THE GENERALIZED WARING PROCESS

Transcription

1 THE GENERALIZED WARING PROCESS Mioz Zogrfi d Evdoki Xeklki Depre of Sisics Ahes Uiversiy of Ecooics d Busiess 76 Pisio s., 434, Ahes, GREECE The Geerlized Wrig Disribuio is discree disribuio wih wide specru of pplicios i res such s ccide sisics, icoe lysis, eviroel sisics ec. I hs bee used s odel h beer describes such prcicl siuios s opposed o he Poisso Disribuio or he Negive Bioil Disribuio. Associed o boh he Poisso d The Negive Bioil disribuios re he well kow Poisso d Póly Processes. I his pper he Geerlized Wrig Process is defied. Two odels hve bee show o led o he Geerlized Wrig process. Oe is reled o Cox Process while he oher is Copoud Poisso Process. The defied Geerlized Wrig process is show o be siory bu ohoogeous Mrkov process. Severl properies re sudied d he iesiy, he idividul iesiy, he Chp-Kologorov differeil equios of i, re obied. Moreover, he Poisso d he Póly processes re show o rise s specil cses of he Geerlized Wrig process. Usig his fc, soe kow resuls d soe properies of he re obied. Keywords d phrses: Póly process, ccide proeess, ccide libiliy, Mrkovi propery, siory icrees, Cox process, rsiio probbiliies, Chp-Kologorov equios, idividul iesiy.

2 . Iroducio-The bsic coceps There re lo of upredicble eves i every perso s life. Oe of his kid eves is o be ivolved i rffic ccide. Vrious heories hve bee developed cocerig he ierpreio of differe siuios of he. Oe of he is he heory of he pure chce which ssues h he probbiliies of hvig ccide re oly resul of he rdo fcors. A url odel for his siuio (see for isce Brei, L 963 is h he uber of ccides is Poisso disribued wih e λ, i.e. λ P{ N } e λ,,,...! where N is he rdo vrible which describes he uber of ccides of sigle perso. Oe oher heory which hs rced uch ieres is he ccide proeess heory, i he coex of which, he idividuls differ fro ech oher i probbiliies of hvig ccide or i heir ccide proeess d he idividul ccide proeess reis cos i ie.. This heory ssues h he fcors of he ccides re of wo kids, rdo d o-rdo, where he o-rdo oes referred o he idividul s psychology, expliig i his wy, ore or less, why he idividuls hve uequl ccide proeess. I his siuio i is obied (Greewood d Wood s 99 h he uber of ccides N hs Negive Bioil disribuio wih preers k d ν, i.e. k + { } ( ( k + + P N ν ν,,,... The Irwi s (968 proeess-libiliy odel which ssues lso h he ordo fcors c be furher spli io psychologicl d exerl fcors provides ore explio s o why soe idividuls i he populio ed o hve ore ccides h ohers. I he coex of his odel, he idividul ccide proeess does o rei cos, becuse he populio is exposed o vrible risk. I his odel, Irwi used he er ccide proeess ν o refer o perso s 2

3 predisposiio o ccides, d he er ccide libiliy (λ ν, i.e. λ for give ν o refer o perso s exposure o exerl risk of he ccide d he derived he uivrie geerlized Wrig disribuio s he disribuio of he uber of ccides. This disribuio ws pplied by Irwi (968, 975b o d o ccides susied by e i sop fcory, providig iproved fi s copred o he Negive bioil. I is oe ieresig eber of he fily of ixed Poisso disribuio. I Irwi s odel he codiiol disribuio of he uber of ccides N give λ is Poisso disribuio wih preer λ, where λ for give ν follows iself G disribuio wih preers k d, i.e. λ for give ν is ν rdo vrible wih desiy u give by ( ul ν k k l k e ( ( lν, l, resulig hus i Negive Bioil disribuio wih preers k d. He, lso ν llowed he preer ν of he Negive Bioil o follow BeII disribuio of he secod ype (see for isce Johso d Koz 969 wih preers d ρ, i.e. ν is rdo vrible wih desiy φ give by ( ϕν ( ρ ( ( ρ ( ( ( + ρ + v + v, ρ obiig hus he Geerlized Wrig disribuio wih preers, k d ρ. The Poisso d he Poly processes hve lso bee used i he Accide Theory o describe he ccide per. Uder he hypoheses of he pure chce, he Poisso process wih iesiy λ hs bee proposed s odel which c describe he uber of ccides susied by idividul durig severl yers. The Poly process which is of Negive Bioil for, is defied by srig fro Poisso process, which he, is ixed wih G disribuio. I hs bee obied (Newbold 927 s odel which c describe he ccide per of populio of idividuls durig severl yers, uder he hypoheses of ccide proeess.i.e h idividuls differ i heir probbiliies of hvig ccide which rei cos i ie. (We hve o deoe h his is o he oly schee which leds o he Poly process, see for isce Grdell (997, p.5-7 for he Poly schee d he discussios of Bes d Ney (952 d Ce(974 for he differecies 3

4 bewee hese wo schees. Boh of hese processes sisfy he Mrkovi propery becuse his is propery of he ccide per, i.e. he uber of ccides durig he ex period ( h], + depeds oly o he uber of ccides he prese ie. I his diserio, ew process is defied d sudied. This process is ssocied wih he geerlized Wrig disribuio which, s i is eioed bove, is discree disribuio wih wide specru of pplicios (see, e.g. Irwi (975, Xeklki (983b. This ew process is ered i he sequel s he geerlized Wrig process. Alogously o he cse of Poisso d Poly process, he geerlized Wrig process is posuled s Mrkov process. The srig poi is process of Negive bioil for bu differe fro Poly process. We cosider group of idividuls uder he hypoheses of he Irwi s proeess-libiliy odel durig severl yers. We lso ssue h ccide libiliy, i iervl of ie (, + h], is he oucoe of rdo vrible Λ( h wih disribuio Uh ( which depeds o he iervl size h, d i wo o-overlppig ie periods he respecive rdo vribles re idepede. If hese disribuios for ech h re G, he resulig process is of Negive bioil for, bu differe of Poly process. This is Cox process d we cll i he Negive bioil process. This process is he ixed wih BeII disribuio of he secod ype. This schee is show o led o he geerlized Wrig process. We re i i he secod chper. A proof h his process is siory bu o-hoogeous Mrkov process is provided i his chper, oo. We use he fourh srucurl propery of ulivrie geerlized Wrig disribued rdo vecor, reed by Xeklki E. (986, i order o prove i. We hve cosidered, oher schee which leds o he Geerlized Wrig process, referrig o Cresswell d Frogg s (963 spells odel is proposed i he frework cosidered by Xeklki (983b. This is oher odel h describes he uber of ccides, which rejecs he coceps of ccide proeess d cogio. This schee, oo, llows for siory icrees d he vlidiy of he Mrkovi propery, s lso show i chper 2. I he sequel he defiiios of he Negive bioil d he Geerlized Wrig disribuios d Byesi ierpreio of he secod oe, hve bee used 4

5 o prese soe properies d o ierpre soe esies of he ccide proeess ν. The defiiio of he Geerlized Wrig process is he subjec of he fourh chper. We defie i s siory bu o-hoogeous Mrkov process. Expressios for he firs wo oes of his process, s well s he resuls of he iesiy d he idividul iesiy of i, re give i he fourh chper. The defiiio of he Negive bioil process d equivle defiiio of he Geerlized Wrig process re used here i order o fid he idividul iesiy of i. I he sequel we prese he rsiio probbiliies derived i he secod chper, d we clcule he forwrd d he bckwrd Chp-Kologorov differeil equios of he Geerlized Wrig process. The Poisso process d he Poly process re specil cses of he Geerlized Wrig process. Usig his fc, i he fifh chper, soe ew defiiios of he s liiig cses of he Geerlized Wrig process hve bee proposed, d he derivios of soe kow heoreicl resuls bou he re preseed. The oes, he rsiio probbiliies d he Chp-Kologorov differeil equios of hose processes, re derived i his coex. Filly, wo furher geesis schees re cosidered i chper 6. The resuls re bsed o Zogrfi d Xeklki (2 d hve bee obied i he coex of odels h hve widely bee cosidered for he ierpreio of ccide d. However, he coceps d eriology used c esily be odified so h he obied resuls c be pplied i severl oher fields rgig fro ecooics, iveory corol d isurce hrough o deoery, bioery d psychoery. 2. The bsic hypoheses of he Geerlized Wrig process Two differe geerig schees which give rise o process of Geerlized Wrig for re cosidered i his secio. The firs oe is exesio of he proeesslibiliy odel developed by Irwi. I cosiders populio of idividuls exposed o rdo ccide risk whose disribuio vries durig he ie. The secod oe is vri of he spells odel due o Cresswell d Frogg (963, reed lso i he pper of Xeklki (984. This odel ssues h ech perso is lible o spells (periods of ie d ll of his ccides occur wihi he. The hypoheses of proeess d cogio re o prese here. 5

6 2. The descripio of he ccide per by Cox Process. This odel cosiders severl idividuls exposed o he se exerl risk (e.g. drivers ll drivig bou he se disce i siilr rffic eviroe d h here re irisic differeces og differe idividuls (e.g. differeces i ccide proeess. Supposig h, he uber of ccides i iervl of ie (, + h for ech idividul is Poisso process wih persol re λ, where his λ sds for he respecive ccide proeess d regrdig persol λ i iervl of ie (, + h], s he oucoe of rdo vrible Λ( h wih disribuio Uh ( which depeds o he iervl size h where Uh ( is kh (, ν kh d, where ( ( h ν( h re i geerl soe fucios of h, he i is cler h he uber of ccides N ( will be sochsic process of Negive bioil for which fulfills he followig ssupios (I N( (II N ( + h N ( hs he followig disribuio { ( ( } P N + h N ( λh k( h λh ν( h ( k( h e ( λν( h e λ +! ( kh ( dλ,,... (2..2 We fid, { ( + ( } P N h N d usig he firs ssupio we fid lso, { } ( ( P P N k( h kh ( + - vhh ( + vhh ( + vhh (,,... k( k ( + - v ( + v ( + v (.,,... This does e h N ( for y hs Negive Bioil disribuio wih preers k ( d ν(. Hece, we c cofir h he followig relio sds 6

7 { ( } P N ( λ k( λ ν( e ( k +! λ ( k ( ( λ ν( e dλ,,... This ells us precisly h N ( is Cox Process (see e.g. Grdell, J. 997, p A priculr cse Assue h he ccide proeess vries fro idividul o idividul wih e h does o deped o ie. This is equivle o cosiderig preer pir ( kh (, ν ( h wih kh ( ν( h cos. So, leig ν( h ν h, d kh ( kh, i.e., llowig Λ( h o hve g disribuio h chges wih ie so h is expecio reis cosly equl o νk, we obi d lso kh v P{ N( + h N( } v + v,,... kh k v P ( P{ N( } v + v k,,... (2.2. This does e h N ( is NB k, ν -disribued. The followig digr illusres how he disribuio of of he rdo vrible Λ( h depeds o he size h Series Series2 Series3 Series4 Series Figure 2.2. Disribuios of Λ( h, for differe vlues of h, whe ν 6 d k 2 (he e 2. Respecively, series, 2, 3, 4, 5, sd for h, 2, 3, 5,. 7

8 I c be see fro he figure 2.2. h s h icreses, he shpe of he curves becoes hier. I fc for big h he vlue of he vrice k ν 2 is sll. Hece, h wh we hve ssued is h he ccide s proeess i iervl of ie ( h, + vries i such wy h he vlues of i for big size h re oo er o he cos e νk. 2.3 A exesio of Irwi s ccide odel. I he sequel we cosider he hypoheses of he Irwi s ccide odel. This odel cosiders populio which is o hoogeeous wih respec o persol d eviroel ribues h ffec he occurrece of ccides I his odel, Irwi used he er ccide proeess ν o refer o perso s predisposiio o ccides, d he er ccide libiliy ( λν, i.e λfor give ν o refer o perso s exposure o exerl risk of ccide. I his odel he codiiol disribuio of he uber of ccides N ( give λ is Poisso disribuio wih preer λ, where λ for give ν follows iself G disribuio wih preers k d. The codiiol disribuio of ν he rdo vrible Λ for give ν explis here he vriio of he ccides libiliy fro idividul o idividul. As eioed bove, he er ccides libiliy, here, refers o he idividul s psychology. The codiiol disribuio of he rdo vrible Λ for give ν describes differeces i exerl risk fcors og idividuls. As before we suppose h, libiliy flucuios over ie iervl ( h ( kh h, + deped o he legh h of he iervl d re described by, ν disribuio for Λ ν. Moreover, ssuig idepedece i wo ooverlppig ie periods, he uber of ccides N, ( for give ν, will be sochsic process of Negive bioil for wih preers k d ν.this srs d hs siory icrees wih disribuio give by (2..2. Le us furher llow he preer ν of he Negive Bioil o follow BeII disribuio wih preers d ρ, obiig hus for he disribuio of he uber of ccides N (: 8

9 PN ( h N ( E kh kh + ν ( ν ν + + kh + p v ν ν ν ( ( ( p+ ( ( kh ( + p [ + ] p( kh kh + p + p+ kh! ( ( ( ( ( kh ( (,,... d ρ ( ( ( ( ( k k ( PN ( P,,... (2.3. ( + ρ ( + ρ+ k! k ( ( I he sequel, we refer o he process defied by N ( s he Geerlized Wrig Process. Rerk 2.3. If we cosider idividuls of proeess ν d libiliy λi ν i 2, respecively for ech of he wo o-overlppig iervls of ie, by he odel s ssupios he ubers N, N2 of ccides icurred by hese idividuls follow double Poisso disribuio wih preers ( λ ν λ2 ν,. The for idividuls wih he se proeess bu vryig libiliies he joi disribuio of ccides over he wo iervls re double Negive Bioil wih preers kh kh, ; ν 2,, ν where h, h re he respecive sizes of hese iervls. If we llow ow he 2 proeess preer ν o follow BeII disribuio wih preers d ρ he he joi disribuio of he ubers of ccides over he wo iervls is bivrie Geerlized Wrig disribuio wih preer ( kh, ρ ;( kh,, ρ, 2 (see Xeklki, E Now, i is cler h, if we cosider uber of iervls greer h wo, he he joi disribuio of he ubers of ccides over hose iervls, will follow ulivrie Geerlized Wrig disribuio. I he sequel we re goig o use he bove rerk o prove h he resulig process of his firs geerig schee is Mrkov process i.e. o prove h 9

10 ( ( + (, (, s < coicides wih ( ( PN h N Ns s every o-egive iegers,, s s<. Firs we deoe h { ( (, (, s< } { ( + ( ( ( s ( ( s, < } P N + h N N s s PN h N N Ns Ns N s Cosider, ow, he rdo vecor ( ( ( ( ( ( ( N+ h N, N Ns, Ns N, s< ( PN+ h N for I follows fro he rerk 2.3. h his vecor hs MGWD( ; k;ρ where ( ( k kh, k h s, ks. We refer ow o he hird srucurl propery of he ulivrie Geerlized Wrig disribuio proved by Xeklki E. i her pper The ulivrie Geerlized Wrig disribuio, (p.54 d we fid ( ( ( ( ( ( ( ( ( N+ h N N Ns, Ns N ~ MGWD+ ; kh; ρ + k, where ( is he vlue of N(. Hece { ( ( ( ( s, ( ( s} PN+ h N N Ns Ns N ( ρ + k ( ( + ( ( kh + ( ( ρ+ k + kh ( ( ρ+ k + kh + + ( ( + ( + ( kh ( ( ρ + k ( + ( + (! ( ρ + k + kh ( +! (2.4. [ + ] [ ( ( ( ( ] ( ( PN+ h N N N P N h N which proves h he geerlized Wrig process hs he Mrkovi propery, i.e. he codiiol disribuio of he fuure N ( + h give he prese se N ( d he ps Ns (, s, depeds oly o he prese d is idepede of he ps. 2.4 The Spells Model I he sequel, lerive schee geerig process of Geerlized Wrig for is cosidered. This is vri of Cresswell d Frogg s (963 Spells odel h hs bee cosidered i he pper of Xeklki (984. Accordig o his

11 odel, ech perso is lible o spells, ( spell is period of ie durig which he perso s perforce is wek. For ech perso, o ccides c occur ouside spells. Le Sdeoe ( he uber of spells up o give oe. I is ssued h S ( is hoogeeous Poisso process wih re k,, k >, he uber of ccides wihi spell is rdo vrible wih give disribuio F d h he uber of ccides risig ou of differe spells re idepede d lso idepede of he uber of spells. So, he ol uber of ccides ie is S ( X(, where ( X k k S is hoogeous Poisso process wih re k d { } X k idepede disribued ( i.i.d rdo vribles fro he disribuio F. re ideicl Whe { X k } is logrihic series disribuio wih preers (, ν, i.e. d P( X log( + v i v P( Xi,, >, v >, + v he rdo vrible X, ( is Negive Bioil rdo vrible wih preers k, (see Kep, 967 d Chfield d Theobld, 973. Here ν is ν regrded s he exerl risk preer, oo. The, if he differeces i his exerl risk c be described by BeII(,ρ disribuio, he resulig ccide disribuio is of Geerlized Wrig for wih preers, k, d ρ. Le us cosider, ow, he couig process { N } represeed, for, by S ( X k Xk,, k k (, where N ( c be

12 where ( S is hoogeous Poisso process wih re k, { } X k is logrihic series disribuios wih preers (, ν h is lso idepede of he process S, ( d ν is o egive rdo vrible wih BeII(,ρ disribuio. Theore 2.5. For he process N ( (I N( { } { N } { }, defied s bove he followig codiios hold: (II N (, posseses siory icrees (III (, is Mrkov process. Proof The proof of (I is srighforwrd. To prove codiio (II, deoe by ϕ he probbiliy disribuio fucio (p.d.f of he rdo vrible ν. The we c wrie: ( ϕ( + pn+ h N PN+ h N v vdv ( ( ( ( ( + S ( + h P ( X ϕ k v dv k S( + + i P X ( ( + ( ( k p S h S i ϕ v dv i k kh kh + + i exp P X k i k i! ρ( kh kh ρ+ + ρ+ kh! ( ( ( ( ( kh ( ( i ϕ ( vdv 2

13 S ( To prove he Mrkovi propery, le Nν ( X k { ν( } k for give ν. The process N N. is copoud Poisso process. Hece, i is Mrkov process. ν We ow oe h: ( ( (, ( s PN+ h N Ns for s + where P ( A ( ( +, (, ( s ϕν ( P N h N N s for s dν ν + ( (, ( s ϕν ( P N N s for s dν ν ν sds for he codiiol probbiliy of eve A give he vlue ν of he rdo vrible ν. The we fid: ( (, (, ( ( Pν N+ h N N s s for s ( ( ( p ( s p p h s s s ( ( s ( ( ( kh+ k s+ s ks + s v s I he se wy we fid h ( ( khk ( ν(, ν( (, s ( s k ( s + s ( ks + ( s v ( ( + k ν ( s s + ( ( ( ( PN N s s s p s p s Hece, we ke ( ( +, (, ( ( PN h N Ns s for s ( + ( kh ( ( ρ+ k ( + ( + (! ( ρ+ k + kh ( + (2.4. The ls resul proves he Mrkovi propery of he process d provides rsiio probbiliies. 3

14 3. The Geerlized Wrig disribuio. This chper is devoed o he Geerlized Wrig disribuio The os of resuls obied refer o he ides h Grdell (997 hve used o obi siilr resuls i he cse of Poly process d i geerl of ixed Poisso process. I he firs pr he defiiio of his disribuio, Byesi ierpreio, soe properies of i, d soe esies of he ccides proeess, re provided, while he relio bewee he Geerlized Wrig d he Negive bioil disribuios wih he se e is illusred by soe digrs i he secod pr. 3. The defiiio of he Geerlized Wrig disribuio. Defiiio 3.. A discree rdo vrible N is sid o be Negive Bioil disribued wih preers k d ν, k >, ν > i.e. N ~ NB k, ν, if k k + ν P{ N },,,... (3.. ν + + ν Defiiio 3..2 A discree rdo vrible N is sid o be Geerlized Wrig disribued wih preers k, d ρ, >, k >, ρ >, i.e. GW( k,,ρ if P P{ N } E k k + ν + + ν ν,,... where ν~ (, ρ BeII. (3..2 So we c wrie for hese probbiliies: ρ( k ρ+ P P{ N } E k k + ν + + ν ν ( p ( ( p + ( ( ( ( ( ρ+ + k ( + k + ν ( ( + v + v dv ν + + ν k k!,,... k p 4

15 I boh he bove defiiios k eed o o be ieger. Le us cosider rdo vrible N ~ NB k, ν, i.e. k k + ν P{ N },.,... (3..3 ν + + ν If we suppose h he preer ν is oucoe of rdo vrible BeII(,ρ - disribued, we c ierpre (3..3 s he codiiol disribuio of he rdo vrible N, give he vlue ν. The ucodiiol disribuio of N is i h cse give by P ( P{ N } E k k + ν + + ν ν k + k + ν v ( v ( p + + dv ν + + ν ( p ( ( p ρ( k ρ+ ( ( ( ( ( ( ρ+ + k ( So, N GW( k ~,,ρ. I follows fro (3..4 h Sice P ( of N. k k! ( + ( k + ( ( ( + ( + ρ + k + P P + ρ( ρ + k ( (,,... (3..4 (3..5 we ge by (3..5 recursive lgorih for he disribuio Usig his ierpreio of he Geerlized Wrig disribuio s he Byesi versio of he Negive bioil disribuio, we c regrd for y eve B B ( { ν } U x P x N B x + ( ν ( ν PN B du ( ν ( ν PN B du 5

16 where U sds for he probbiliy fucio of he rdo vrible ν~ (, ρ s he poserior disribuio of ν give B or, ore precisely, { N B} + P N B P N B l du l >. { } ( ( Proposiio 3.. Le N be GW(, k,ρ -disribued. The Proof { ν ( } P x N s x + + ( + + ρ+ ks ν ν dν ( + ( ( ρ ks ν ν dν P{ ν x, N( s } { ν ( } P{ N( s } x ( ( ( x + ρ ( ls ( ls ( s d ( ( ρ ν ν + ρ ν exp ν +! ( ks P x N s + ( + ρ ( ( ks ks ( ( + ρ ( ls exp( ls ( s + d! ( ks ks ks l ρ ν ν + ν ν l ls exp dl. ν ls exp dl. ν BeII,, provided h x ks + + ( + ( + + ρ+ ks ν ν dν + ks + + ( + ( + + ρ+ ks ν ν dν Corollry. Le N be GW(, k,ρ -disribued. The Proof { ν } E N( s + + x ( ( ρ ks ν ν dν ks ( ( ρ ks ν ν dν ( ( ρ + ν ν dν ( ( ρ ks ν ν dν + ρ + ks The proof follows fro he bove propery d fro he relio ( { ν ( } { ν ( } E N s xdp x N s The oher clculios re 6

17 { ν } E N( s + + ( + + ( ρ + ks ( + + ρ + ks ( ( ρ ks ν ν dν ( ( ρ ks ν ν dν ( + + ρ + ks ( + ( ρ+ ks + ρ+ ks + s he bes ρ + ks Fro (3..6 i sees url o ierpre ν * B E{ ν N( s } esie, or he Byes esie of ν. Aoher siple d obvious esie of ν is N. ( We fid k E N ( ( ( k k E N ν ν νk ν k d we hve lso 2 2 E N ( ν EE N ( ν ν k k E E N k 2 ( k E k + 2 ( k ( ( ν ( νν ( 2 ν [ νν ( ] E k + Thus, for lrge vlue of, we hve N ( ν k Usig he lier expressios of his esie, we c fid oher siple esie ν * L b + c N (, where b d c re chose so h E ν * L ν k iiized. This esie c be clled he bes lier esie. We fid ν * L E E[ νν ( + ] k vr ν+ E ν( ν+ ( + + N vr ν vr ν+ E ν( ν+ k vr ν + k E k N [ ] vr ν+ [ ν( ν+ ] [ νν ] ( k [ ] ( 2 is 7

18 3.2 A copriso hrough digrs I figures we illusre he relio bewee he disribuios of N i he wo cses: he series, N is GW(, k,ρ -disribued d he series 2, N is NB k, -disribued, bu such h he e for boh of he is he se. We hve ν chose he relively lrge vlue, i order o be ble o see how P ( behves for sll vlues of. I he figures 3.2., he Geerlized Wrig vriio doies he Negive bioil vriio while i he figure hey re los he se Series Series Figure 3.2. Disribuio of N i wo cses. The vlues of he preers re: 6, k 2, ρ 4, ν 2, d. The e is 4. 8

19 Series Series Figure Disribuio of N i wo cses. The vlues of he preers re: 6, k 8,, ρ 4, ν 2, d. The e is Series Series Figure Disribuio of N i wo cses. The vlues of he preers re: 6, k 2,, ρ 4, ν 2, d. The e is 4. 9

20 4. Geerlized Wrig Process. I his chper we shll give he defiiio of he Geerlized Wrig process. This defiiio is bsed o he wo schees reed i he secod chper. The wo firs oes he iesiy d he idividul iesiy of his process re preseed i he sequel. Here we eed o give he defiiio of he Negive bioil process d equivle defiiio of he Geerlized Wrig process, oo. The ls pr of he chper res he rsiio probbiliies d he Chp-Kologorov equios of his process. 4. The defiiio of he Geerlized Wrig process. The Geerlized Wrig process c, ow, be defied i he followig wy: Defiiio 4.2. The couig process { N } Wrig Process wih preers ( k,,ρ, k (I N( (II N ( is Mrkov process (III N ( + h N ( is GW( kh (, is sid o be Geerlized >, >,ρ > if:,,ρ -disribued for ech h>,. Codiios (I, (II d (III ell us h his process srs, i hs siory ρ( k k icrees d PN ( (, i.e. ( ρ+ + ρ+ k! N is GW(, k, ρ -disribued. ( ( ( ( ( k ( ( Give h he defied Geerlized Wrig process is Mrkov process he relios (2.3. d (2.4. give he rsiio probbiliies of i. P { N( s + N( s } We deoe ( + ( k ( ( + (! ( ( s, s + P N ( s + N ( s p, ( ρ + ks ( + ( ρ + ks + k ( + 2

21 d recll h i represes he probbiliy h process presely i se will be i se ie ler. This probbiliy i his cse depeds o he prese ie so he defied Geerlized Wrig process is o-hoogeous Mrkov process. ( P N ( I is cler h p (, P N ( N ( ( p (, I order o show h such process does exis i s sufficie o prove h he rsiio probbiliies sisfy he Chp-Kologorov equios i.e. p, ( s, pi, ( s, τ pi, ( τ,, for s τ, (4.2. We clcule firs: i i ( s τ p ( p,, i, i, τ ( + i ( k( τ s ( i ( ρ + ks ( ( ( ( + + k τ ( i i ( + ( i! ( ρ + kτ ( + i ( + i ( i! ( ( + + ρ ks ( ( k( τ ( ( ( + i k τ s ( i ( ( + ρ + k ( ( i! ( i! + i ( ρ + kτ ( + i ( ρ + k ( + Usig he ideiy k ( s + ( ( we obi h i k ( τ + ( i ( i k ( τ s + ( i ( i i ( s τ p ( p,, i, i, τ ( + ( ρ + ks ( + k( s + ( ( + ( ρ + k ( ( + ( s p,, which proves I order o deerie if rbirry couig process is cully Geerlized Wrig process, we us show h he codiios (I, (II, (III re sisfied. The firs wo codiios c usully be direcly verified fro our kowledge of he process. However, i is o ll cler how we would deerie h he codiio (III is sisfied. 2

22 4.2 The oes d soe oher properies Le N be Geerlized Wrig process wih preers ( k,,ρ. For y, N ( is Geerlized Wrig disribued rdo vrible, Hece, (see for isce [ ] Xeklki 994, for y, EN ( k, Vr N( ρ [ ] ( ρ + ( ρ + 2 ( ρ ( ρ 2 k k Followig Irwi (975, oe y show h he vrice c be divided io hree ddiive copoes, hus [ (] ( ( k Vr N Λ ν 2 R, σ + σ + σ. where, σ 2 ρ ρ is he copoe due o libiliy ( k( + ( ( 2 Λ d. ( ( ( ν σ + ρ ρ ρ σ ( ρ 2 R k is he copoe due o proeess is he copoe of he rdoess. The Geerlized Wrig process is siory process. For siory process N, EN ( η, where η is ered he iesiy of N (see e.g. Grdell, [ ] 997, p.53. I is cler h he iesiy of he Geerlized Wrig process is k η. For his process (like for ll siory processes, here lwys exiss, ρ rdo vrible N wih EN ( η, clled he idividul iesiy, such h N ( p. N c s + (see e.g. Grdell, 997, p.53. The iesiy η is fiie. Hece, i follows h he idividul iesiy N is fiie s.. d We give ow defiiio of he Negive bioil process i order o give equivle defiiio of he Geerlized Wrig process which shll be useful o prove he heore 4.3. below, which gives he idividul iesiy of he Geerlized Wrig Process. c The sybol p. here sds for he covergece i probbiliy of rdo vrible 22

23 Defiiio The couig process { N } Process wih preers k, (I N( (II Nis ( Mrkov process (III N ( h N ( k >, ν >, if ν + is NB kh, (, is sid o be Negive bioil -disribued for ech h>,. ν The firs codiio ogeher wih he codiio (III llow us o ke lso h N ( is NB k, -disribued. ν Defiiio A Negive bioil process wih preers k d ν is clled Sdrd Negive Bioil Process. Defiiio Le ν be rdo vrible BeII(,ρ -disribued d sdrd Negive bioil process ~ N idepede of i. Le k > be cos. ~ The poi process N N o k, ν, where ~ ~ N o k, N k, d for every, ν ν ~ N k, ~ NB k,, is clled he Geerlized Wrig Process. ν ν I is lredy cler h defiiio 4..3 is equivle o defiiio 4... By defiiio 4..3 oe c prove he followig propery def Theore 4.3. Le N be Geerlized Wrig process. The Proof N ( p. k. ν ~ (, (, li N N k k N k Nk ν νk νk ν ν ν ~ ENk (, ν li li ~, li ~ [ ] d The sybol s (.. (los sure iplies h P ω N( ω Ω is fiie 23

24 ~ Nk We use ow he Chebishev Iequliy i order o fid li ENk We hve ~ Nk E ENk ~ Nk vr ENk (, ν (, ν ~ [ ] (, ν (, ν ~ [ ]. ~ { Nk (, ν } ~ N( k, ν vr ν E 2 [ ] k( + ν 2 ( ν Hece ~ Nk (, ν ν p ~ ε ENk [ (, ν ] +. ενk which ells h ~ Nk (, ν p. ~ ENk, ν d [ ( ] N ( p. k ν (, ν (, ν ~ [ ] ν + k νk. Cosiderig his resul d he fc h if ν is BeII(,ρ -disribued he ρ E( ν, we ke E( νk idividul iesiy of he Geerlized Wrig process. k. Hece, he rdo vrible N ν k is he ρ 4.3 The rsiio probbiliies d he Chp-Kologorov equios of he Geerlized Wrig process. The rsiio probbiliies of Mrkov process sisfy he Chp- Kologorov equios p, ( s, pi, ( s, τ pi, ( τ,, for s τ, i I order o fid he forwrd d bckwrd Kologorov differeil equios we hve used he ides used by Ross (996, p o obi he respecive equios of hoogeous Mrkov process. 24

25 We obi he forwrd Kologorov differeil equios srig he clculios fro he equios We fid ( ( (, p, s, + h pi, s, pi,, + h for s,, h i pi, h p', ( s, pi, ( s, li h h We clcule li li h i, (, + ( + h ( i p h (, + p (, + h li + h h ( kh ( ( ρ + k i ( + i h ( i! ( ρ + k + kh ( + h p, ( s, d ( + ( ρ+ k ( + k ( + ( ρ+ k ( + ( + k( ( ρ+ k i ( + i ( + i ( i! ( ρ+ k ( + i - oherwise k ( + ( + ρ + k + ( + k ( ρ + k ( + i ( + i ( i( i ( ρ + k ( + - i - i > ( (,, ρ + k ( + li h h h ( ρ + k + kh ( + ( ρ + k ( + ( ρ + k + kh ( + p + h li h li h li h k If we deoe ( ρ k ( + i h + k + i ( ρ + k + kh ( + ρ + k + i + ρ + k + kh + i 25

26 q (, k ( + ( + ρ + k+, q ( i, ( + k ( ρ+ k ( + i ( + i ( i( i ( ρ+ k ( + i< d v ( k + i ρ + k + i Geerlized Wrig process re: p p,, ( s, ( s, ν he he forwrd Chp-Kologorov equios for he ( p ( s,, ν( p, ( s, + qi, ( pi, ( s,, < i The bckwrd equios, follows fro he Chp-Kologorov equios wih τ s+ h. Hece we sr he clculios fro he followig equios p, ( s, pi, ( s, s+ h pi, ( s+ h,, for s+ h,, h i We fid ( +, (, p s h p s,, h p s s+ h p, ( s+ h, h (, pi( s, s+ h,, i + h ( +, p s h i, We clcule siilrly d fid li i, (, + p s s h h k ( + ( + ρ + ks+ ( + i k ( ρ + ks ( ( + ( i ( i ( ρ + ks ( + i h + d (, p, s s+ h + li k h h ρ + ks + i i i- oherwise If we deoe q ( s, + k ( + ( + ρ + ks+, 26

27 q ( s i, ( + i k ( ρ + ks ( + ( + ( i ( i ( ρ + ks ( + i i > d ( v s k + i ρ + ks + i he he bckwrd equios for he Geerlized Wrig process re: p p,, ( s, ( s, ν ( p ( s,, ( ( ( ( ν p s, q p s,, <, i, i, i + 27

28 5. Soe priculr cses of he Geerlized Wrig Process. We refer o he relio bewee he Poly, he Poisso d he Geerlized Wrig disribuios, reed by Irwi (968, o coe o he coclusio h he Poly d he Poisso processes re liiig cses of he Geerlized Wrig process. So he clssicl heory of hose processes is priculr cse of h preseed here. The ew defiiios of he s specil cses re give i his secio. The Mrkovi propery for boh of hose processes, he clculios of he rsiio probbiliies d he Chp-Kologorov equios, he proof h hey re birh processes, he propery of hvig idepede icrees of Poisso process d kow equivle defiiio of i, re provided here, usig hese defiiios. A brief discussio bou coiuous logue is ieded i he ls prgrph of his chper. Two cses re cosidered: whe d whe boh,ρ. Boh i hese cses he coiuous logue of he Geerlized Wrig Process c be regrded s liiig fors of i. 5. The Poly d he Poisso processes s specil cses of he Geerlized Wrig process. I is show (Irwi 968 h, if k d ρ so h w k k k +ρ reis cos he Geerlized Wrig disribuio eds o he egive bioil disribuio wih geerig fucio uk w u θ k k where u k wk ρ k + ρ. Alogous o he proof used by Irwi (968 o obi he Poly disribuio s liiig cse of he Geerlized Wrig disribuio, he followig heore c ow be prove: Theore 5.. If k d ρ c k where c > is cos i.e. w ( k k k +, he, he Geerlized Wrig process eds o he Negive ρ + c bioil process wih geerig fucio 28

29 where u ( w ( k uk ( w u k k ( ( θ ρ c k k +. ρ + c Hece, i his cse ρ( k k + c li ρ+ ρ+ + k! + c k ( ( ( ( ( k ( ( + c (5.. Hece, we c give he followig defiiio of he Poly process. Defiiio 5.. The Geerlized Wrig process wih preers (, k, ρ i which k d ρ preers,. c ck where c > is cos, is clled Poly process wih I is show lso h he Poisso process is liiig for of Poly process. Le us cosider he bove Poly process, d le us ssue h c d where λ> is cos. I his cse we fid λ c + c li + c c li c li c λc ( λc+!! ( λc! c+ ( + c + c ( ( (! c / λ λc λc +! ( λc! c ( + c 29

30 ( λc ( ( + li + c c c ( λ exp( λ! c / which prove he followig heore λ! Theore 4.2 Cosider ow he Poly process, defied bove, d le c d λ c, where λ > is cos. The, + c li + c c + c ( λ exp ( λ The resul of his heore ells us h he Geerlized Wrig Process eds o hoogeous Poisso process wih re λ.!. So, siilr wih bove, we c defie Poisso process s specil cse of he Poly process, d he we c defie i s specil cse of Geerlized Wrig process. Defiiio 5..2 The Poly process wih preers, i which c d c λ c where λ> is cos, is clled Poisso process wih re λ. Defiiio 5..3 The Geerlized Wrig process wih preers (, k, ρ i which k, ρ k d λ ρ where λ > is cos is clled Poly process k wih preers,. c I is cler h he defiiios 5..2 d 5..3 re equivle. 5.2 The oes d soe oher properies I his prgrph we fid he firs wo oes of he Poly process d he Poisso process, usig he respecive resuls for he Geerlized Wrig process. Theore 5.2. Le N be Poly process wih preers,, he for y c 3

31 Proof [ ]. EN ( [ ] 2. Vr N( c c + c 2 2 Fro he defiiio 5.. he Poly Process, is Geerlized Wrig process c wih preers (, k, ρ where k d ρ c k. For his reso we oly eed o fid he lii of he expressios of [ ( ] Wrig process, whe k d ρ [ ] EN d Vr N( c k. We fid foud for Geerlized li k [ (] EN li k k ρ c which proves. We clcule firs li σ k d 2 Λ li k ( k ( ρ ( ρ 2 li σ k 2 R li k k ( ρ c [ ] ( ( k α, li ( k k R 2 2 σ ν li k The fro Vr N( σ + σ + σ 2 we prove 2. ν 2 ( ( + ρ 2 ( ρ ( ρ 2 k c Theore Le N be Poisso Process wih preer λ. The for y Proof [ (] ( [ ] EN VrN λ Fro he defiiio 5..2 he Poisso Process wih preer λ is Poly process wih preers, i which c d λ c. For his reso we oly eed c o fid he lii of [ (] We fid li [ (] EN [ ] EN d ( li c λ li Vr[ N( ] li c c λ which proves he heore. Vr N, foud bove, whe c d λ c

32 5.3 The rsiio probbiliies of he Poly d Poisso processes. The Poly d he Poisso Processes defied by us re boh siory Mrkov processes. Le us fid he rsiio probbiliies of he. We sr fro he Poly process wih preers,. We eed oly o clcule c We fid li k li k li k ( + ( kh ( ( + (! ( ρ + k ( ( ρ + k + kh ( + + ( + ( kh ( ( ρ + k ( + ( + (! ( ρ + k + kh ( + ( ρ + k ( + ( ρ + k + kh ( + kh( ρ + k ( + ( ( ρ + k + kh ( + + ( + ( kh ( ( ρ + k ( + ( + (! ( ρ + k + kh + + ( ( ρ + k + kh ( ( + c+ c+ + h ( ( ( + hc+ + ( + + ( + + c h ( ( + ( + ( + h c ( + ( ( + + ( + c h > +! > + Now, for he Poisso process wih preer λ, we eed o clcule ( + c+ c+ + h ( ( ( li hc ( ( + + c+ + h ( ( ( ( + + h c+ ( (! ( ( c h > c 32

33 We fid ( c + + λh í c+ h h c c li li exp( h c c+ + h + + c c + λ c+ + h c+ + h λh ( + c+ li li λhexp( λ h c c+ + h li c ( ( ( ( hc + + h+ + ( ( c h c ( c+ + h ( h ( + ( + ( ( ( c+ λh ( ( ( (! (! exp λ h c+ + h c+ + h 5.4 The Chp-Kologorov equio for he Poly d Poisso processes. We c fid he forwrd Kologorov differeil equios of he Poly process fro he respecive equios of he geerlized Wrig process p p,, ( s, ( s, ν ( p ( s,, ν( p, ( s, + qi, ( pi, ( s,, < We eed o clcule he ( fid li q k, ( li k i li qi,, i < k k ( + ( + ρ + k+ d li ( k ( + c+ ν where ρ c k. We d li q k i, ( ( + k ( ρ + k ( + i ( + i ( i( i ( ρ + k ( + li k i < - li k ν ( li k k + i + k c+ + k + i + ρ c+ If we deoe k ( li ν ( we see h q ( k ( li, k Hece, he forwrd Chp-Kologorov equios for he Poly Process wih preers, re c 33

34 p p,, ( s, ( s, ( (, k p s, ( ( ( ( k p, s, + k p, s,, > I he se wy we fid he bckwrd Chp-Kologorov differeil equios of he Poly process: li q k, + ( s li k k ( + ( + ρ + ks+ + k c+ s + li ν + k k i ρ + ks + i c+ s So, hese equios re: p, ( s, k( s p, ( s, s ( s li k k( s ( s, p, k( s [ p, ( s, p+, ( s, ], > s So, we hve proved he followig: Theore 5.4. The Poly Process is siory o-hoogeous birh process wih he rsiio iesiies k ( Rerk 5.4. c + + ( s where, re he preers of i. c The rsiio iesiies of he Poly process c be preseed i he followig for: ( ( ( 2( ( k h+ o h P, (, + h k h+ o h + o3 h > + where o ( h, o ( h, o ( h 2 3 e re respecively e The sybol oh ( es h ( oh li h h 34

35 ( o h o 2 ( h ( o h 3 ( + c c h c h ( ( h + + c+ c+ G G ( + ( + c+ + h ( ( ( + h c+ ( ( + c+ + h I he se wy, we c fid he forwrd d he bckwrd Chp-Kologorov differeil equios of he Poisso Process wih preer λ, usig he respecive equios of he Poly Process. We oly eed o fid ow, he li k ( where c λ : + li k ( li c+ λ So, we hve proved he followig: Theore The Poisso Process is siory hoogeous birh process wih he rsiio iesiies k ( λ where λ is he re of i. Rerk The rsiio iesiies of he Poisso process c be preseed i he followig for: ( ( λh+ o h P, (, + h λh+ o 2 h + o3( h > + where o ( h o ( h o ( h,,, re respecively 2 3 ( exp( λ + λ ( λ [ exp( λ ] ( h ( ( λ ( h (! exp λ o h h h o h h h 2 o h 3 Theore The Poisso process hs idepede icrees. Proof { } Le Y Y(, be Poisso process. We eed oly o prove h 35

36 ( ( ( ( s, ( ( ( PY+ h Y Y s s PY+ h Y I follows fro he Mrkovi propery h ( ( ( ( s, ( ( ( ( PY+ h Y Ys s PY+ h Y Y ( + ( ( + + s ( Bu ( ( ( PY h Y Y PY h Y ( λh exp ( λh! ( ( ( PY+ h Y We re ow i posiio o give lerive defiiio of he Poisso process which is he se o he defiiio 3. of Poisso Process, give i he book of Ross (993, p.2-2. Defiiio 5.4. The couig process Y ( hvig re λ, λ >, if (I N( { } (II I hs idepede icrees (III For ll s. { ( ( } pn+ s Ns, is sid o be Poisso process ( λ exp( λ!,,, Coiuous logues. Two oher liiig fors of he Geerlized Wrig Process. We refer here o he coiuous logue of he Geerlized Wrig Disribuio obied by Irwi(975c. He hs show h he coiuous logue of he Geerlized Wrig Disribuio is Perso s Type VI. Followig hi we ried o obi he coiuous logue of he Geerlized Wrig Process. If we deoe i by { ( } X X,, we fid h for every he probbiliy desiy fucio of X ( is where q ( ( + 2 ( + y X C X X, 2 2 q 2 re he roos of he equio X + { ρ+ 2( α + k + } X + α k 2 2 (

37 C ρ( k + ρ ( ( k d q 2 ( k + ( + αk + ρ+ 2 α ρ, q 2 2 I he priculr cse whe α, by chge of he origi d chge of scle which kes he iervl bewee successive vlues of he vrie ed o zero, X + 2 (wriig ξ ised of he coiuous logue becoes y ( ξ ( ρ + k ( k ( 2 ( ( k + k ρ ξ ξ ρ+ (5.5.2 This for c be regrded s liiig for of he Geerlized Wrig process. We hve o eio here h he bove rsforio of he vrie depeds o. Hece, ech relizio of his priculr cse of he Geerlized Wrig process is rdo vrible whose disribuio c ke he for of BeII oe wih preers k d ρ. If i (5.5.2 we wrie ξ Irwi(975c gi, we obi y ( k X c k e X c ρ d le ρ, c beig fiie, followig Xc (5.5.3 I c be cosidered s liiig for of he Geerlized Wrig process, oo.this is { p} λ lso he liiig for of he Negive Bioil ( qa ρ,, q (See Irwi 975c. k whe 37

38 6. Soe lerive geesis schees The Geerlized Wrig process hs bee defied s o-hoogeous siory Mrkov Process risig s Be ixure of he Negive Bioil process i proeess coex. I he sequel we cosider wo furher geesis schees where he uderlyig echis is idicive of cogio rher h proeess i he sese of Irwi (94 d Xeklki (983b.The cogio odel ssues h, ie, he idividuls hve hd o ccides d h, durig ie period ( d], +, he probbiliy of perso hvig oher ccide depeds o ie d o he uber of ccides x susied by hi/her by ie. So his probbiliy is fucio f ( x ν,, wih ν referrig o he idividul s risk exposure. 6. A ixed Poly process. Assuig h f ( x ν, k + x k x ν + +, he disribuio of ccides for ech + ν ( ν (λ fixed is Negive bioil wih preers ( k, ν (he ccide per is described i h cse by Poly process. As show by Xeklki (983b, he overll disribuio is he Geerlized Wrig wih preers ( k,,ρ, whe λ vries fro idividul o idividul, ccordig o expoeil disribuio, i.e., λ λ ~ e, > for. Bu for, we obied γ + ( ( ( ( ( γ + ( p+ [ ] ( ( p v ( + p d + + p ( p+ γ + ( ( ( ( ( p γ + ν ν ν ( ν ( p ( p r + γ + + ( r + + r ( ( ( + p+ γ + + r ( ( p r r! + ( p+ γ + ( + p ( ( ( r r + + r p+ γ ( ( p r r! ( + p+ γ + + r PN P ν ν ν ν d ν ν dν 38

39 ( p ( ( p ( p ( ( p ( p ( ( p p ( + p r ( ( + ( + ( ( p+ γ r r r r! ( + p+ γ + ( + p+ γ + ( r ( ( + p ( + p ( ( + r γ γ r ( r ( p γ ( r p γ ( r! r ( ( p γ ( γ + ( ( + p ( ( + r r ( r ( + p+ γ + ( γ! r ( + p+ + ( r! γ r + γ γ + p +! p r! ( ( ( ( ( ( ( γ r ( r ( ( ( + p + p+ γ ( ( r + + γ + γ ( r p( ( γ γ ( ( ( + p ( ( + p+ γ (! γ F + p, +, + p+ γ + ; r where (6.. Fb (,, γ; z ( b( z γ (!. 6.2 The Mrkovi propery of he ixed Poly process. { } I c be show h he couig process Y Y( Y( ; > ;, where Y, ( for ech, hs he disribuio give by (6.., is birh process, bu o of Geerlized Wrig for. I is cler h birh process is deeried by is iesiies d we kow lso h he oe-diesiol rgil disribuios deerie he iesiies hrough he followig relios ( Ludberg (964, p.8: ( ( ( ( ( ( ( ( P' k P, P' k P + k P, > So we hve o prove oly he exisece of i. Cosider he followig iegrl ( ( ( ( γ + + [ + ] + p I ν ν v ν ( d ν (6.2. d oice h ( P We clcule ( p+ ( γ + ( ( p! ( γ! I ( 39

40 ( ( ( ( γ+ [ ] ( ( ( ( ( γ ( ( ( ( γ [ ] I' ν + ν v + ν dν+ γ + ν + ν v + ν dν J + + J where ( + p + + ( + p ( ( ( ( + + J + p ν + ν γ v [ + ν ( ] dv (6.2.2 Hece, differeiio of he probbiliy yields P' ( p+ ( γ + 2! ( ( ( p (! ( γ P( k ( P ( k ( + I ( ( γ + J ( I ( ( p ( γ ( ( p! ( γ + +! + + I( I follows fro he bove relios, provided N is birh process, h, d where ( k P ( γ + J ( I ( ( p+ ( ( ( ( ( ( p I γ J I( P( k ( ' ( k ( ( J γ I ( γ J ( I ( (6.2.3 I follows fro (6.2., (6.2.2 d (6.2.3 h he iesiies re posiive, coiuous d differeible for >. Thus here exiss well-defied birh process wih is iesiies give by ( We deoe h i is lso difficul o clcule he vlues of he fucio F + p + + p+ +,, γ ;, d he respecive probbiliies. ( 6.3 A o-rkovi sochsic process of Geerlized Wrig for. Assuig h f ( x λ( k x λ, +, he disribuio of ccides for ech is k Negive bioil wih preers, e λ, whe λ is fixed (Irwi, 94 d Geerlized Wrig wih preers k,,, whe λ λ ~ e, > (Xeklki, 98. 4

41 Followig Irwi (94 d Arbous d Kerrich (95, d cosiderig he icree Y ( h N( + h N( he oe, s firs sep, we verify h durig he period h, h+ dh for idividul of libiliy λ hvig Y ( h y, give h N ( where x, his icree, c ke he vlues: wih probbiliy wih probbiliy ϕ λ > wih probbiliy ϕ λ (, ( yhdh yhdh, (6.3. ϕλ( yh, fλ( y+ xh, λ( k+ y+ x λ( ( k+ x + y Hece, i is esy o verify h he disribuio of he icree Y ( h N( + h N( ie, give h N ( k disribuio wih preers + x, e λh x, hs Negive bioil whe λ is fixed, d Geerlized Wrig disribuio wih preers k x +,,, whe h λ λ ~ e, So, i his cse, >. ( ( ( ( ( ( ( ( P s PN+ s i N j PN+ s N i j N j i, j, ( s ( k j + + s ( k j + ( i j. k j s ( i j Fro he ls relioship, oe y esily fid h (, τ ( τ, + (, τ ( τ, (, p s p p s p p s 2, i j, 2 3, i j, 3 j, i for soe vlues of s,,,, τ, i, j. This iplies h his process does o sisfy he Chp-Kologorov equios d hus is o Mrkov Process. 4

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