Fluctuation Results For Quadratic Continuous-State Branching Process

Transcription

1 IOSR Journal of Mathematcs (IOSR-JM) e-issn: , p-issn: X. Volume 13, Issue 3 Ver. III (May - June 2017), PP Fluctuaton Results For Quadratc Contnuous-State Branchng Process Hongwe B Unversty of nternatonal busness and economcs, , Bejng, Chna. Emal: bhw2009@gmal.com Abstract. In ths note, some fluctuaton results for renormalzed number of ancestors of a statonary quadratc contnuous-state branchng process are gven. We consder three dfferent cases: same tme, dfferent step wdth; dfferent tme, the same step wdth; adjacent tme wth the same step wdth. The Laplace transform of some related quanttes s derved to prove ths result. Key words and phrases.fluctuaton, statonary CB process, number of ancestor Mathematcal Subject Classfcaton.60J80,60J85. I. Introducton Contnuous state branchng processes (CB processes) are non-negatve real-valued Markov processes frst ntroduced by Jrna [4] to model the evoluton of large populatons of small partcles. Contnuous state branchng processes wth mmgraton (CBI processes) are generalzatons of those descrbng the stuaton where mmgrants may come from outer sources, see e.g. Kawazu and Watanabe [5]. For a survey n ths drecton, you may refer to L [7] and the reference theren. Snce then they have been powerful tools n bology. In ther semnar work, Chen and Delmas [2], statonary contnuous-state branchng process s consdered and they use the mmortal decomposton to calculate some quanttes assocated to the most recent common ancestor (MRCA) and the number of ancestor. They also consder the quadratc CB process as an example and then the renormalzed fluctuaton result for the number of ancestor s partally consdered, and some nterestng phenomenon s revealed. In ths note, we are gong to consder some fluctuaton results assocated to M t s (to be specfed later) for fxed tme s n Secton 2. To state our man result, we wllprovde n the ntroducton some background n the followng subsectons. In partcular, wefrst recall the defnton of CB process n Subsecton 1.1, and then the tree formulaton n Subsecton1.2, the statonary CB process and the ancestor process assocated to statonary CB process are respectvely ntroduced n Subsecton 1.3 and Quadratc CB process. Consder a sub-crtcal branchng mechansm DOI: / Page

2 Fluctuaton Results For Quadratc Contnuous-State Branchng Process 0 M r 1.2. The real tree formulaton. Defne for r >0,, the number of ancestors (the mmortal ndvdual excluded) at tme r of the current populaton at tme 0. In order to defne precsely ths quantty, some fner structure for CB process s needed. We recall n what follows the genealogcal tree for the CB process whch s studed n Le Gall [6] or Duquesne and Le Gall [3]. Snce the branchng mechansm s quadratc, the correspondng Levy process s just the Brownan moton wth drft. Let B Bt, t R ) be a standard Brownan ( moton. We consder the Brownan moton B ( B t, t R ) wth negatve drft and the correspondng reflected process above ts mnmum H H( t), t R ) : ( We deduce from equaton (1.7) n [3] that H s the heght process assocated to the branchng mechansm. For a functon H, set max(h) = max(h(t), t R ). Let N[dH] be the excurson measure of H above ( l x ( H), x R, t R 0normalzed such that N[max(H) r] = c(r). Let t ) be the local tme of H at tme t and level x. Let ζ = nf{t >0;H(t) = 0} be the duraton of the excurson H under N[dH]. We recall that l r ( H), r R ) under N s dstrbuted as Y under N. From now on we shall dentfy Y wth l r ( H), r R ) and ( wrte N for N. We now recall the constructon of the genealogcal tree of the CB process Y from H. ( 1.3. Statonary CB process. Let D be the space of cadlag paths havng 0 as a trap. Consder underp a Posson pont measure Let E be the correspondng expectaton. Forwardng to [2], Z s a CB process condtonally on non-extncton and moreover a statonary CB process (Certanly t can be treated as a CB process wth mmgraton from L [8]). Usng the property of the Posson pont measure, we have: DOI: / Page

3 Fluctuaton Results For Quadratc Contnuous-State Branchng Process 1.4. The ancestor process.let be a Posson pont measure wth ntensty I ( t, H ) 2dt N[dH]. We wll wrte allows to code (on an enlarged space) the genealogy of Z defned by (5). Y for l a ( H ) for I a. Thus Let r<t. Defne explctly the number of ancestors (excludng the mmortal partcle) at tme r of the populaton lvng at tmet, M, by t r We wll always dentfy M r wth 0 M r to most recent common ancestor (TMRCA) we have when there s no rsk of confuson. Notce that for r larger than the tme M r = 0. It s easy to deduce the followng denttes. The next result s n a sense a consequence of the tme reversblty of the process Y wth respect to ts lfetme ζ whch was derved from [1]. Our frst result concerns wth the same current tme s and dfferent step wdth (Theorem2.1), that s, condtonal on converges n the sense of fnte dmensonal to wth a Gaussan process. It s somewhat ant-ntutve. The second result consders wth dfferent current tme s whle wth the same step wdth(theorem 2.2): Subsecton 2.2 and 2.3 are devoted to provng the two results usng some nducton tools. II. Man Results On The Number Of Ancestors In ths secton, we focus on the fluctuatons on the number of ancestors M assocated toa fxed tme s. In partcular we shall deal wth the fluctuatons from three pont of vew: thesame startng tme wth dfferent step wdth; dfferent startng tme wth same step wdth;adjacent tme wth the same step wdth Man Results. Frst let us recall some known results from [2]. For r >0, we note s t DOI: / Page

4 Fluctuaton Results For Quadratc Contnuous-State Branchng Process We shall consder the normalzed fluctuatons for the number of ancestors n fnte dmensons.the frst man result assocated to fxed tme s wth dfferent step wdth s as follows: The second type of fluctuaton s gven as follows: Remark 2.3. Theorem 2.2 mples the processes Remark 2.4. For the thrd type of fluctuaton on the process twodmensonal convergence for ths process, that s for α, η 0, U s are ndependent for dfferent s., we can only get the It s not straghtforward to get the fnte dmensonal convergence Proof of Theorem 2.1. By statonarty, we may fx s = 0. We frst get the Laplacetransform assocated to M and Z n the n-dmensonal case through nducton. 0 Let b<a,c.recall that R b (H) denote the number of ndvduals (coded by H) at tmea (born at tme b) whch c ab s stll alve at tme c. Then we have Lemma 2.5. For = 1, 2,, n, let 0 and t1 tn. Then DOI: / Page

5 Fluctuaton Results For Quadratc Contnuous-State Branchng Process An applcaton of (2) yelds that By decomposton wth respect to the brth tme, for n = j + 1, we have: It s clear that for the frst j terms, they appear just as n the n = j case. Besdes the ntegrand n the (j + 1)-th term s the same as that of the last term n n = j case but wth dfferent ntegratng range. We denote the frst j + F 1 terms as j1. Thus we have: DOI: / Page

6 Fluctuaton Results For Quadratc Contnuous-State Branchng Process Moreover we have: where t can be deduced from the n = j case that For 1 n 1, t s elementary to deduce that as r 0+, The result thus follows Proof of Theorem 2.2. Recall n the quadratc case, for, t 0, DOI: / Page

7 We can further generalze ths to [ 2, ) Fluctuaton Results For Quadratc Contnuous-State Branchng Process snce for anyt>0, u(, t) can be defnedon [ ( t), ), whle f ( t) 2. We consder the n-dmensonal case as r >0 small enough, and the smlar calculatons as that of Lemma 2.5, we have: Lemma 2.6. For 1 n,, 0, we get: (13) s s 1 n and note s n1 f wth. By statonarty Then we are ready to prove Theorem 2.2. The followng s dedcated to ths. Proof of Theorem 2.2.Note frst that equaton (13) can also be generalzed to the negatve case as llustrated above for u(λ, t). Now let / c( r), c( r) for 1 n wth the condton that 2 4 convergence asr 0+ from (13) through approxmaton. Frst we deal wth and 1 1. We haveths yelds n 1. We can get the DOI: / Page

8 Fluctuaton Results For Quadratc Contnuous-State Branchng Process where we use the condtonal on Z s n n the second equaton. An applcaton of Mukherjea et al [9] wll gve the result for any 0. The result thus follows. Acknowledgement H. B was supported by the Fundamental Research Funds for the Central Unverstes n UIBE (16QN04) References [1]. H. B and J.-F. Delmas. Total length of the genealogcal tree for quadratc statonary contnuous-state branchng processes. Ann. Inst. H. Poncare Probab. Statst., 52(3): , [2]. Y.-T. Chen and J.-F. Delmas. Smaller populaton sze at the MRCA tme for statonary branchng processes.ann. Probab., 40(5): , [3]. T. Duquesne and J.-F. Le Gall. Random trees, Levy processes and spatal branchng processes, volume 281. Astersque, [4]. M. Jˇrna. Stochastc branchng processes wth contnuous state space. Czechoslovak Mathematcal Journal, 8(2): , [5]. K. Kawazu and S. Watanabe. Branchng processes wth mmgraton and related lmt theorems. Theory of Probablty & Its Applcatons, 16(1):36 54, [6]. J.-F. Le Gall. Ito s excurson theory and random trees. Stochastc Process. Appl., 120(5): , [7]. Z. L. Measure-valued branchng Markov processes. Sprnger Scence & Busness Meda, [8]. Z. L. Contnuous-state branchng processes. arxv preprnt arxv: , [9]. A. Mukherjea, M. Rao, and S. Suen. A note on moment generatng functons. Stoch. Prob. Letters, 76(11): , DOI: / Page