Balanced multicolour Pólya urns via smoothing systems analysis

Transcription

1 ALEA, Lat. Am. J. Probab. Math. Stat. 15, (2018) DOI: /ALEA.v15-16 Balane multolour Pólya urns va smoothng systems analyss Céle Maller Department of Mathematal Senes, Unversty of Bath, Claverton Down, BA2 7AY Bath Unte-Kngom. E-mal URL: Abstrat. The present paper ams at esrbng n etals the asymptot omposton of a lass of -olour Pólya urns: namely balane, tenable an rreuble urns. We eompose the omposton vetor of suh urns aorng to the Joran eomposton of ther replaement matrx. The projetons of the omposton vetor onto the so-alle small Joran spaes are nown to be asymptotally Gaussan, but the asymptot behavour of the projetons onto the large Joran spaes are not nown n full etals up to now an are esrbe by a lmt ranom varable alle W, epenng on the parameters of the urn. We prove, va the stuy of smoothng systems, that the varable W has a ensty an that t s moment-etermne. 1. Introuton A Pólya urn s a srete tme stohast proess whh was orgnally ntroue by Eggenberger an Pólya (1923) to moel the sprea of epems. Sne then, they have been useful n many fferent areas of mathemats an theoretal omputer sene an are therefore broaly stue. We an for example te applatons to the analyss of ranom trees (AVL 1 Mahmou, 1998, 2 3 trees Flajolet et al., 2005), to the analyss of the Bant algorthm Lamberton et al. (2004), or to the renfore ranom wals (see for example the survey of Pemantle, 2007). The range of methos use to stuy ths ranom objet s also very large. Hstorally stue by enumeratve ombnators (see for example Bagh an Pal, Reeve by the etors May 4th, 2016; aepte Otober 8th, Mathemats Subjet Classfaton. 60F15. Key wors an phrases. Pólya urns, smoothng systems, ensty, moments, Fourer analyss. 1 The AVL s a balane searh tree name after ts nventors, Ael son-vel sĭ & Lans (see Ael son-vel sĭ an Lans, 1962). 375

2 376 C. Maller 1985), Pólya urns have been effently stue by embeng n ontnuous tme (or possonzaton) sne the wors of Athreya an Karln (1968) (see for example Janson, 2004). In parallel, sne the semnal paper by Flajolet, Dumas & Puyhaubert (Flajolet et al., 2006), the analyt ombnators ommunty suessfully tales the problem n the two-olour ase. A Pólya urn proess s efne as follows: an urn ontans balls of fferent olours, let us enote by the number of fferent olours avalable. Fx an ntal omposton α t pα 1,..., α q P N, meanng that there are α balls of olour at tme zero n the urn, for all P t1,..., u. Fx a ˆ matrx R pa,j q 1ď,jď wth nteger oeffents. At eah step of the proess, p a ball unformly at ranom n the urn, enote ts olour by ( P t1,..., u), put ths ball ba n the urn an a nto the urn a, balls of olour for all 1 ď ď. The stanar queston ase s how many balls of eah olour are there n the urn at tme n? when n tens to nfnty? Of ourse, the answer epens on the ntal omposton vetor an on the replaement matrx hosen, an many fferent behavours are exhbte n the lterature. The am of the present paper s to nvestgate nto more etal the asymptot behavour of a very large lass of Pólya urns: the -olour, balane, tenable an rreuble urns. Ths framewor s very general an nlues, among others, Pólya urns moellng m-ary trees (Chern an Hwang, 2001; Chauvn an Pouyanne, 2004), page bnary trees (Chern an Hwang, 2001; Mahmou, 2002) or B-trees (Chauvn et al., 2016). Our prese assumptons are the followng: ptq The non-agonal oeffents of R are non-negatve, an, for all 1 ď ď, ether a, ě 1 or a, s the g of ta 1,,..., a,, α u. pbq The urn s balane, meanng that there exsts an nteger S, alle the balane, suh that, for all P t1,..., u, ř a, S. piq The replaement matrx R s rreuble, meanng that for all 1 ď, ď, there exsts n ě 0 suh that pr n q, ą 0. These assumptons ensure us that S s the egenvalue of R havng maxmal real part. We then onser the Joran eomposton of the matrx R an fx one Joran blo: ths Joran blo s assoate to a stable subspae E (ontanng a unque egenlne) an to an egenvalue λ. Note that several Joran blos an orrespon to the same egenvalue. We are ntereste n the behavour of the projeton of the urn omposton vetor on E, asymptotally when n tens to nfnty. When σ : Reλ {S ď 1 {2. Ths projeton, approprately renormalse, onverges n law to a Gaussan strbuton, nepenent of the ntal omposton α of the urn. However n the ase of large egenspaes,.e. f σ ą 1 {2, we observe a fferent behavour, whh we am at esrbng better n the present paper. The projeton on a large egenspae E of the omposton vetor of the urn at tme n, renormalse approprately, onverges almost surely an n all L p pp ě 1q to a omplex ranom varable Wα tmes an egenvetor v assoate to E. Moreover, f we embe the urn proess n ontnuous tme, the obtane Galton-Watson proess U CT ptq, projete onto E, an orretly renormalse, onverges almost surely an n all L p pp ě 1q to a omplex ranom varable Wα CT tmes v. Our man goal s to gather nformaton about the two ranom varables Wα an Wα CT. The srete tme an the ontnuous tme proess are losely relate an one of our man tools wll be to transport nformaton from one settng to the other. The three man results of the paper are: for all ntal omposton α,

3 Balane multolour Pólya urns va smoothng systems analyss 377 the support of the ranom varables Wα plane f λ P CzR an the whole real lne f λ P R; W α W α an W CT α an Wα CT onverges on the whole plane). an W CT α s the whole omplex both amt a ensty on C (resp. R f λ P R); are moment-etermne (the Laplae transform of Wα These results are the frst results n the lterature about the varables W nue by mult-olour urns. To prove them, we frst use the branhng property of the urn proess to prove that t s enough to onser the atom ntal ompostons e ( P t1,..., u) efne as follows: all the oeffents of e are zero exept the th whh s equal to 1 f a, ě 0, an to a, otherwse. The branhng property of urn shemes s largely use n the lterature; the novelty of our approah s that we an nlue the ase of Pólya urn shemes wth agonal oeffents possbly less than 1. Usng agan the branhng property, we prove that pwe 1,..., We q (resp. pwe CT 1,..., We CT q) s soluton of a system of smoothng equatons. To our nowlege, ths artle ontans the frst analyss of suh a smoothng system: the lterature ontans examples of smoothng systems of two equatons wth real ranom varable solutons (see Chauvn et al., 2015 for an example) or one smoothng equaton wth omplex solutons (see e.g. Chauvn an Pouyanne, 2004), but no smoothng system wth omplex solutons as n the present artle. Suh an analyss neee the evelopment of new arguments an ths approah woul be useful n any other ontext where suh smoothng systems woul appear. In a very reent artle (appeare on arxv.org whle the urrent paper was uner revew), Leey (2016) gves a survey about smoothng systems an ther applatons, as well as general results about them. From the smoothng system n srete tme, we eue a system of equatons verfe by the Fourer transforms of the W s, an prove that these Fourer transforms are ntegrable, nung the exstene of enstes. We then show how to eue the same result for the ontnuous tme W s, an for any ntal omposton. From the system n ontnuous tme, we eue an nuton formula for the moments of the W CT s. We then prove by nuton that these moments are small enough to apply Carleman s rteron to onlue that they are moment-etermne. We then transport the result n srete tme an for all ntal ntal omposton. Plan of the paper: We frst esrbe n Seton 2 our framewor an the state of the art onernng the asymptot behavour of the mult-olour urns; at the en of Seton 2, we also state our man results (namely Theorems 2.9 an 2.10). Seton 3 s evote to haraterse the ranom varables W CT as a soluton of a system of fxe pont equatons n law. In Seton 4, we use these systems to stuy the moments of the W s an prove that they are moment-etermne, both n ontnuous an n srete tme. In Seton 5, we mply, from the ontnuous tme systems an from the moment stuy, that the ranom varables W are also haraterse as a soluton of a smoothng system. Fnally, Seton 6 ontans the proof that the W s amt a ensty, both n srete an ontnuous tme. It s very nterestng to see how we wll travel from srete to ontnuous tme all along the paper an how gong from one worl to the other s very frutful. But frst, let us suss our set of hypothess: ptq, pbq an piq.

4 378 C. Maller 1.1. Dsusson of the hypothess. Dfferent hypothess are mae n the lterature, n orer to ontrol the behavour of Pólya urns: pt1 q The oeffents of R are all non-negatve exept the agonal oeffents whh an be equal to 1. ptq The non-agonal oeffents of R are non-negatve, an, for all 1 ď ď, ether a, ě 1 or a, s the g of ta 1,,..., a,, α u. pbq The urn s balane, meanng that there exsts an nteger S, alle the balane, suh that, for all P t1,..., u, ř a, S. Assumng pt1 q or ptq permts to avo non-tenable urns,.e. urn shemes n whh somethng mpossble s ase: an example woul be that you must subtrat 3 balls of olour 1 from the urn whle there s only 1 suh ball n the urn. Allowng only non-negatve oeffents n the replaement matrx an oeffents at least 1 on the agonal permts to ensure that the urn s tenable. It s proven n Pouyanne (2008) that assumpton ptq has the same effet although t s muh weaer an allows to nlue a muh wer lass of Pólya urns nto our framewor. Some authors prefer worng wthout suh an assumpton but then mae all reasonng ontone to tenablty,.e. ontone on the event no mpossble onfguraton our (see for example Janson, 2004, Remar 4.2). The balane assumpton pbq s qute stanar n the lterature, though t s not always neessary (ths assumpton s not neee n Janson (2004), for example). Ths assumpton mples that the total number of balls n the urn s a etermnst funton of tme an ths property s the founaton of ombnators approahes whle ontnuous tme analyss of Pólya urns an be one for non-balane urns. Fnally, we wll assume that the urn s rreuble, meanng that any olour an be proue from an ntal omposton wth one unque ball: for a two olour urn, beng rreuble means havng a non-trangular replaement matrx. The followng efntons efne the noton presely. Defnton 1.1 (see Janson, 2004, page 4). Let, P t1,..., u, we say that omnates f there exsts n ě 1 suh that pr n q, ą 0. A olour P t1,..., u s sa to be omnatng f t omnates every other olour n t1,..., u. Defnton 1.2 (see Janson, 2004, page 4). We say that an urn of replaement matrx R s rreuble f an only f every olour s a omnatng olour. piq The replaement matrx R s rreuble. Note that t s sometmes enough (see, for example, Janson, 2004 an Pouyanne, 2008) to only assume the followng weaer verson of the rreublty assumpton. Frst note that the omnaton relaton s transtve an reflexve an thus parttons the set of olours nto some equvalene lasses: Defnton 1.3 (see Janson, 2004, page 4). We say that two olours an j are n the same lass f omnates j an j omnates. A omnatng lass s a lass of omnatng olours. An egenvalue λ belongs the one of the equvalene lass D f the restrton of R to D amts λ as an egenvalue. We are now reay to assume ths weaer verson of rreublty, whh we all psq for smplty: psq The largest real egenvalue λ max of R s postve an s a smple egenvalue of R. Furthermore, there s at least one ball of a omnatng olour n the urn at tme 0, an λ max belong to the omnatng lass.

5 Balane multolour Pólya urns va smoothng systems analyss 379 Note that uner pbq, S λ max. We refer the reaer to Janson (2004, page 5) for a susson of ths weaer assumpton. Cases of non-rreuble urns are stue n the lterature: the agonal ase R SI s the orgnal Pólya-Eggenberger proess an ts behavour s well esrbe n Athreya (1969); Blawell an Kenall (1964); Johnson an Kotz (1977); Chauvn et al. (2015), an trangular urn shemes are eveloppe n Janson (2006); Bose et al. (2009). Note that a large two-olour urn annot have negatve agonal oeffents, whereas there exsts olour Pólya urns wth possbly negatve oeffents havng large egenvalues. We thus have to nlue suh urns n our stuy, that s why we only assume ptq an not the more restrtve assumpton pt1 q. In the present paper, we thus hoose to assume ptq, pbq an psq (although t s neessary to assume piq for Theorem 2.10): we an te many examples of urn proesses that fall n ths framewor (see for example m-ary trees (Chern an Hwang, 2001; Chauvn an Pouyanne, 2004), page bnary trees (Mahmou, 2002; Chern an Hwang, 2001), B-trees (Chauvn et al., 2016) an we wll see all along the paper how eah of these assumptons s use n the proofs. We are ntereste n the asymptot behavour of an urn uner these three ontons. The present settng s fferent from Janson (2004) s settng for olour Pólya urns ( ě 3), where pt1 q s assume wth further assumptons, but pbq s not: t s however mentone n Janson (2004, Remar 4.2) that Janson s man results hol uner our set of hypothess. We wll thus be able to apply Janson s results before gong further n the stuy of the asymptot behavour of the urn. The followng seton s evote to summarsng the results of the lterature neee as prelmnares to state our man results. 2. Prelmnares an statement of the man results The behavour of the urn proess s alreay qute well nown: We reall hereby the man results of the lterature (manly by Athreya, 1969, Janson, 2004 an Pouyanne, 2008) an thereafter state our man results Joran eomposton. In vew of pbq an psq, the matrx R amts S as a smple egenvalue, an every other egenvalue λ of R verfes Reλ ă S. The matrx R an be wrtten on ts Joran normal form, meanng that t s smlar to a agonal of blos agpj 1,..., J r q where eah J s a matrx shape as follows: λ λ 1... J , λ λ where λ s an egenvalue of R. Note that several Joran blos an be assoate to the same egenvalue. In the followng, we hose a Joran blo an stuy the behavour of the projeton of the omposton vetor onto the subspae assoate to ths Joran blo. Note that the fat that R s rreuble mples that S s a sngle egenvalue of R.

6 380 C. Maller Defnton 2.1. Let λ be an egenvalue for R an σ Reλ {S. We all λ a large egenvalue of R f 1 {2 ă σ ă 1, or a small egenvalue f σ ď 1 {2. A large Joran blo s a Joran blo of R assoate wth a large egenvalue of R an a small Joran blo s a Joran blo assoate wth a small egenvalue of R. We enote by U α pnq Upnq P N the omposton vetor of the urn at tme n (the subsrpt α s the ntal omposton of the urn): ts th oornate s by efnton equal to the number of balls of olour at tme n n the urn. We are ntereste n the behavour of Upnq when n tens to nfnty. It s showe n the lterature that U pnq s easer to esrbe when eompose aorng to the Joran blo eomposton of R. For every stable subspae E assoate to a Joran blo of R, we wll enote by π E the projeton on E relatve to the ret sum of all Joran subspaes of R, an we wll stuy separately eah projeton on a Joran subspae E. It s stanar to embe urn proesses n ontnuous tme (see for example Athreya an Karln, 1968): eah ball s seen as a lo that rngs after a ranom tme wth exponental law of parameter one, nepenently from other los n the urn. When a lo rngs, t splts nto a,j ` δ,j balls of olour j (@j P t1,..., u) f the lo ha olour (where we uses Kroneer s notaton: δ,j equals 1 f j an 0 otherwse). We enote by τ n the tme of the n th rng n the urn an by U CT ptq the omposton vetor of the (ontnuous tme) urn at tme t. We have the followng stanar onneton: almost surely, pupnqq ně0 pu CT pτ n qq ně0. (2.1) In aton, the proess pupnqq ně0 s nepenent of the sequene of stoppng tmes pτ n q ně0. The asymptot behavour of the fferent projetons of Upnq an U CT ptq s partally esrbe n the lterature: In ontnuous tme (see Janson, 2004), small projetons have a Gaussan behavour, an renormalse large projetons onverge almost surely to a ranom varable W CT. In srete tme, f R has only small egenvalues apart from S, f E s one of the largest Joran blo assoate to the egenvalue λ realsng the seon hghest real part (after S), an f σ Reλ {S 1 {2 then projetons onto E have a Gaussan behavour (see Janson, 2004, Theorems 3.22 et 3.23); an renormalse large projetons onverge almost surely to a ranom varable W (see Pouyanne, 2008). As one an see, the behavour of small projetons (.e. projetons on a small Joran blo) n srete tme s not nown yet n full generalty: Subseton 2.2 s evote to esrbng the behavour of π E pupnqq for all small Joran spae of R. However, the man am of the paper s to esrbe the unexplore W an W CT : Subseton 2.3 wll state the results onernng the projetons on large Joran spaes (.e. assoate to a large Joran blo), as a prelmnary to our man results.

7 Balane multolour Pólya urns va smoothng systems analyss Projetons on small Joran spaes. As explane above, the behavour of the projetons of the omposton vetor (n srete tme) onto the small Joran blos s not nown yet n full generalty. Although our man am s to fous on the less unerstoo projetons onto large Joran blos, we seth here a proof of a general result for small Joran blos n orer to omplete the theory of Pólya urns uner pbq, ptq an psq. Theorem 2.2. Uner assumptons pbq, ptq an psq, f E s a blo of sze ν ` 1 assoate to a small egenvalue λ of R, then there exsts a ovarane matrx Σ suh that If Reλ S 2 then π E pu α pnqq? Ñ N p0, Σq, Sn ln 2ν`1 n n strbuton, asymptotally when n tens to nfnty. If Reλ ă S 2 then π E pu α pnqq? Sn Ñ N p0, Σq, n strbuton, asymptotally when n tens to nfnty. Moreover, Σ oes not epen on α. Let E an λ as n Theorem 2.2. The followng result by Janson s the ey of the proof of Theorem 2.2: Theorem 2.3 (Janson, 2004, Theorem 3.15 () an ()). Uner assumptons pbq, ptq an psq, for all vetor b P R, efne τ b pnq nftt ě 0 xb, Uα CT ptqy ě nu. Then, () If Reλ S 2, then 1? Sn ln 2ν`1 n π EpUα CT pτ b pnqqq Ñ N p0, σq, n strbuton, where σ s a ovarane matrx. () If Reλ ă S 2, then 1? π E puα CT pτ b pnqqq Ñ N p0, σq, Sn n strbuton, where σ s a ovarane matrx. Theorem 3.15 n Janson (2004) s slghtly fferent than the above verson. The above verson orrespons to the speal ase z n n Janson s Theorem The pq of Janson (2004, Theorem 3.15) onerns the projeton on the unon of the small Joran spaes, an t mples the pq above by projeton on a spefe small Joran spae. The pq n Theorem 3.15 Janson (2004) s more general than the above verson, whh s the speal ase ν of Janson s result. The matrx σ s gven by Equatons (3.11) an (2.15) n Janson (2004) for ase pq above, an by Equatons (3.12) an (2.16) n Janson (2004) for ase pq above. It s mportant to note that σ oes not epen on α. Theorem 2.2 an be prove by usng the ummy balls ea use n the proof of Theorems 3.21 an 3.22 n Janson (2004):

8 382 C. Maller Conser the ontnuous tme urn wth ` 1 olours, suh that the ` 1 frst olours evolve as the orgnal olour proess exept that eah tme a ball splts, one ball of olour ` 1 s ae to the proess. When a ball of olour ` 1 splts, t splts nto tself, ang no new balls n the proess. Apply Theorem 2.3 to ths ` 1 olour proess (ths proess satsfes psq). Go ba to the orgnal olour proess by approprate projeton. We o not evelop the proof sne no new ea s neee from there Projetons on large Joran spaes. Exept for ths gresson on small egenvalues, we are ntereste n the present paper n the behavour of Upnq along large Joran blos. We wll from now on fx E a Joran subspae of R assoate to a large egenvalue λ. We enote by ν ` 1 the sze of ts assoate Joran blo (beng also the menson of E) an we enote by v one egenvetor of E assoate to the egenvalue λ State of the art. The asympot behavour of U pnq projete onto the subspae E s esrbe by the followng theorem: Theorem 2.4 (f. Pouyanne, 2008). Uner pbq, ptq an psq, f 1 2 ă σ ă 1, then, π E pupnqq lm nñ8 n λ {S ln ν n 1 ν! W v, a.s. an n all L p (p ě 1), where π E pupnqq s the projeton of the omposton vetor at tme n onto E (aorng to the Joran eomposton of R). Remar 2.5. Note that fferent hoes for v are possble, an that the ranom varable W epens on ths hoe. The ranom varable W shoul atually be enote by WE,v sne t epens on the Joran subspae E fxe an on the hoe of v. For larty s sae, we wll st to the ambguous but smpler notaton W ; there s no ambguty sne E an v are fxe all along the present paper. Note that one oul also hoose to nlue the 1{ν! nto the efnton of W ; we hoose to leave t outse as one n Pouyanne (2008). In ontnuous tme, the omposton vetor projete on a large stable subspae satsfes Theorem 2.6 (see Janson, 2004). Uner pbq, ptq an psq, f 1 2 almost surely an n all L p (p ě 1), ă σ ă 1, then, π E puptqq lm tñ`8 t ν e λt 1 ν! W CT v, where π E s the projeton on the large stable subspae E. Moreover, the ranom varable W CT amts moments of all orers. Remar 2.7. Note that Theorem 2.6 s proven n Janson (2004) uner pt1 q an not ptq; but Janson explans n hs Remar 4.2 how to mae t hol uner ptq. We are ntereste n the two ranom varables W an W CT efne n Theorems 2.4 an 2.6. These ranom varables atually epen on the ntal omposton of the urn, enote by α t pα 1,..., α q, meanng that there are, for all P t1,..., u, α balls of type n the urn at tme 0. It s thus more rgorous to enote by Wα (resp. Wα CT ) the ranom varable assoate to the ntal

9 Balane multolour Pólya urns va smoothng systems analyss 383 omposton α, emphaszng that we have to stuy two nfnte famles of ranom varables. Conneton (2.1) mples onnetons between the ranom varables W nue by the srete an ontnuous proesses. We nee the followng result to eue them: lm nesτn ξ, (2.2) nñ8 almost surely, where ξ s a ranom varable wth Gamma law of parameter pα 1`...` α q{s. Ths result s shown for two olour urns n Chauvn et al. (2011), an an be straghtforwarly aapte to the present ase, usng the balane hypothess pbq. We o not evelop ths proof, whh s very stanar n the stuy of Yule proesses (see for example Athreya an Ney, 1972, page 120). We have π E `U CT pτ n q τne ν π E `U pnq λτn n λ {S ln ν n nλ {S ln ν n τ ν ne λτn. Moreover, Equaton (2.2) mples ln n τ n Ñ S when n tens to `8, n λ {S ln ν ν n ˆln n τne ν pnesτn q λ {S Ñ S ν ξ λ {S. λτn τ n Thus, for all ntal omposton α, we have (alreay mentone n Janson, 2004): W CT α plawq S ν ξ λ {S W α, (2.3) where ξ s a Gamma-strbute ranom varable wth parameter α1`...`α S, an where ξ an W are nepenent. We also have that pu pnptqqq tě0 pu CT ptqq tě0 almost surely, where ř α ` Snptq s the total number of balls n the ontnuous tme urn at tme t. It mples that, for all ntal omposton α, W α plawq Sν ξλ {S W CT α, (2.4) where ξ s a Gamma-strbute ranom varable wth parameter α1`...`α S but where ξ an Wα CT are not nepenent, whh an be verfe va a ovarane alulaton Statement of the man results. The am of the present paper s to gather nformaton about Wα an Wα CT. We wll prove the two followng theorems, whh happen to be the frst results on the varables W nue by a mult olour urn. Defnton 2.8. A omplex-value ranom varable Z s moment-etermne f for all ranom varable Y, E Y p Ȳ q E Z p Zq p@p, q ě 1q ñ Y plawq Z. Theorem 2.9. Uner assumptons pbq, ptq an psq, for all ntal omposton α, () the ranom varable W CT α s moment-etermne. onverges on the whole plane, whh mples that s moment-etermne. () the Laplae seres of W α W α

10 384 C. Maller Note that t s an open problem to etermne whether the Laplae transform of W CT onverges n a neghbourhoo of 0. A smlar result s alreay prove n Chauvn et al. (2015) n the two olour ase uner assumptons pbq, pt1 q an piq. Our proof s smlar to the one evelope there, although more nvolve ue to the hgher menson an to the less restrtve tenablty assumpton: new hgher level arguments are neee here when smple alulatons were sometmes enough n the two olour ase. Theorem Uner assumptons pbq, ptq an piq for all ntal omposton α: If λ P CzR, then the ranom varables Wα CT an Wα both amt a ensty on C, an ther support s the whole omplex plane. If λ P R, then, the ranom varables Wα CT an Wα both amt a ensty on R, an ther support s the whole real lne. A smlar result s prove n Chauvn et al. (2011) or n Chauvn et al. (2015) for two olours urns wth the tenablty onton pt1 q where the ranom varables W CT an W are real. Note that n menson 2, an urn wth negatve entres n ts replaement matrx amts no large Joran blo. Sne the ranom varables W CT an W an be non-real n the mult olour ase, the proof of Theorem 2.10 nees atonal nput; for example, provng that the support of W s the whole omplex plane beomes a non trval step n the present artle whereas the fat that the support was the whole real lne t was straghtforwar n the real ase. 3. Contnuous tme branhng proess smoothng system In ths seton, we fous on the ontnuous tme proess an show how to use the tree le struture of the proess. It s the frst step of the proofs of our two man results Theorems 2.9 an As alreay mentone, seeng urn shemes as branhng proesses s stanar n the lterature; the novelty of our approah s that we exten ths analogy for urns that o not fall uner the strong tenablty onton pt1 q but only uner the weaer ptq. Frst, we reue the stuy to only ntal ompostons (nstea of an nfnte number), namely the ntal ompostons wth a unque ball. Sa fferently, t s enough to stuy the ranom varables W e1,..., W e where, for all P t1,..., u, e s the vetor whose oornates are all 0 exept the th whh s 1 f a, ě 1 an a, otherwse. We all e 1,..., e the atom ntal ompostons of the urn. We then show, agan usng the tree-le struture of the proess, that the ranom varables W e1,..., W e satsfy a system of smoothng equatons. Frst ntroue further notatons: For all P t1,..., u, let us enote $ & α f a, ě 1, α α % (3.1) otherwse, a, an for all, P t1,..., u, let us enote $ & a, f a, ě 1, ã, a %, a, otherwse. In vew of Assumpton ptq, for all, P t1,..., u, ã, an α are ntegers. (3.2)

11 Balane multolour Pólya urns va smoothng systems analyss 385 Fgure 3.1. Deomposton of the urn proess n ontnuous tme example. Remar 3.1. If we suppose further that pt1 q hols, then α α for all P t1,..., u an ã,j a,j for all, j P t1,..., u Deomposton. To explan how to eompose the ontnuous tme urn proess, we wll fous on an example, before generalsng to any urn proess that satsfes pbq, psq an ptq. Assume for example that R One an verfy that the urn proess efne by R satsfes pbq, piq an ptq. Moreover, ts egenvalues are 8, 6 an 4. In partular, 6 s a large egenvalue whh allows us to apply Seton 2 an efne Wα CT through Theorem 2.6 for any ntal omposton α. In the followng, we wll enote by E 6 the one-mensonal Joran stable subspae assoate to ths egenvalue 6 an by π 6 the Joran projeton onto t. Note that we an eompose the multtype branhng proess as shown n Fgure 3.1, whh gves the followng Up2,4,1q CT plawq 2ÿ 1 U pq p1,0,0q ` 4ÿ 3 U pq p0,2,0q ` 5ÿ 5 U pq p0,0,1q, where the U ppq are nepenent urn proesses. Let us qut the example an mae the same reasonng as above n full generalty uner pbq, psq an ptq. Reall that, for all P t1,..., u, e has all ts oornates equal to zero exept for the th, whh s equal to 1 f a, ě 0 an to a, otherwse. We get Uα CT ptq plawq ÿβ 1 pβ 1`1 where β 0 0 an β ř jď α j, an where the Ue ppq ptq, nepenent of eah other. U CT e Ue ppq ptq, ptq are nepenent opes of Dvng ths equalty n law by t ν e λt, projetng onto the fxe large Joran subspae E va π E, an applyng Theorem 2.6 gves

12 386 C. Maller Proposton 3.2 (alreay mentone n Janson, 2004, Remar 4.2). For all replaement matres R an ntal omposton α satsfyng pbq, psq an ptq, W CT α plawq ÿβ 1 pβ 1`1 W ppq e, where the We ppq are nepenent opes of We CT, nepenent of eah other an nepenent of U. Proposton 3.2 allows to reue the stuy to only ranom varables, namely pwe CT 1,..., We CT q nstea of havng to stuy an nfnte famly of suh varables. Any nformaton gathere about those ranom varables wll a pror gve us some nformaton about any Wα CT Dsloaton. In vew of Proposton 3.2, t s enough to fous on the atom ntal ompostons e 1,..., e. Reall that for all P t1,..., u, e s the vetor whose all omponents are zero, exept the th whh s equal to 1 f a, ě 0 an to a, otherwse. We wll from now on enote by θ P t1, 2,...u the non zero omponent of e : # 1 f a, ě 0 θ (3.3) a, otherwse. Let us agan stuy frst the partular example gven by R The three atom ntal ompostons are gven by e 1 p1, 0, 0q, e 2 p0, 2, 0q an e 3 p0, 0, 1q, sne a 1,1, a 3,3 ě 0 an a 2,2 2. In all three ases, the frst step s etermnst: we now the olour of the frst ball to be rawn an we therefore now what s the omposton of the urn after the frst splt tme (f. Fgure 3.2). We therefore have that, U CT e 1 U CT e 2 plawq plawq 7ÿ 1 5ÿ 1 U pq e 1 U pq e 1 pt τ p1q q ` pt τ p2q q ` 8ÿ 8 ÿ10 6 U pq e 2 pt τ p1q q, U pq e 3 pt τ p2q q, U CT e 3 plawq 1ÿ 1 U pq e 2 pt τ p3q q ` 8ÿ 2 U pq e 3 pt τ p3q q, where the U pq are nepenent ontnuous tme urn proesses wth replaement matrx R, where τ p1q, τ p2q an τ p3q are nepenent ranom varables exponentally strbute of respetve parameters 1, 2 an 1. The ranom varables τ p1q, τ p2q an τ p3q are the frst splt tmes that our n urns of respetve ntal ompostons e 1, e 2 an e 3.

13 Balane multolour Pólya urns va smoothng systems analyss 387 Fgure 3.2. Dsloaton of a ontnuous tme urn proess the fferent atom ntal ompostons an ther omposton after the frst rawng. The same reasonng n full generalty, for all replaement matres R satsfyng pbq, psq an ptq gves that, for all P t1,..., u, U CT e plawq γÿ pq U γ 1`1 e pt τ q, where γ 0 0 an γ ř jď ã, ` δ,, τ s an exponentally strbute ranom varable of parameter θ (whh s a postve nteger), an the U pq are nepenent ontnuous tme urn proesses wth replaement matrx R. We reall that E s a fxe Joran spae of menson ν ` 1 assoate to a large egenvalue λ. Dvng the prevous equalty n law by t ν e λt, projetng onto E va π E, an applyng Theorem 2.6 gves Proposton 3.3 (alreay mentone n Janson, 2004, Theorem 3.9). Uner assumptons pbq, psq an ptq, for all P t1,..., u, W CT e plawq U λ {θ γÿ pq W γ 1`1 e, (3.4) where γ 0 0, γ ř jď pã,j ` δ,j q (usng Kroneer s notaton δ,j 1 f j, 0 otherwse), where U s a unform ranom varable on r0, 1s, an where the W pq e are nepenent opes of W CT e, nepenent of eah other an of U. Remar 3.4. If we assume pt1 q nstea of ptq n the result above, we get W CT e plawq U λ ÿ γÿ pq We, γ 1`1

14 388 C. Maller where γ 0 0, γ ř jď pa,j ` δ,j q, where U s a unform ranom varable on r0, 1s, an where the We pq other an of U. are nepenent opes of We CT, nepenent of eah Remar 3.5. One an prove that the soluton of System (3.4) s unque at fxe mean an uner a onton of fnte varane. A very smlar proof n one n Janson (2004, Proof of Theorem 3.9(), page ). 4. Moments Proof of Theorem 2.9 Ths seton s evote to the proof of Theorem 2.9. We stuy here the moments of the ranom varables pwe q Pt1,...,u an pwe CT q Pt1,...,u. Frst note that the onvergene n all L p (p ě 1) state n Theorems 2.4 an 2.6 ensures us that those ranom varables amt moments of all orers. The frst step n the proof s the followng lemma, whh onerns the ontnuous tme proess: Equaton (2.4) wll then allow us to nfer results about the srete tme proess. Lemma 4.1. Let px 1,..., X q be a soluton of System (3.4) wth moments of all ˆ E X p 1 p orers. Then the sequenes, for all P t1,..., u, are boune. p! ln p p Proof : Let px 1,..., X q be a soluton of System (3.4), let ϕppq : ln p pp ` 2q an let, for all P t1,..., u, u pq p : E X p p!ϕppq. Let us prove by nuton on p ě 1 that, for all P t1,..., u, the sequene E X p p!ϕppq 1 p s boune. Rase the equatons of System (3.4) to the power p an apply the multnomal formula. Sne E U µp 1 preµ`1 for all µ P C (wth U unformly strbute on r0, 1s), for all P t1,..., u, ˆ E X p 1 ď p Reλ pã, ` δ, qe X p θ ` 1 whh means p Reλ E X p ď θ ` ÿ p 1` `p p γ p jďp1 ã, E X p ` It mples that, for all P t1,..., u, preλ u p ď θ ã, u pq p ` ś p! 1ďjďγ ÿ p 1`...`p p γ p jďp1 ÿ p 1`...`p p γ p jďp1 ś p j! ś ź 1ďjďγ γź γ 1`1 E X p! 1ďjďγ ϕppq ϕpp j q p j! ź ź p, γź γ 1`1 E X γź pq u γ 1`1 p. p. (4.1)

15 Balane multolour Pólya urns va smoothng systems analyss 389 Let Φ ppq : ÿ p 1` `p p γ p jďp1 ś ϕpp j q. (4.2) ϕppq 1ďjďγ A slght generalsaton of Chauvn et al. (2015, Lemma 1) ensures that Φ ppq ď p1 ` 8 lnpp ` 2qq γ, (4.3) for all p ě 2, as soon as γ ě 1; reall that γ ř pã, ` δ, q ě 1. Denote by p the etermnant of preλθ R where Θ agtθ1 1,..., θ1 an R pã,j q 1ď,jď. Ths etermnant s non-zero for all p ě 2 sne ˆ ź p etppreλθ Rq θ1 etppreλi Rq an sne preλ ą S (as, by assumpton, Reλ ą S{2). Note that, uner pt1 q, we have Θ I. Reall that θ ě 1 for all P t1,..., u (see Equaton (3.3)), an note that }preλθ R} 8 preλ 1 Ñ mn 1ďď θ when p tens to `8, mplyng that there exsts p 0 ě 1 suh that, for all p ě p 0, }preλθ R} 8 preλ ď 2 mn θ : ρ. (4.4) In aton, let us enote by p pj, q the etermnant of preλθ R n whh the th olumn an the j th lne have been remove. For all 1 ď, j ď, the polynomal p pj, q has egree at most 1 n p, whh mples p pj, q ˆ1 sup O, 1ď,jď p p when p goes to nfnty, an there exsts a onstant η ą 0 an an nteger p 1 ě p 0 suh that, for all p ě p 1, p pj, q sup ď η 1ď,jď p p. Fnally, let us enote by p the etermnant of the matrx preλθ R n whh the th olumn has been replae by a olumn of 1. We now that p has egree n p whereas p s a polynomal wth egree at most 1 n p. It mples that there exsts an nteger p 2 ě p 1 suh that, for all p ě p 2, for all P t1,..., u, p p p1 ` 8 lnpp ` 2qq γ 1 ď ρη 2 Reλ, (4.5) where the hoe of the rght-han se onstant wll beome lear later on. Let us efne A : maxtpu pq q q 1 q, 1 ď q ď p2, 1 ď ď u, an prove by nuton on p ě p 2 that, for all q ď p an P t1,..., u, pu q q 1 q ď A. Fx p ą p 2 an assume that the nuton hypothess s true for p 1; then u

16 390 C. Maller Equatons (4.3) an (4.1) mply preλθu p ď ã, u pq p ` A p Φ ppq. Let pv 1,..., v q be the soluton of the system preλθv ã, v ` A p Φ ppq. We thus have that v ě u p for all 1 ď ď. Solvng ths Cramr system, we get, n vew of Equatons (4.3) an (4.5), v A p Φ ppq p p ď A p ρη 2 Reλ. For all p ě p 2, usng the fat that for all -mensonal matrx M an all vetor v, }Mv} 8 ď }M} 8 }v} 8, we get ppreλθ Rqu ppq ď ppreλθ Rqv ď }preλθ R} 8 ρηreλ ω, where u ppq pv q 1ďď, where ω s the vetor whose all oornates are equal to 1, an where the sgn ď between two vetors s to be rea oornate by oornate. In partular, we have an v enote the vetors of respetve oornates pu pq p q 1ďď an }ppreλθ Rqu ppq } 8 ď A p }preλθ R} 8 ď A p p ρηreλ η, where we have use Equaton (4.4). Let us enote M preλθ R. The oeffents of M 1 are gven by pm 1 q,j p1q `j ppj, q, p where p s the etermnant of M, an p pj, q s the etermnant of the matrx M n whh the j th lne an the th olumn have been remove. By efnton of p 2, for all p ě p 2, }M 1 } 8 sup pm 1 q,j ď η 1ď,jď p, whh mples that, for all p ě p 2, }u ppq } 8 ď }M 1 } 8 A p p η ď Ap. Fnally, for all P t1,..., u, u ppq ď A p, whh onlues the proof. Proof of Theorem 2.9:()To prove that a real-value ranom varable X s momentetermne, one an show that t satsfes Carleman s rteron: 8ÿ E X 2 1{ For a omplex-value ranom varable Z, one an for example apply Janson an Kajser (2015, Theorem 10.3), whh states that, f all moments of Z are fnte an f Z satsfes Carleman s rteron, then Z s moment-etermne n the sense of Defnton 2.8. Lemma 4.1 mples that, for all P t1,..., u, the ranom varable We CT, whh amts moments of all orers n vew of Theorem 2.6 satsfes A p

17 Balane multolour Pólya urns va smoothng systems analyss 391 Carleman s rteron. Therefore, We CT s moment-etermne. Proposton 3.2 eventually allows us to generalse Lemma 4.1 to any ntal omposton: For all ntal omposton α, Wα CT also satsfes the Carleman s rteron. () By Lemma 4.1, we have the followng nequalty: there exsts a onstant C suh that, for all nteger p ě 2, for any ntal omposton, E W CT p ď C p ln p p. p! It mples, va Equaton (2.3), there exsts a onstant D suh that, for all nteger p ě 2, E W p ď D p ln p p, p! Γ preλ`1 where we reall that Reλ ą S{2 ą 0. Ths mples that the Laplae seres of W has an nfnte raus of onvergene. Sne xt, W y ď 2 t W, ths mples that the Laplae transform of W onverges on the whole omplex plane. S Theorem 2.9 gves an upper boun for the moments of W CT that no lower boun s nown up to now. an W ; note 5. Dsrete tme urn proess Smoothng system 5.1. Smoothng system n srete tme. Ths subseton s evote to eue from Setons 3 an 4 that the ranom varable pwe 1,..., We q s a soluton of a smoothng system: Proposton 5.1. Uner assumptons pbq, ptq an psq, for every olour 1 ď ď, W e plawq γÿ γ 1`1 V λ{s W pq e, (5.1) where γ 0 0 an γ are nepenent opes of W pv 1,..., V γ of parameter π, gven by ř j1 pã,j ` δ,j q for all P t1,..., u; where the We pq e, nepenent of eah other; an where V q s a Drhlet-strbute ranom vetor nepenent of the W an π θ {S f γ 1 ă ď γ. Remar 5.2. If we assume pt1 q nstea of ptq n the theorem above, we obtan the followng system: W e plawq γÿ V γ 1`1 λ{s We pq, where γ 0 0 an γ ř j1 pa,j ` δ,j q for all P t1,..., u; where the We pq are nepenent opes of We, nepenent of eah other; an where V pv 1,..., V S`1 q s a Drhlet-strbute ranom vetor of parameter ` 1 S,..., 1 S, nepenent of the W.

18 392 C. Maller Proof : We wll present two proof for ths statement: the frst one s a moment proof, evelope hereafter, the seon one s the lassal one base on the branhng property of the urn proess, we wll etal ths other proof n Subseton 5.4. Let us prove that, for all p, q ě 1, ˆ qı E `W p`w e E e γÿ V γ 1`1 pˆ λ{s We pq γÿ λ{s V γ 1`1 W pq e qj. Sne, We s moment-etermne n vew of Theorem 2.9, ths wll onlue the proof. Let us use Conneton (2.1), whh gves, for all P t1,..., u, CT CT q E `W p`w e S pp`qqν E e ξ pλ`q λ S ı E `W e p`w e q, (5.2) where ξ s a Gamma-strbute ranom varable, of parameter θ {S. Let V be a Beta-strbute ranom varable of parameter p θ {S, 1q. Note that f U s unformly strbute on r0, 1s, we have V plawq U S {θ. Let pζ, q 1ďď;ě1 be a sequene of nepenent, Gamma-strbute ranom varables of parameter θ {S. The ranom varable ζ V ÿ γÿ γ 1 ζ, s Gamma-strbute wth parameter θ {S (t an be verfe by alulatng ts moments). Fnally, let V, ζ, ř ř. γ ζ γ, 1 Then (see for example Berton, 2006, Lemma 2.2), the γ -mensonal ranom vetor V whose oornates are gven by V V, f γ 1 ă ď γ s Drhlet-strbute of parameter π pπ 1,..., π γ q, where π θ {S f γ 1 ă ď γ,

19 Balane multolour Pólya urns va smoothng systems analyss 393 an nepenent from ζ. Thus, System (3.4) together wth Equaton (5.2), gves that, for all P t1,..., u, ı S pp`qqν E ξ pλ`q λ S q E `W p`w e (5.3) ˆ E V λ {S ˆ E γÿ e W γ 1`1 γÿ V γ 1`1 ˆ S pp`qqν E pˆ CT,pq e V λ{s λ{s S ν ζ λ {S, W,pq e pˆ γÿ γ 1`1 γÿ W γ 1`1 γÿ V γ 1`1 pˆ `ζ, V λ{s We,pq We also have that, by nepenene of V an ζ, E ζ pλ`q λ S ˆ E ˆ E ı ˆ E γÿ γ 1`1 γÿ γÿ γ 1`1 ζ V, γ 1`1 pζ,v q V, λ{s W,pq λ{s W,pq e e pˆ λ{s We,pq pˆ pˆ γÿ γÿ γ 1`1 qj CT,pq e qj λ{s S ν ζ λ{s, W,pq e γÿ γ 1`1 γÿ γ 1`1 γ 1`1 pζ,v q `ζ, V λ{s W,pq e qj. ζ V, V, λ{s W,pq e λ{s W,pq λ{s W,pq e qj (5.4) qj e qj. (5.5) The result follows from Equatons (5.4) an (5.5), usng the fat that W moment-etermne. s 5.2. Unty. The man goal of ths seton s to prove that the soluton of System (5.1) s unque. We therefore use the so-alle ontraton metho. Ths metho, presente for example n Nennger-Rüshenorf s survey (Nennger an Rüshenorf, 2006) onssts n applyng the Banah fxe pont theorem n an approprate omplete Banah spae. It has alreay been use n a Pólya urn ontext n the lterature. In Knape an Nennger (2014) the ontraton metho s use as a new approah to prove an equvalent of Theorem 2.4 for large an small egenvalues (for srete tme two-olour urns). In Chauvn et al. (2015), t s use as n the present paper, to prove the unty of the soluton of a two-equaton system, n the stuy of large two olour Pólya urns. In Janson (2004), t s also use to prove the unty of the soluton of system (3.4): therefore, we wll only evelop the proof for the srete ase. Smlar proofs an be foun n Knape an Nennger (2014) or Chauvn et al. (2015). Let M 2 be the spae of omplex-value square ntegrable probablty measures. For all A P C, let M C 2 paq be the subspae of measures n M 2 wth mean A. We

20 394 C. Maller onser the Wassersten stane as follows: for all µ, ν two measures n M C 2 paq, W pµ, νq nf }X Y } 2, X µ,y ν where } } 2 s the L 2 -norm on C. For all A 1,..., A P C. Let us enote by Ś MC 2 pa q the Cartesan prout of the spaes M C 2 pa q. We efne the Wassersten stane on ths spae as follows: for all µ pµ 1,..., µ q an ν pν 1,..., ν q two elements of Ś MC 2 pa q, pµ, νq max 1ďď t W pµ, ν qu. We now that pm C 2 paq, W q an thus Ś MC 2 pa q are omplete metr spaes (see for example Duley, 2002). The ranom vetor pwe 1,..., We q s a soluton of System (5.1): W e plawq γÿ γ 1`1 V λ{s W pq e. For all µ pµ 1,..., µ q P Ś MC 2 pm q, for all P t1,..., u, let γ ÿ K pµq L, γ 1`1 V λ{s X pq where γ 0 0, γ ř jď pã,j ` δ,j q an for all P t1,..., u, the px pq q 1ďď are nepenent ranom varables, nepenent of eah other an of vetor V, whh s Drhlet-strbute of parameter ˆ π 1,... π γ, an, for all 1 ď ď an 1 ď ď γ, γ 1 ` 1 ď ď γ ñ X pq µ an π θ {S. (5.6) We efne the funton K as an prove the followng result: Kpµq pk 1 pµq,..., K pµqq, Proposton 5.3. For all large egenvalue λ of the replaement matrx R, for all A pa 1,..., A q P C, enote by ΨpAq the vetor whose oornates are gven by A ΨpAq λ ` θ where λ ` θ 0 sne Reλ ą S{2 ą 0 an θ ě 1. for all P t1,..., u, () For all A pa 1,..., A q P C suh that ΨpAq P KerpR λi q, the funton K s an applaton from Ś MC 2 pa q nto tself. () Moreover, the law of pwe 1,..., We q s the unque square-ntegrable soluton of (5.1) at fxe mean. Remar 5.4. If we assume pt1 q n aton, then, remar that ΨpAq P KerpRλI q f an only f A P KerpR λi q.

21 Balane multolour Pólya urns va smoothng systems analyss 395 Proof : pq Frst remar that ż EK pµq : C x K pµqpxq γÿ pγ 1`1 E V p λ{s EX pq beause pv 1,..., V S`1 q s nepenent of px p1q,..., X ps`1q q 1ďď. Sne, for all p P t1,..., S ` 1u, for all P t1,..., u, EX ppq A, we have EK pµq A γÿ pγ 1`1 E γ A γ 1 1 ` λθ1 A a, ` δ, θ θ ` λ V p λ{s ã, ` δ, A 1 ` λθ1 pa, ` δ, θ qb, where B ΨpAq. The above alulatons are true beause V p λ{s plawq U λ {θ for all p P t1,..., S ` 1u, where U s a ranom varable unformly strbute on r0, 1s, an beause Reλ ą S {2, whh mples λ θ sne θ ě 1 (see Equaton (3.3)). Sne λ s an egenvalue of R, an B pb 1,..., B q P KerpR λi q, we have for all 1 ď ď. It mples a, B λb EK pµq pλ ` θ qb A for all µ P Ś MC 2 pa q. Moreover Kpµq s square-ntegrable, whh mples that K s nee a funton from Ś MC 2 pa q nto tself, for all A suh that ΨpAq P KerpR λi q. pq Let µ pµ 1,..., µ q P Ś MC 2 pa q an ν pν 1,..., ν q P Ś MC 2 pa q be two solutons of System (5.1), meanng that Kµ µ an Kν ν. Let us prove that pµ, νq 0, usng the total varane law: t s enough to prove that, for all P t1,..., u, W pµ, ν q 0. Kantorovth-Rubnsten s theorem mples that, for all 1 ď ď, there exsts two ranom varables X µ an Y ν suh that W pµ, ν q }X Y } 2. ˆFx 1 ď ď, an let V be a Drhletstrbute ranom varable of parameter π 1,... π (as n Equaton (5.6)), X px 1,..., X q an Y py γ 1,..., Y q be two sequenes of nepenent γ ranom varables suh that X an Y are both nepenent of V (but not neessarly nepenent of eah other), an, for all 1 ď ď an 1 ď ď γ, γ 1 ` 1 ď ď γ ñ LpX γ, Y q LpX, Y q an π θ {S.

22 396 C. Maller We thus have, for all P t1,..., u, W pµ, ν q 2 W pk pµq, K pνqq 2 ď mplyng that ď γÿ E γ 1`1 E X Y 2 W pk pµq, K pνqq 2 ď γ ˇ V γÿ E γ 1`1 γ 1 γÿ γ 1`1 2λ{Sˇˇˇˇ EˇˇpX pq ˇ V V Y pq qˇˇ2 2λ{Sˇˇˇˇ, 2Reλθ1 ` 1 W pµ, ν q 2 λ{s px pq Y pq q ã, ` δ, 2Reλθ1 ` 1 W pµ, ν q 2. If we let W pµ,ν q 2 θ `2Reλ, for all P t1,..., u, we get that, for all 1 ď ď, 2Reλ ď a, pr q. Let v pv 1,..., v q be a horzontal vetor wth postve entres suh that vr Sv, then vpr q Sv. The exstene of suh a v s a onsequene of the rreublty of the urn, namely hypothess psq, as explane n Janson (2004). It mples that, S v 1 v pr q ě 2Reλ v. 1 Sne Reλ {S ą 1 {2, ths last nequalty mples that v 0, 1 an thus, by postvty of the v an non negatvty of the, we get, for all P t1,..., u, 0. It thus mples that, for all P t1,..., u, whh onlues the proof. 1 µ ν, an thus µ ν, 5.3. Deomposton n srete tme. The argument use to prove Proposton 5.1 an also be use to prove the followng result from Proposton 3.2 an Theorem 2.9: Proposton 5.5. Uner assumptons pbq, ptq an psq, for all ntal omposton α, Wα plawq ÿβ Z λ {S p We ppq, 1 pβ 1`1 2 2

23 Balane multolour Pólya urns va smoothng systems analyss 397 where β 0 0, β ř jď α j; where the W ppq e are nepenent opes of W e, nepenent of eah other; an where Z pz 1,..., Z β q s a Drhlet-strbute ranom vetor nepenent of the W an of parameter η gven by η θ {S f β 1 ă ď β. Remar 5.6. If we assume pt1 q nstea of ptq n the theorem above, we obtan the followng equaton: for all ntal omposton α, W α plawq ÿβ 1 pβ 1`1 Z λ {S p We ppq, where β 0 0, β ř jď α j; where Z pz 1,..., Z β q s a Drhlet-strbute ranom vetor of parameter ` 1 S,..., 1 ppq S ; an where the W e are nepenent opes of We, nepenent of eah other an of Z Tree struture n srete tme. Propostons 5.5 an 5.1 an be proven from srath by an analogue of the proof of Propostons 3.2 an 3.3, usng the unerlyng tree struture of the urn. The analyss of the tree struture n srete-tme s however more ntrate sne the subtrees of the onsere forest are not nepenent. Suh a tree eomposton n srete-tme s alreay propose n Chauvn et al. (2015) (for two-olour urns) or Knape an Nennger (2014) but both papers assume pbq, piq an pt1 q. We wll evelop ths alternatve proof of Proposton 5.5 beause t happens to be more omplate ue to the possbly negatve agonal oeffent of the replaement matrx. Alternatve proof of Proposton 5.5: The srete-tme urn proess an be seen as a forest whose leaves an be of fferent olours: we refer to Fgure A tme zero, the forest s ompose of α roots of olour (for all P t1,..., u), eah of these roots ontans θ balls of olour. At eah step, we p up unformly at ranom a ball: ths ball belongs to a leaf. The pe leaf then beomes an nternal noe, whh has γ hlren, amongst them ã,j ` δ,j ontan θ j balls of olour j (for all j P t1,..., u) f the pe up leaf was of olour. The omposton of the urn s thus esrbe by the set of leaves of the forest. Let us number the subtrees of the forest from trees roote by olour 1 up to trees roote by olour. If we enote by D p pnq the number of balls n leaves of the p th subtree of the forest, then ths p th subtree of the forest at tme n represents the omposton vetor of an urn proess wth ntal omposton of arnal θ f β 1 ă p ď β, taen at nternal tme S p pnq Dppnqθ S. Inee, the nternal tme n the p th tree s the number of ts nternal noes; the fat that the urn s balane means that all nternal noes of the tree have gven brth to S ` 1 balls, whh gve the above relatonshp between leaves an nternal noes n the p th subtree. We thus have U α pnq plawq ÿβ 1 pβ 1`1 U ppq e ˆDp pnq ω p S, (5.7) 2 Fgure 5.3 s an example of a two-olour urn. The example taen s hosen for t smplty even f the hosen urn only has small Joran blos: t s alle a small urn n the lterature. It oes not affet the arguments evelope n the proof.

24 398 C. Maller Fgure 5.3. A realsaton at tme n 7 of the forest assoate to the urn proess wth ntal omposton t p2, 2q an wth replaement matrx R. Remar that S ˆ p7q 3, S 2 p7q 3 an S 3 p7q 1. Moreover, D 1 p7q 8, D 2 p7q 3 an D 3 p7q 1. where β 0 0, where for all ě 1, β ř α ; where ω p θ f β 1 ă p ď β ; an where the urn proesses Ue ppq are nepenent opes of the proess U e, nepenent of eah other. We are thus ntereste n the asymptot behavour of the vetor pd 1 pnq,..., D β pnqq when n grows to nfnty. Ths vetor happens to be the omposton vetor of a β ř α olour Pólya urn wth ntal omposton ω t pω 1,..., ω β q, where ω θ f β 1 ă ď β. Inee, forget the ntal olourng of the leaves an olour the leaves of the th subtree wth olour ; at eah step, a leaf of the forest pe up unformly at ranom beomes an nternal noe an gves brth to S ` 1 leaves of ts same olour. Suh a agonal urn s alle a Pólya-Eggenberger urn an has been long stue n the lterature. We an for example te ths result by Athreya (1969) (for a omplete proof, see Berton, 2006 or Chauvn et al., 2015): Theorem 5.7. Let p an K be two postve ntegers an pd 1 pnq,..., D p pnqq the omposton vetor at tme n of an urn proess of ntal omposton ν t pν 1,..., ν p q an wth replaement matrx KI p, then, asymptotally when n tens to nfnty, almost surely, 1 nk pd 1pnq,..., D p pnqq Ñ Z pz 1,..., Z p q where Z s a Drhlet-strbute ranom vetor of parameter ` ν 1 K,..., νp K. Thus, projetng Equaton (5.7) onto E va π E, renormalsng t by n λ {S ln ν n an tang the lmt when n Ñ `8 gves Proposton Denstes Proof of Theorem 2.10 In ths seton, we prove Theorem 2.10 va an analyss of the Fourer transforms of the ranom varables W CT an W. We generalse the metho evelope by Lu (1998, 2001) for smoothng equatons wth postve solutons an refer to Chauvn et al. (2014) for a smlar proof n the ase of m-ary trees where a omplex fxe pont equaton (but not a system) s stue. A smlar result s also prove for

25 Balane multolour Pólya urns va smoothng systems analyss 399 the two olour ase n Chauvn et al. (2015), but ue to both the hgher menson an the weaer tenablty onton, the present proof follows a fferent route. The strategy of the proof s the followng. Frst fous on the srete tme ase,.e. on W : We prove that the support of We ontans some non-srete set. It mples that the Fourer transform of W s ntegrable an thus nvertble (see Lemmas 6.4, 6.5 an 6.6), so that the varable W has a ensty on C. We fnally eue from the exstene of a ensty an from Lemma 6.2 that the support of W s the whole omplex plane. Va the martngale onneton (2.3), one an onsequently nfer that, for all P t1,..., u We CT has a ensty on C. Propostons 5.5 an 3.2 permt to generalse to any ntal omposton. We assume pbq, ptq an piq an reall that the egenvalue λ, the Joran subspae E, an ts menson ν ` 1 are fxe. Lemma 6.1. There exsts 1 ď ď suh that EW e 0. Proof : It s nown (see for example Pouyanne, 2008) that, for all P t1,..., u, 1 ν! W e v lm nñ`8 `1 ` tr E 1 ˆ 1 ` t 1 ˆ R E 1 ` 1 ` S t 1 R E π E pu e 1 ` pn 1qS pnqq, as a martngale almost sure lmt, where t R E stans for the restrton of the enomorphsm nue by R to the Joran subspae E (note that the unque egenvalue of R s thus λ an sne Reλ ą S{2, the nverses above are well efne). Thus, EWe v ν!π E pe q. Note that, f EWe v ν!π E pe q 0, for all 1 ď ď, then, π E 0, whh s mpossble. Lemma 6.2. For all z pz 1,..., z q P Ś SupppWe q, for all P t1,..., u, for all pv 1,..., v γ q n the support of a Drhlet- strbute ranom vetor of parameter ω pω 1,..., ω γ q, where ω θ {S for all P t1,..., u, we have γÿ v γ 1`1 λ{s z P SupppW e q. Note that the support of the Drhlet strbuton of parameter ω s! pv 1,..., v γ q P r0, 1s γ : γ 1 ) v 1. Proof : Reall that for a gven omplex ranom varable Z, for all z P C, z P SupppZq ą 0, Pp Z z ă εq ą 0.