N = P Z T < > N = = = δ 4. Suppos ha: ˆ δ δ T J = P { } X { } < Z T < > N = = ˆ δ δ T J = P X { } { } < > N = P Z T < > N = = = Whch s udrsood as Equao 4.: Jˆ lm sup Λ[T] = Jɶ Proof Scc Publcaos 4. W d o prov h wo quals Equao 4.3 ad 4.4: Jˆ lm sup Λ[T] Jɶ Jˆ lm f f Λ[T] Jɶ Frsly w ar rady o prov h rlao 4.4 rcallg h dfo of h l sasfyg h.7 ad Lmma 3. w hav Equao 4.5: δ X { ˆ } < J P = = δ T Z { < } > T δ X { < } Z { T < } > l N = δ X { < } Z { T < } > l N = δ T δ T δ X { < } δ T Z { } < > T l By obsrvao w add h qualy: δ X { < } δ T Z > T { < } l δ δ { < } T X Z { T < oh } > = l N Thus w g: δ X { < } P = δ T Z { < } > T δ δ { < } T X Z { T < } > = l N δ X { < } P T Z { T < } = δ δ { } δ T < T < > l X Z { } > l N = If w apply h basc probably formula PB A c = PB -PB A ad PAB = PA BPB w hav: Jˆ = δ δ { < } T X Z { T < P } > = l N δ X { < } P = δ T Z T δ δ { < } T X Z { T < } > = l N { < } l

12 L Wag / Joural of Mahmacs ad Sascs 3: < δ P X { < } δ T δ Z > l X { T < } { < } δ T Z { } T < l N > = > = P X Z l J l J l J l 3 Bcaus of H ad F blog o h class L h dpd rlaoshp bw X ad Z ad h Lmma 3.5 w fd Equao 4.6: J l lm f f Λ[T] Jɶ J lm f f Λ[T] J 4.6 Thus w ca coclud ha h rlao 3.9 ca lad o h rlao 4.7. For J l cosdrg h dpd rlaoshp amog X Z ad N ad h dpd prcpl of probably PA A B PA B PA B w drv: δ δ T l { } X < Z { T } < J l P = < δ δ { } T X { } < Z T < > l N = δ l δ δ { } { } { } T P X X Z < < T < > l N = = < δ T l δ δ { } { } { } T P Z T < X < Z T < > l N = = < δ l δ l P X { < } X { } < > N = δ l δ { } T l P X { } < Z T < > N = δ T l δ l P Z { } { } T < X < > N = δ T l δ { } T l P Z T < Z { T } < > N = δ l δ l P X { } { } X < < > N = = < δ l δ { } T l P X { } < Z T < > N = δ T l δ l P Z { } { } T < X < > N = δ T l δ { } T l P Z { } T < Z T < > N = For h las four rm of rlao 4.8 w ca apply h formula PAB = PA BPB ad h dfo of psqai rspcvly: δ l δ l δ l sup sup P X { } { } > P { } > = Λ[ T] X X lm < < < δ l δ δ sup sup { } T l { } > P T l P X Z T { } > = Λ[ T] Z T lm < < < Scc Publcaos 35

13 L Wag / Joural of Mahmacs ad Sascs 3: lm lm δ T l δ l δ l sup sup P Z { T < } { } > P { } > = X X Λ[ T] < < δ T l δ sup sup { } T l P Z T { } > Z T Λ[ T] < < δ P T l Z { T < } > = Fally J l also gos o zro. Ad h w ur o J 3 l. < < T < δ δ δ 3 { } { } T J l P X l X Z { } > l N = < Z l X P P X Z > l > = δ T δ { } T < { < } δ T Z { } T < l N δ δ P X { } < l X { } < < > l N = δ δ { } T P X { } < l Z T < l N δ T δ P Z { } { } T < l X < l N δ T δ { } T P Z { } T < l Z T < l N > = > = > = W ry o apply h dfo of psqai ad h codo of H L ad F L aga: sup sup P X l X > l = Λ[ T] δ δ { } { } lm < < δ δ T { } { } lm < < sup sup P X l Z T > l = Λ[ T] δ T δ { } { } lm < sup sup P Z T < l X > l = Λ[ T] lm sup sup P Z δ T δ T T l Z < T < > l = Λ T [ ] { } { } Cosquly J 3 l also gos o zro. Thus a combao of J l J l ad J 3 l ca hold h rlao 4.4. I h co of h abov proof w ca cou our proof of rlao 4.3. If w cosdr h followg oao rlao 4.3. w df: δ δ T A = X { < } = Z { T < } > = Z > { } { } < T < δ δ T B X l = W ca a h sam logc rasog of h gral cas of h probably ha PA = PAB PAB c o h followg rlao w fd Equao 4.7: Scc Publcaos 35

14 L Wag / Joural of Mahmacs ad Sascs 3: ˆ δ δ T J = P X { } { } < = Z T < > = N δ δ T = X Z > = < l N < { } { } T δ δ T { } { } < T < P X Z > N = = X Z l { } { } N = T δ δ T = < < 4.7 Afr applyg h P = B P B AB = = o rlao 4.9 Thrfor w oba: ˆ δ δ T J P X Z > l N = = { } { } < < T δ δ T { } { } < T < P X Z > N = < X Z l = Afr h rasformao of h qualy from: < X Z l o X Z l < Ad addg o h s of : = W drv: Jˆ = Scc Publcaos δ δ T X { } < Z { T < } > δ δ T X { < } Z { T < P } > = l N δ X { < } δ T P { } = Z T < < l N = δ δ T X Z > { } { } = < N T < Bcaus of h fac ha: δ δ T { X { } { } } < Z T < > l δ { X { } } < Z > l Thrfor: 35 δ δ T X { < } Z { T < P } > = l N δ P X Z > l { < } I addo w all hav ow ha h Bool s qualy ca lad o h cocluso: δ X { < } P δ T l Z { } < < = T N P δ X { < } δ T Z { T < } < l N = Th w apply h smpl logc rasog ad D Morga rul w may fd: P δ X { < } δ T Z { T < } > l N = δ X { < } P δ T l Z { } T < > = N Hc w g:

15 L Wag / Joural of Mahmacs ad Sascs 3: δ δ { < } { < } > ˆ δ { } < > = T l X Z T P X Z l P = δ δ T X Z > { } { } = < N < T Accordg o h Bool s qualy: Th w hav: δ δ T l P { } { } < X Z T < > = N δ δ T l P X { } { } < Z T < > = N δ δ T { } { } < T < Jˆ P X Z > l = = δ δ T l P X { } { } < Z T < > = N δ δ T X Z > { } { } = < N < T δ δ T T < > l P X Z { } { < } = δ l δ δ T P X { } { } { } < > X Z < > N = < T = δ δ T l δ T { } { } { } Z < > > = T < < X Z N T = δ < Z > l P X { } δ l δ P X { } { } < > X > N = < δ l δ T { } { } P X < > > = Z N < = T δ δ T l P Z { } { } < > > = T < X N δ T l δ { } T { } P Z T < > > = = 4 5 < Z N J l J l T By h H ad F blogg o h class L w should cosdr Lmma 3.6 for vry : J4 l lmsup sup lmsup sup Λ[T] Jɶ Λ[T] Jˆ lmsup sup Λ[T] Jɶ = δ P X { } < Z > l Jɶ Scc Publcaos 353

16 L Wag / Joural of Mahmacs ad Sascs 3: For J 5 f h H ad F blog o h class L ad h propry of psqai dpd srucur w drv ha: δ l δ lmsup P X { } { } < > X > N = < = δ l δ lmsup T P X { } { } < > Z > N = T < = δ δ lmsup T l P Z { } { } T < > > X > N = < = δ δ lmsup T l T P Z { } { } T Z < T < N > > > = Thus combg h J 4 l ad J 5 l w hold h rlao 4.3. Th followg sco wll dscuss h asympoc bhavor of P Rδ > dal. Bcaus w ar o Scc Publcaos oly rsd h asympoc bhavor of P Rδ > volvd h dpd srucur of h clam sz ad h arrval ms of succssv clams bu also h dpd srucur h rwal rs modl. I ds h proof of Lmma 4.. Lmma 4. Assum ha h {X Y ; }{ ; } ad {T ; } ar muually dpd. Cosdr h rwal rs modl w also assum ha X ad Y o cssarly dpd. L clam sz {X ; } ad by-clam pars {Z ; } follows psqai dpd srucur wh commo dsrbuo fucos H L D ad F L D. Th r-arrval m {Y ; } wh commo dsrbuo V ar WLOD ad also suppos h rlao. ad. hold. Th for ay fd T Λ Equao 4.8: > δ δ λ P R H d δ s F d G s dλ Holds uformly Λ [T]. Proof: δ m = m P R > = δ δ T X { < } Z P = I I { } < > = T N Frsly w dal wh h I accordg o h Lmma 4. holds uformly for Λ [T]. W go: I ~ m P X = = m = = δ { } < > N = δ T { } T < N P Z > = m δ P X { } N = = m < δ T Z { } T < N I3 I4 = > = P >> = = By usg h Lmma 3. ad h dpd rlaoshp bw X Z ad w ca chag h ad m m = = = = δ 3 P X = δ T P Z δ H d λ w ca g: I = > = { < } N > = = { T < } N δ s F d G s dλ Ad h for I 4 bcaus h dpd rlaoshp amog X Z ad ad H F D : δ { < } > = δ T P Z { } T < > = N δ = P X { } < > I4 P X N = m = = m = δ T P Z = { T < } > P N =

17 L Wag / Joural of Mahmacs ad Sascs 3: > > = m P X P Z P N = = = H P N F = m = m P N = { N > m } E { N > m } = H EN F N By h codo H D ad Lmma 3.3 w drv: P > a E E N a / d Ad combg h rlao.6 w oba: F / v v c c F H / v H ad c c v I lm 4 m sup sup Λ[T] = δ H dλ δ s F d G s dλ Bfor w ur o h I w should roduc som basc probably hory rlad o I. Bcaus of h fac ha Py> P>/ Py>/ w g: P X Z P X > > / = = = P Z > / = W may also fd wo cosas c ad d such ha for γ J ad γ J udr h codo of H D ad > H > F rlao.6 m /d holds uformly for Λ[T]. Thrfor from orgal qualy I should yld: m < /d I H / P N = m < /d F / P N = P N > / d m < / d γ m < / d γ γ = γ {N > / d } cf ch P N = cf d P N EN γ γ {N > m } {N > m } ch EN EN Scodly h basc rul probably s ha ay v of probably should o grar ha : I Thus: Scc Publcaos P X Z > = = m < / d / d < P X Z > P = = = P X > / m < /d = N P Z > / P N = P N = = / d < Applyg h Marov s qualy ad ma corrspodg subsuo w drv: 355 Hc by h guarad codo of H ad F blogg o h class D ad applyg Lmma 3.3 o h followg qualy aga w oba: lm lm [ ] m Λ T F δ s F dλ δt δt Λ[ ] γ {N > m } m Λ T T T sup EN γ {N m } cf EN > m sup [ ] δ δ H λ F λ c lm H H F λ m EN δ H dλ lm sup m T γ {N > m } = d G s ch I

18 L Wag / Joural of Mahmacs ad Sascs 3: Cosquly w compl h proof of Lmma 4.. Proof of Thorm 3.8 Followd by h L 3 approach w foud ha: ɶ δ Rδ = Rδ = δ s δ c ds X { } = δ T Z { } = T Th w rwr as: Scc Publcaos Rɶ δ = cɶ Rδ Cosdrg h ru probably.5 f m Rs Modl sco: Φ = P Rδ s < for som s Thrfor: δ Φ = P Rδ s < for som s I follows ha: δ < ɶ Φ δ > P R c P R From whch w ca s ha: δ δ ɶ c Rδ Rδ R δ I follows ha: δ P Rδ c P Rδ > Φ > δ W hav alrady prov h rlao 4. so w ca rwr : δ c δ δ ~ λ c P Rδ > H d δ δ δ δ s λ c F d G s d δ 356 Th for ay ε> du o h codo.6 prlmary sco w fd ha: ε δ δ ε λ c P Rδ > H d δ δ s F ε d G s d λ η δ H d λ δ F d G s d λ By h arbrarss of ε> follows ha: c > δ δ δ λ δ F d G s d λ P Rδ H d Ad h smlarly w ca also oba h rmag par: > δ y δ δ F d G s dy P R H dy Ths compls h proof of Thorm CONCLUSION Elghd by h rsuls of L 3; Wag al. 3 w obad som ovl rsuls rgardg h psqai ad WLOD radom varabls wh h class L D. Our ma rsuls cocrd h appromao for cosa rs ra ad by-clam modl. I addo w furhr prov h ma rsuls ad corrspodg assumpos. Th asympoc bhavor of P Rδ > s a y rol our proof par. Fally w apply h obad rsuls o a d of clam-dpd rs modl ad drv a mor prcs ad mor gral asympoc formula for ru probably f m. 6. ACKNOWLEDGEMENT Th rsarchr would l o ha som comms ad cosrucv suggsos of my advsor ad all h

19 L Wag / Joural of Mahmacs ad Sascs 3: mmbrs of comm. Svral smulag dscussos allowd m o dvlop orgal das ad mprov my papr. 7. REFERENCE Ch Y. ad K.W. Ng 7. Th ru probably of h rwal modl wh cosa rs forc ad gavly dpd havy-ald clams. Isur. Mah. Eco. 4: DOI:.6/.smahco Embrchs P. C. Klupplbrg ad T. Mosch 997. Modllg Ermal Evs: For Isurac ad Fac. s Ed. Sprgr Scc ad Busss Mda Brl ISBN-: pp: 645. Hao X. ad Q. Tag 8. A uform asympoc sma for dscoud aggrga clams wh subpoal als. Isur. Mah. Eco. 43: 6-. DOI:.6/.smahco Klupplbrg C. ad U. Sadmullr 998. Ru probabls h prsc of havy-als ad rs ras. Scad. Acuar. J. 998: DOI:.8/ Lhma E.L Som cocps of dpdc. Aals Mah. Sa. 37: L J. K. Wag ad Y. Wag 9. F-m ru probably wh NQD domad varyg-ald clams ad NLOD r-arrval ms. J. Sys. Sc. Compl. : DOI:.7/s L J. 3. O parws quas-asympocally dpd radom varabls ad hr applcaos. Sa. Probab. L. 83: DOI:.6/.spl Lu X. Q. Gao ad Y. Wag. A o o a dpd rs modl wh cosa rs ra. Sa. Probab. L. 8: DOI:.6/.spl...6 Ludbrg F. 93. Cofgurao appromad fro of h probably fuco. II. FBK orsargav collcv rss Almqvs ad Wsll Uppsala. Sh X. ad Z. L 8. Prcs larg dvaos for radomly wghd sums of gavly dpd radom varabls wh cossly varyg als. Sa. Probab. L. 78: DOI:.6/.spl Tag Q. ad G. Tssashvl 3. Radomly wghd sums of subpoal radom varabls wh applcao o ru hory. Erms 6: DOI:.3/B:EXTR Tag Q. 6. Issvy o gav dpd of h asympoc bhavor of prcs larg. Elcro J. Probab. : 7-. Tag Q. 5. Th f m ru probably of h compoud Posso modl wh cosa rs forc. J. Appld Pro. 4: Wag D. 8. F-m ru probably wh havyald clams ad cosa rs ra. Soch. Modls 4: DOI:.8/ Wag K. Y. Wag ad Q. Gao 3. Uform asympocs for h f-m ru probably of a dpd rs modl wh a cosa rs ra. Mhodol. Compu. Appld Probab. 5: 9-4. DOI:.7/s9--96-y Wg C. Y. Zhag ad S. Ta. Tal bhavor of posso sho os procsss udr havy-ald shocs ad acuaral applcaos. Mhodol. Compu. Appld Probab. 5: DOI:.7/s Yag Y. ad Y. Wag. Asympocs for ru probably of som gavly dpd rs modls wh a cosa rs ra ad domadly-varyg-ald clams. Sa. Probab. L. 8: DOI:.6/.spl Yu K. J. Guo ad K. Wag 5. O ulma ru a dlayd-clams rs modl. J. Appld Prob. 4: DOI:.39/ap/38378 Wars H.R. ad A. Paparadafylou 983. Ru probabls allowg for dlay clams slm. Isur. Mah. Eco. 4: 3-. DOI:.6/ Kabaov Y.M. R.S. Lpsr ad A.N. Shryav 986. O h varao dsac for probably masurs dfd o a flrd spac. Probab. Thory Rlad Flds 7: DOI:.7/BF3667 Scc Publcaos 357