Decomposition of Supercritical Linear-Fractional Branching Processes

Transcription

1 Appled Maeac, 203, 4, p://dxdoorg/04236/a Publed Ole February 203 (p://wwwcrporg/oural/a) Decopoo o Supercrcal Lear-Fracoal Bracg Procee Ser Sagov, Alyay Saerdeova 2 Maeacal Scece, Caler Uvery o Tecology ad Uvery o Goeburg, Goeburg, Swede 2 Faculy o Mecac ad Maeac, Al-Farab Kaza Naoal Uvery, Alay, Kazaa Eal: er@calere, alyayazu@galco Receved Noveber 29, 202; reved Deceber 29, 202; acceped Jauary 6, 203 ABSTRACT I well ow a a upercrcal gle-ype Beayé-Galo-Wao proce ca be vewed a a decopoable bracg proce ored by wo ubype o parcle: oe avg e le o dece ad oe wo ave e uber o deceda I paper we aalyze uc a decopoo or e lear-racoal Beayé-Galo- Wao procee w couably ay ype We d explc expreo or e a caracerc o e reproduco law or o-called eleo ad dooed parcle Keyword: Harr-Sevayaov Traorao; Dual Reproduco Law; Bracg Proce w Couably May Type; Mulvarae Lear-Fracoal Drbuo; Beayé-Galo-Wao Proce; Codoed Bracg Proce Iroduco Te Beayé-Galo-Wao (BGW-) proce a bac odel or e ocac dyac o e ze o a populao ored by depedely reproducg parcle I a a log ory [] w org dag bac o 837 T paper devoed o e BGW-procee w couably ay ype Oe o e ouder o e eory o ul-ype bracg procee B A Sevayaov [2,3] A gle-ype BGW-proce a Marov ca Z w couably ay ae 0,, 2, Te 0 evoluo o e proce decrbed by a probably geerag uco p, p, () 0 were p ad or e probably a a gle parcle produce exacly oprg I parcle reproduce depedely w e ae reproduco law (), e e ca Z repree coecuve geerao 0 ze I paper, o peced oerwe, we aue a, e bracg proce e ro a gle parcle, 0 Z Due o e reproducve depedece ol- Z low a E e - erao o Sce zero a aborbg ae o e BGW-proce, P Z 0 oooely creae o a l called e exco probably Te laer plcly deered a a al o-egave oluo o e euao x x (2) A ey caracerc o e BGW-proce e ea oprg uber M I e ubcrcal M ad crcal M cae e proce boud o go exc, wle e upercrcal cae M we ave Clearly 0 ad oly p 0 0 I e upercrcal cae e uber o deceda o e progeor parcle eer e w probably or e w probably Recogzg a e ae rue or ay parcle appearg e BGWproce we ca dgu bewee eleo parcle avg a e le o dece [4] ad dooed parcle avg a e le o dece Grapcally we ge a pcure o e geealogcal ree lar o a gve Fgure I we dregard e dooed parcle, e eleo parcle or a BGW-proce w a raored reproduco law excludg exco (3) Copyrg 203 ScRe

2 S SAGITOV, A SHAIMERDENOVA 353 Fgure A exaple o a BGW-ree up o level 0 Sold le repree e e le o dece ad doed le repree e e le o dece ad avg e ae ea M M Forula (3) uually called e Harr-Sevayaov raorao O e oer ad, e dooed parcle or aoer bracg proce correpodg o e upercrcal bracg proce codoed o exco Te dooed parcle produce oly dooed parcle accordg o aoer raorao o e reproduco law, wc uually called e dual reproduco law ad a ea M Te upercrcal BGW-proce a a wole ca be vewed a a decopoable bracg proce w wo ubype o parcle [5] Eac eleo parcle u produce a lea oe ew eleo parcle ad alo ca gve re o a uber o dooed parcle I Seco 2 we decrbe deal decopoo or e gle ype upercrcal BGW-procee I e pecal cae we e reproduco geerag uco () lear-racoal ay caracerc o e BGW-proce ca be copued a explc or [6] I Seco 3 we uarze explc reul cocerg decopoo o a upercrcal gle-ype BGWprocee Seco 4 pree e BGW-procee w couably ay ype Our ocu o e lear-racoal cae recely uded [7] Te a reul o paper are colleced Seco 5 ad er dervao gve Seco 6 Te rearable ac a a upercrcal bracg proce codoed o exco aga a bracg proce wa recely eabled [8] a very geeral eg I geeral, e raored reproduco law are caracerzed a plc way ad are dcul o aalye T paper pree a cae were e propere o e eleo ad dooed parcle are very rapare 2 Decopoo o a Supercrcal Sgle-Type BGW-Proce Te BGW-proce a e oogeeou Marov ca w rao probable ayg 0 P I e upercrcal cae w ea M co probably, ug e propery P, 0 ad ex- we ca ge aoer e o rao probable pug P : P Te raored rao probable alo poe e bracg propery P, 0 were e - erao o e o-called dual geerag uco p, p p, 0 0 Te correpodg dual BGW-proce a ubcrcal bracg proce w oprg ea M, ee Fgure 2 Te dual BGW-proce drbued a e orgal upercrcal BGW-proce codoed o exco: p 0 0, 0 0, 0 0 PZ 0 Z P Z Z Z P Z Z Z P Z 0 Z P P P () / ()/ Fgure 2 Dualy bewee e ubcrcal ad upercrcal cae Le: a upercrcal geerag uco () w wo pove roo, or e Euao (2) Rg: e dual geerag uco draw o a dere cale Copyrg 203 ScRe

3 354 S SAGITOV, A SHAIMERDENOVA Te wo par o e curve o e le pael o Fgure 2 repree wo raorao o e upercrcal bracg proce Te lower-le par o e curve, replcaed o e rg pael o Fgure 2 ug a dere cale, gve e geerag uco o e dual proce Te upper-rg o e curve o e le pael correpod o e Harr-Sevayaov raorao (3) Te uco (3) e geerag uco or e probably drbuo p0 0, p p w e ae ea M M a e orgal oprg drbuo I eay o ee a e - erao o gve by Loog o e uure o e ye o reproducg parcle we ca dgu bewee wo ubype o parcle: eleo parcle w e le o dece (buldg e eleo o e geealogcal ree); dooed parcle avg e le o dece Tee wo ubype or a decopoable wo-ype BGW-proce S, D 0 w S D Z Te o reproduco law or e eleo parcle a e ollowg geerag uco S D F, E A cec o e bracg propery or e decopoed proce gve by S D E Z 0 S D D E E Z 0 S E P Z D Z 0 F 0, Te orgal oprg drbuo ca be recovered a a xure o e o reproduco law o e wo ubype F, Oberve alo a e oal uber o oprg or a eleo parcle a a drbuo gve by F, p, p p, w ea M M M I ollow, M M M M 0 ad we ca uarze e relaop aog dere oprg ea a M M M M 3 Lear-Fracoal Sgle-Type BGW-Proce A pora exaple o BGW-procee e learracoal bracg proce I reproduco law a a lear-racoal geerag uco 0 (4) ully caracerzed by wo paraeer: e probably 0 p0 o avg o oprg, ad e ea o e geoerc uber o oprg beyod e r oe Here 0 ad or e probably o avg a lea oe oprg Noce a w 0, e geerag uco (4) decrbe a Geoerc drbuo w ea I 0 0 e geerag uco (4) gve a Sed Geoerc drbuo w ea I 0, we arrve a a Beroull drbuo Sce e erao o e lear-racoal uco are aga lear-racoal, ay ey caracerc o e lear-racoal BGW-procee ca be copued explcly er o e paraeer 0, For exaple, we ave M, ad M, we ge M 0 Te dual reproduco law or (4) aga learracoal 0, 0, 0, w M = M Te Harr-Sevayaov raorao e lear-racoal cae correpod o a ed geoerc drbuo Copyrg 203 ScRe

4 S SAGITOV, A SHAIMERDENOVA 355, M Ieregly, e o reproduco law o eleo parcle F, a ree depede copoe: oe parcle o ype (e e leage); a Geoerc uber o oprg eac coog depedely bewee e eleo ad dooed ubype w probable ad ; a Geoerc uber o dooed oprg Oberve a eve oug bo argal drbuo ad are lear-racoal, e decopo- able BGW-proce S, D o a wo-ype learracoal BGW-proce Te drbuo o e oal uber o oprg or e eleo parcle o lear-racoal ad a ea M M M 4 BGW-Procee w Couably May Type A BGW-proce w couably ay ype 2 Z Z, Z,, 0,,2, decrbe deograpc cage a populao o parcle w dere reproduco law depedg o e ype o a parcle, 2, Here Z e uber o parcle o ype exg a geerao I e ul-ype eg we ue e ollowg vecor oao: x x2 e 2 x,,,,,,,,, xy x y we wre x xy xy 2 2 x x x2 y y2 x x2,, x,,,,,,, we eed a colu vero o a vecor A parcle o ype ay produce rado uber o parcle o dere ype o a e correpodg o reproduco law are gve by e ulvarae geerag uco Te oprg ea Z 0 E Z e (5) M 0 E Z Z e are covee o uarze a arx or M, M For e - geerao e vecor o geerag uco w copoe Z 0 E Z e are obaed a erao o w copoe (5), ad e arx o ea gve by M Te vecor o exco probable, 2, a - copoe deed a e probably o exco gve a e BGW-proce ar ro a parcle o ype Te vecor oud a e al oluo w o-egave copoe o euao x x, wc a uldeoal vero o (2) Fro ow o we rerc our aeo o e pove recurre (w repec o e ype pace) cae we ere ex a Perro-Frobeu egevalue or M w pove egevecor u ad v uc a ad vm v Mu u vu v,,, M u v, I e upercrcal cae,, all ad we ca pea abou e decopoo o a upercrcal BGW-proce w couably ay ype: S, D Now eac ype decopoed wo ubype: eer w e or e le o dece Te decopoed upercrcal BGW-proce aga a BGW-proce w couably ay ype woe reproduco law gve by e expreo F,, Lear-racoal BGW-procee w couably ay ype were uded recely [7] I cae e o probably geerag uco (5) ave a rerced lear-racoal or Copyrg 203 ScRe

5 356 S SAGITOV, A SHAIMERDENOVA 0 (6) g Te deg paraeer o bracg proce or a rple,, H a H g, were, ub-ocac arx, g g, g, 2 a proper probably drbuo, ad a pove coa Te ree er (6) deed a 0 Te deoaor (6) are ecearly depede o e oer ype o eure a e erao are alo lear-racoal T a aor rerco o e ulype lear-racoal BGW-proce excludg or exaple decopoable bracg procee I ow [7] a e lear-racoal cae e Perro-Frobeu egevalue, ex, e uue pove oluo o e euao gh (7) I e pove recurre cae, we e ex u e gh, (8) e Perro-Frobeu egevecor vu, ca be ora lzed uc a way a vu v Tey are copued a u H, (9) 0 v gh (0) I e upercrcal pove recurre cae w ad e exco probable are gve by Oberve a u () gu ad g (2) Te oal oprg uber or a ype parcle a ea M (3) 5 Ma Reul 0 I eco, we uarze explc orulae a we were able o oba or e decopoo o e uper- crcal lear-racoal BGW-procee w couably ay ype Te dervao o ee reul gve e ex eco Coder e pove recurre upercrcal cae w ad We deorae a e dual reproduco law are aga lear-racoal 0, (4) g w 0 0,, (5) g, g (6) I ur ou a e ollowg rearably ple orulae old or e ey caracerc o e dual bracg proce,, (7) gh (8) For e Perro-Frobeu egevecor we oba e ollowg expreo u u, (9) v gh (20) 0 We ow a e Harr-Sevayaov raorao reul ulvarae ed geoerc drbuo, (2) g were g, (22) 0, g g (23) Moreover, we deorae a,, (24) ad u, 0 v g H H g (25) Teore 5 Coder a lear-racoal BGW-pro- Copyrg 203 ScRe

6 S SAGITOV, A SHAIMERDENOVA 357 ce caracerzed by a rple H, g, Aue upercrcal ad povely recurre over e ae pace, a ad I dual BGW-proce ad eleo are alo lear-racoal BGW- procee w e raored paraeer rple H, g, ad H, g, w copoe gve by Euao (5), (6), (22) ad (23) Te o oprg geerag uco or a eleo parcle o ype a e or F, g g (26) 0 g were 0 g, g 0 g 0 Slarly o e gle-ype cae, we ca dgu (26) ree copoe bu ow w depedece: a rebor eleo parcle o ype ay cage ype o w probably ; depede o ad a ulvarae geoerc uber o oprg o bo ubype; a lear-racoal uber o dooed oprg w e ae o e r oprg beg depede o, Te oal uber o oprg o a eleo parcle o ype a geerag uco F, o e ex or, were 0 u belog o e erval 0, Te correpodg ea oprg uber larger a a gve by (3): M 0 M 0 6 Proo o Teore 5 I eco we derve e orulae aed Seco 5 Proo o (4) Fro 0 g ragorward o oba Euao (4) w Eua- o (5) ad (6) We ave o very a 0 g ad Te r reuree ollow ro (2) Te ecod obaed ro H H (27) wc proved ex We ave (relao (6) [7]) H M M g ad ereore M e la wo euale ad () we d r H, wc (3) Ug H M M ad e oba (27) Proo o (7) I vew o Euao (7) deerg e Perro-Frobeu egevalue or a lear-racoal BGWproce, o ow (7) eoug o very a gh Oberve a accordg o Euao (5) gh gh (28) I ollow, gh gh, (29) o a we ave o cec a Turg o Euao (27) we d yeldg gh (30) H H H H (3) H gh gh (32) T ad Euao (2) eal Euao (30) Proo o (8) Sarg ro a couerpar o Euao (8) we d ug Euao (29) Rewre Euao (3) a H H H H H Copyrg 203 ScRe

7 358 S SAGITOV, A SHAIMERDENOVA o oba Tu H H 0 gh 0 Proo o (9) ad (20) Fro (5) we derve H H T ad a couerpar o (9) vew o (32) brg (9) u H H u O e oer ad, a couerpar o (0) ogeer w (28) yeld v gh gh Proo o (26) We ave 0 0 = g g I ollow, g g g g g g g Replacg e la ueraor by g ad dvdg e wole expreo by we ge F, g g g ad e relao (26) ollow Proo o (2) ad (24) Pug (26) we arrve a (2) Noce a accordg o deo (22) ad relao (2), (27) we ave ad g, H Sce e uue pove oluo o gh gh, we derve Tu ad g gh 7 Acowledgee Ser Sagov wa uppored by e Swed Reearc Coucl gra Alyay Saerdeova wa uppored by e Scec Coee o Kazaa Mry o Educao ad Scece, gra 0732/ GF REFERENCES [] C C Heyde ad E J Seea Beayé: Sacal Teory Acpaed, Sprger, New Yor, 977 do:0007/ [2] B A Sevayaov, Te Teory o Bracg Rado Procee, Upe Maeace Nau, Vol 6, 95, pp [3] B A Sewaaow, Verzwegugprozee, Aadee- Verlag, Berl, 974 [4] N O Coell, Yule Proce Approxao o e Seleo o a Bracg Proce, Joural o Appled Probably, Vol 30, No 3, 993, pp do:02307/ [5] K B Areya ad P E Ney, Bracg Procee, Dover, Meola, 2004 [6] F Klebaer, U Roler ad S Sagov, Traorao o Galo-Wao Procee ad Lear Fracoal Reproduco, Advace Appled Probably, Vol 39, No 4, 2007, pp do:0239/aap/ [7] S Sagov, Lear-Fracoal Bracg Procee w Couably May Type, 202, 24 p Copyrg 203 ScRe

8 S SAGITOV, A SHAIMERDENOVA 359 p://arxvorg/ab/4689 [8] P Jager ad A N Lagerå, Geeral Bracg Procee Codoed o Exco Are Sll Bracg Pro- cee, Elecroc Coucao Probably, Vol 3, 2008, pp do:024/ecpv3-49 Copyrg 203 ScRe