RECENT COMMON ANCESTORS OF ALL PRESENT-DAY INDIVIDUALS

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1 RECENT COMMON ANCESTORS OF ALL PRESENT-DAY INDIVIDUALS Joseph T. Chag Departmet of Statistics, Yale Uiversity Abstract Previous study of the time to a commo acestor of all preset-day idividuals has focused o models i which each idividual has just oe paret i the previous geeratio. For example, mitochodrial Eve is the most recet commo acestor (MRCA) whe acestry is defied oly through materal lies. I the stadard Wright-Fisher model with populatio size, the expected umber of geeratios to the MRCA is about 2, ad the stadard deviatio of this time is also of order. Here we study a two-paret aalog of the Wright-Fisher model that defies acestry usig both parets. I this model, if the populatio size is large, the umber of geeratios, T, back to a MRCA has a distributio that is cocetrated aroud lg (where lg deotes base-2 logarithm), i the sese that the ratio T /(lg ) coverges i probability to 1 as. Also, cotiuig to trace back further ito the past, at about 1.77 lg geeratios before the preset, all partial acestry of the curret populatio eds, i the followig sese: with high probability for large, i each geeratio at least 1.77 lg geeratios before the preset, all idividuals who have ay descedats amog the preset-day idividuals are actually acestors of all preset-day idividuals. COALESCENT, WRIGHT-FISHER MODEL, GALTON-WATSON PROCESS, GENEALOGICAL MODELS, POPULATION GENETICS AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 92D25 SECONDARY 60J85 Ruig head: Recet commo acestors Postal address: Yale Uiversity Statistics Departmet Box Yale Statio New Have, CT Phoe: Fax: joseph.chag@yale.edu Versio: Jue 12, 1998

2 1 Itroductio Startig with the set of all of us preset-day humas, imagie tracig back i time through our mothers, our mothers mothers, ad so o. This is the materal family tree of makid, ad we are at its leaves. Recet research has suggested that the woma at the root of this tree lived roughly 100,000 or 200,000 years ago, perhaps i Africa (Ca et al., 1987; Vigilat et al., 1991). This woma has bee dubbed mitochodrial Eve, sice all preset-day huma mitochodrial DNA desceded from hers. Mitochodrial Eve was udoubtedly ot the oly woma alive at her time, so the ame Eve is misleadig, as has bee poited out by a umber of authors; see, e.g., Ayala (1995). However, this misuderstadig aside, questios of the origis of makid ad the ature of our relatioships to each other are still of kee iterest, ad the research o mitochodrial Eve has received a great deal of publicity, geeratig headlies i the popular press as well as i scietific publicatios. Svate Pääbo (1995) explais:...the recet date of our mitochodrial acestor is i a sese the really cotroversial coclusio from these studies. Everyoe agrees that we trace our acestry to Homo erectus, who emerged i Africa ad from there coloized most of Eurasia about a millio years ago or eve earlier. What the mitochodrial data seem to show, however, is that we have a much more recet acestor, oe who lived some 100,000 or 200,000 years ago. What captures the imagiatio is ot the particular choice to trace back through the materal lie, but rather it is the idea that all of preset-day humaity may have a commo acestor who lived as little as 100,000 years ago, a time that seems to may to be surprisigly recet. If we retai this idea while removig the restrictio to the materal lie, the questio becomes: How far back i time do we eed to trace the full geealogy of makid i order to fid ay idividual who is a commo acestor of all preset-day idividuals? I this paper we address this sort of questio i a simple mathematical model. The coalescet model of Kigma (1982) forms the basis of may of the calculatios, formal ad iformal, used i recet treatmets of questios about mitochodrial Eve ad related topics. The coalescet is a large-populatio limit of a umber of the fudametal models of populatio geetics, icludig the Wright-Fisher process. These models are haploid, with each idividual i a give geeratio havig a sigle paret i the previous geeratio. The Wright-Fisher model assumes radom matig, i the sese that the paret of a give idividual is equally likely to be ay of the idividuals i the previous geeratio. The stadard model also postulates a costat populatio size, which may be a effective populatio size whe modelig more geeral situatios. A umber of importat properties of the coalescet model are used i applicatios. For example, the model implies a relatioship betwee coalescece times ad populatio size: the expected coalescece time (measured i geeratios) of a large sample is about twice the populatio size. Hudso (1990) gives a survey of the theory ad applicatios of the coalescet. Here we study a atural two-paret aalog of the Wright-Fisher process. (This process was previously cosidered by Kämmerle (1991) ad Möhle (1994); see the ed of this sectio for a discussio of related work.) We assume the populatio size is costat at. Geeratios are discrete ad ooverlappig. The geealogy is formed by this radom process: i each geeratio, each idividual chooses two parets at radom from the previous geeratio. The 2

3 choices are made just as i the stadard Wright-Fisher model radomly ad equally likely over the possibilities the oly differece beig that here each idividual chooses twice istead of oce. All choices are made idepedetly. Thus, for example, it is possible that whe a idividual chooses his two parets, he chooses the same idividual twice, so that i fact he eds up with just oe paret; this happes with probability 1/. This model is desiged oly as a simple startig poit for thought; of course it is ot meat to be particularly realistic. Still, oe might worry that this simple model igores cosideratios of sex ad allows impossible geealogies. If this seems bothersome, a alterative iterpretatio of the same process is that each idividual is actually a couple, ad that the populatio cosists of moogamous couples. The the radom choices cause o cotradictios: the husbad ad wife each were bor to a couple from the previous geeratio. They could eve come from the same couple i the previous geeratio. Our iterest here is i fidig idividuals who are commo acestors of all preset-day idividuals. For coveiece, we use the abbreviatio CA to refer to a commo acestor of all preset-day idividuals, ad MRCA stads for most recet commo acestor. It turs out that mixig occurs extremely rapidly i the two-paret model, so that CA s may be foud withi a umber of geeratios that depeds logarithmically o the populatio size. I particular, our first mai result says that the umber of geeratios back to a MRCA is about lg, where lg deotes logarithm to base 2. Theorem 1. Let T deote the umber of geeratios, coutig back i time from the preset, to a MRCA of all preset-day idividuals, i a populatio of size. The T lg P 1 as. This cotrasts dramatically with the oe-paret situatio. For example if is 1 millio, the the oe-paret MRCA ( Eve ) is expected to occur about 2 millio geeratios ago, whereas a two-paret MRCA occurs with high probability withi the last 20 geeratios or so. Also, the variability i the oe-paret situatio is such that the actual time to the MRCA may easily be as small as half the expected time or as large as double the expected time, say, eve i arbitrarily large populatios. I cotrast, the time to a MRCA for the two-paret model is much less variable. For example, if the populatio is large eough, it is very ulikely that a radom realizatio of the two-paret MRCA time will differ from lg by eve oe percet. This paper also addresses a secod related questio. Imagie tracig back through the two-paret geealogy. Accordig to Theorem 1, after about lg geeratios, we will reach the most recet geeratio that cotais a CA. That geeratio might cotai just oe CA, or it might cotai more tha oe. I ay case, if we cotiue tracig back further through successive geeratios, the the title of CA becomes much less of a prestigious distictio. For example, both parets of a CA will be CA s, ad all gradparets of a CA will be CA s, ad so o. Evetually, i a give geeratio, may (ad i fact most) of the idividuals will be CA s. At some poit we reach a geeratio i which some idividuals are CA s (havig all preset-day idividuals as descedats) ad some are extict (havig o preset-day idividuals as descedats), but o idividual is itermediate (havig some but ot all preset-day idividuals as descedats). That is, at this poit, everyoe who is ot extict is a CA. This coditio persists forever as we trace back i time: every idividual is a CA or 3

4 S 0/ 0/ S S {4,5} S 0/ S S S S {4,5} 0/ {1,2,3,4} {4,5} {1,2,3,4} {2,3} S 0/ {3,5} {2,3} {4} {5} {1,2,4} Figure 1. A example illustratig the model. Here the fourth idividual i geeratio 2 is a CA. By geeratio 5, all idividuals are CA s or extict: idividuals 1,4, ad 5 are CA s, ad idividuals 2 ad 3 are extict. extict. The ext result shows that this coditio is reached very rapidly i the model studied here. Theorem 2. Let U deote the umber of geeratios, coutig back i time before the preset, to a geeratio i which each idividual is either a CA of all preset-day idividuals or a acestor of o preset-day idividual. Let γ deote the smaller of the two umbers satisfyig the equatio γe γ =2e 2, ad let ζ = 1/(lg γ) The U (1 + ζ)lg P 1 as. Thus, withi about 1.77 lg geeratios, a tiy amout of time i compariso with the order time required to get a oe-paret CA, everyoe i the populatio is either a CA of all preset-day idividuals or extict. Figure 1 shows a small example to illustrate the defiitios ad statemets. The populatio size is 5. At the bottom of the figure is geeratio 0, the preset. Goig up i the graph correspods to goig back i time, so that the top row is geeratio 5. For each idividual I i each previous geeratio, we calculate the set of preset-day idividuals (idividuals i geeratio 0) that are descedats of I. For example, the set of preset-day descedats of idividual #1 i geeratio 1 is{3, 5}. The calculatios propagate backward i time accordig to the rule: the set of descedats of a idividual I is the uio of the sets of descedats of the childre of I. For example, the set of preset-day descedats of idividual #4 i geeratio 2 is the uio {3, 5} {5} {1, 2, 4}, which is the whole populatio S. Thus, idividual #4 i geeratio 2 is a CA of the set S of all preset-day idividuals. Cotiuig backward i time, at geeratio 5 we reach the stage where each 4

5 idividual has as descedats either the whole populatio S or the empty set. That is, each idividual i geeratio 5 is either a CA or extict, havig as descedats either everybody or obody from the set of preset-day idividuals, ad all geeratios prior to geeratio 5 also have this property. I the example show, T 5 = 2 ad U 5 =5. What is the sigificace of these results? A applicatio to the world populatio of humas would be a obvious misuse. For example, we would ot claim that a commo acestor of every preset-day huma may be foud withi the last lg geeratios. Eve if we took to be 5 billio, this would imply a CA just about 32 geeratios ago perhaps 500 years or so. A importat source of the iapplicability of the model to this situatio is the obvious o-radom ature of matig i the history of makid. For example, parets are much more likely to live withi a few miles of their childre tha a thousad miles away or halfway aroud the world. So the model studied here is too simple to be directly applicable to the evolutio of makid as a whole. I such complicated situatios, the results soud a ote of cautio: if the logarithmic time to CA s seems patetly implausible, the at least oe of the assumptios of the model, such as the radom matig assumptio, must be causig a great deal of trouble. O the other had, it would be iterestig to kow whether there are simpler real-life situatios i which the assumptios of the model do apply reasoably well ad the theorems provide reasoably accurate quatitative descriptios. Perhaps a relatively homogeeous populatio lackig discerible structures (geographic or otherwise) that iteract strogly with reproductio would be a promisig cadidate. The radom time aalyzed i Theorem 2 seems of atural iterest i this process ad may also be pertiet to certai questios about species trees or populatio trees (as opposed to gee trees ). I may cotexts the species tree is cosidered to be the real object of iterest, ad we use geetic data ad gee trees to attempt to lear about the species tree. For example, for humas, chimpazees, ad gorillas, is the true species tree (HC)G, (HG)C, or (CG)H? Roughly, the coceptual framework of this questio is as follows. There were two speciatio evets that split a sigle species acestral to humas, chimpazees, ad gorillas ito the three separate moder species. The tree (HC)G, for example, says that the first such split separated the subpopulatio that evetually became moder gorillas from the remaider, which later split to become moder humas ad chimpazees. Ufortuately, more precise defiitios of the cocept of species tree that remai useful i difficult or uclear cases seem hard to come by. Oe might adopt the viewpoit that the proper startig poit for a defiitio of species tree is the full two-paret geealogy of all preset-day idividuals. Give such a defiitio, if we kew all details of this geealogy, the we could read off a aswer to the H, C, ad G questio (the aswer might be oe of the 3 choices above that is, the species tree is ot well defied or at least ot bifurcatig). Oe iterpretatio of the time U is as follows. Suppose we imagie a case where evolutio really proceeded accordig to a eat successio of speciatio evets. Uder a certai reasoable defiitio of a species tree, if the times betwee those speciatio evets exceed U, the the species tree is guarateed to be well defied ad coicide with the history of speciatio evets. This idea will be discussed more fully elsewhere. A caveat to forestall potetial misuderstadig: This paper is ot about geetics. That is, it is ot about who gets what gees; it is about somethig more primitive, amely, the acestor-descedat relatioship. Oe-paret models are appropriate i tracig the history of a sample of orecombiig gees or small bits of DNA; a sigle ucleotide desceds from a sigle ucleotide from either the mother or father, but ot both. Here we are cosiderig 5

6 acestry i the more commo, demographic sese of the word, as applied to people, for example, rather tha gees. Previous geetics research that is somewhat related, although still very differet from the preset study, cosiders models icorporatig recombiatio. This type of model has bee ivestigated i a umber of papers, icludig those of Hudso (1983) ad Griffiths ad Marjoram (1997). The history of a sample of DNA sequeces may be described by a collectio of geealogies, with each ucleotide positio i the DNA havig its ow oe-paret geealogy. The geealogies for two positios that experiece o recombiatio betwee them will be cogruet, with the paths of the two geealogies goig back through the same idividuals, whereas a recombiatio betwee two positios causes the geealogies of those positios to differ. Each of the geealogies i the collectio will have its ow MRCA (the ucleotide at its root), which may occur i differet idividuals. Each of these idividuals will be a CA i the sese cosidered i this paper, but the most recet of these idividuals is geerally ot a MRCA i our sese. Our MRCA is more recet, sice the paths from acestors to descedats cosist of all potetial paths for gees to be trasmitted, ad may iclude paths that did ot happe to be take by ay gees. No previous results about these geetic models have bee similar to the results here, for example, i gettig times of order log. This is ot surprisig, sice the asymptotics would require a assumptio that the sequece legths ad the umber of recombiatios ted to ifiity. This is aother maifestatio of the statemet that the questios we are ivestigatig here are ot fudametally geetics questios. There is some previous work o the process we study here ad related processes. Two papers of Kämmerle (1989, 1991) itroduce a geeral class of two-paret (called bisexual i those papers) versios of the Wright-Fisher ad other processes. These papers focus o two mai questios. First, they aalyze the probability of extictio of a set of idividuals i the preset geeratio, that is, the probability that the set of idividuals evetually has o descedats i some future geeratio. Secod, i a two-paret versio of the Mora model, they study the umber R (t) of idividuals t geeratios ago who have at least oe descedat i the preset geeratio. Kämmerle (1989) fids that the Markov chai {R (t) :t =0, 1,...}, suitably ormalized ad suitably iitialized (with the iitializatio essetially requirig that the chai is started i steady state), coverges weakly as to a discrete-time Orstei-Uhlebeck process. Möhle (1994) both geeralizes ad refies the results of Kämmerle. I particular, Möhle provides a detailed aalysis of the extictio probabilities i a two-paret Wright-Fisher model that approximates the probabilities up to o(1/). He also establishes weak covergece i a geeral class of two-paret models, icludig a Orstei-Uhlebeck limit for the two-paret Wright-Fisher process. Möhle also has a umber of other papers i press, icludig oe that relaxes the assumptio of costat populatio size. These previous results are complemetary to the results i this paper. The previous papers cosidered idividuals who have at least 1 descedat i a give future geeratio. Here we cosider CA s, who have as descedats all members of the future geeratio. The previous results about the process {R (t)} apply to large t, that is, to the behavior the process may geeratios before the preset, with the process i steady state. Here we focus o the behavior of a related process at small (i.e. recet) times, startig far away from steady state. We show that at about t =1.77 lg geeratios before the preset, with high probability the R (t) idividuals who have at least 1 preset-day descedat are all i fact CA s. 6

7 2 Simulatios Table 1 presets a small simulatio study cosistig of 25 trials each for = 500, = 1000, = 2000, ad = Two umbers are reported for each trial: T, the umber of geeratios back to a MRCA, ad U, the geeratio at which every idividual is either a CA or extict. = 500 = 1000 = 2000 = Table 1. A small simulatio study. For each of four populatio sizes, the two times T ad U are reported for 25 trials. I these simulatios the distributio of the time back to a MRCA is ideed quite cocetrated aroud the value lg, which is early 9 for = 500, early 10 for = 1000, ad so o. Thus, the simulatio results show that the asymptotic ( ) statemet of Theorem 1 is ot so asymptotic, i that it describes the situatio well eve for rather small values of. The behavior predicted by Theorem 2 is also reflected reasoably well i the simulatios, although oe might have guessed a umerical costat closer to 2 rather tha 1.77 from this small study. 7

8 3 Proofs 3.1 Geeral ideasad tools We start with the observatio that although Theorems 1 ad 2 are phrased i terms of coutig geeratios back i time from the preset util some coditio obtais, these results may be proved by coutig forward i time from a fixed geeratio. For example, the evet {T m} requires that a CA of all idividuals i geeratio 0 may be foud amog geeratios 1, 2,..., m. This is equivalet to requirig that if we start with geeratio m ad trace forward i time, the some idividual i geeratio m becomes a CA of all idividuals i some geeratio t { m +1, m +2,...,0}. So we will cout geeratios forward i time, ad for coveiece let us reumber geeratios so that the iitial geeratio is geeratio 0. The populatio at geeratio t 0 cosists of idividuals deoted by I t,1,i t,2,...,i t,. We ca picture I t,1,i t,2,...,i t, as dots i a array as i Figure 1, with I t,j beig the jth dot i row t. The associatio of a umber j to idividual I t,j is a arbitrary labelig of the idividuals withi geeratio t. Assiged oly as a meas of referrig to idividuals, the labels have o sigificace i the model, which does ot order the idividuals withi a geeratio. Let µ t,1,ν t,1,µ t,2,ν t,2,...,µ t,,ν t, be idepedet ad uiformly distributed o the set {1,...,}. We iterpret µ t,j ad ν t,j as labels of the parets of idividual I t,j ; that is, the parets of I t,j are I t 1,µt,j ad I t 1,νt,j. Defiig a sequece of radom sets G0 i, Gi 1,... recursively by Gi 0 = {i} ad G i t = {j : µ t,j G i t 1 or ν t,j G i t 1}, G i t is the set of labels of the descedats of I 0,i i geeratio t. Let G i t deote the cardiality of G i t. The coditioal probability that idividual I t+1,j has at least oe paret amog the G i t members of G i t is P ({µ t+1,j G i t} {ν t+1,j G i t} G i t)=(g i t/)+(g i t/) (G i t/)(g i t/). The process {G i t : t =0, 1,...} is a Markov chai with trasitio probabilities (G i t+1 G i t) Bi, 2Gi t ( G i t ) 2, (1) where Bi(, p) deotes the biomial distributio for the umber of successes i idepedet trials each havig success probability p. Throughout the proof, {G t } will deote a Markov chai with trasitio probabilities as i (1), although i differet parts of the proof we will cosider differet possible iitial values G 0. For example, takig G 0 = 1 correspods to followig the descedats of a particular idividual i geeratio 0. I the early stages of the process, while G t remais small relative to, i view of (1) the coditioal distributio of G t+1 give G t is early Poisso(2G t ), that is, the Poisso distributio with mea 2G t. I other words, while {G t } remais small, it evolves early as a Galto-Watso brachig process {Y t } with offsprig distributio Poisso(2). Kämmerle (1991) gave a formal statemet of a result of this ature. A special case of his result says that for fixed u, the joit distributio of (G 0,G 1,...,G u ) coverges to that of (Y 0,Y 1,...,Y u )as. For our purposes, we will use the followig result that allows us to approximate 8

9 probabilities for the G process by those for the Y process up to a higher order of accuracy ad over loger itervals of time that may have radom legths. Lemma 3. Let Y 0,Y 1,... deote a Galto-Watso brachig process with offsprig distributio Poisso(2). Suppose that Y 0 = G 0 =1. Defie τb Y = if{t : Y t b} ad τ0b Y = if{t : Y t =0or Y t b}, with correspodig defiitios for τb G ad τ0b G.As,ifm ad b satisfy mb 2 = o(), the P {τ G b >m} = P {τ Y b >m}(1 + o(1)) (2) ad P {τ G 0b >m} = P {τ Y 0b >m}(1 + o(1)). (3) Proof. A straightforward calculatio bouds the likelihood ratio { ) } L(y x) := P {G t+1 = y G t = x} P {Y t+1 = y Y t = x} = P Bi (, 2x ( x2 = y 2 e 2x 1 2x ) y P {Poisso(2x) =y} + x2 2, so that ( log L(y x) 2x +( y) 2x ) + x2 2 (x 2 +2xy)/. This holds wheever the deomiator P {Y t+1 = y Y t = x} is positive, that is, for all x>0 ad y 0, ad also for x = y = 0. Thus, for all such pairs of x ad y satisfyig x<bad y<b,wehave log L(y x) 3b 2 /. A similar calculatio gives the lower boud log L(y x) 5b 2 /(2)[1 + O(b/)], so that log L(y x) 3b 2 / for sufficietly large. Soifx 1,...,x m are all less tha b, the P {G 1 = x 1,...,G m = x m } = P {G 1 = x 1 G 0 =1} P {G m = x m G m 1 = x m 1 } = P {Y 1 = x 1,...,Y m = x m }L(x 1 1) L(x m x m 1 ) P {Y 1 = x 1,...,Y m = x m }e 3mb2 / ad Thus, P {G 1 = x 1,...,G m = x m } P{Y 1 = x 1,...,Y m = x m }e 3mb2 /. P {τb G >m} = P {G 1 = x 1,...,G m = x m } (4) 0 x 1 <b 0 x m<b 0 x 1 <b 0 x m<b = P {τ Y b >m}e 3mb2 / P {Y 1 = x 1,...,Y m = x m }e 3mb2 / 9

10 ad, similarly, P {τ G b >m} P {τ Y b >m}e 3mb2 /, so that, by the assumptio that mb 2 = o(), we obtai P {τ G b >m} = P {τ Y b >m}(1 + o(1)). This proves (2). The proof of (3) uses the same reasoig, with the summatios i (4) ragig over 0 <x t <brather tha 0 x t <b. The previous result will be useful because the Poisso Galto-Watso process is simple ad well uderstood. The ext lemma records a few well kow items for future referece. Lemma 4. Let Y 0,Y 1,... deote a Galto-Watso process with offsprig distributio Poisso(2). Defie the momet geeratig fuctio ψ(z) =E(z Y 1 )=e 2+2z. The extictio probability ρ = P {Y t =0for some t} is the smaller of the two solutios of ψ(ρ) =ρ, ad ρ = γ/2, where γ is as defied i Theorem 2. The t-fold compositio ψ t = ψ ψ satisfies ψ t (z) ρ for all 0 z ρ. The relatio ρ = γ/2 is cofirmed by comparig the defiitios of ρ ad γ. Despite the simple relatioship, we will keep the two differet letters i our otatio for coceptual clarity. Defiig g t = G t /, wehave E(g t+1 g t )=2g t g 2 t = g t (2 g t ). (5) That is, if the fractio of descedats of a give idividual is curretly g t, it is expected to multiply by a factor of 2 g t i the ext geeratio. For example, i the early stages of the process whe the fractio g t is small, it early doubles i expectatio i the ext geeratio. For very small g t (of the order 1/, for example) the radom variability is large; for example, the process could easily go extict. This is whe it is most useful to approximate the G process by the Poisso(2) Galto-Watso process. O the other had, for larger values of g t, the multiplicatio factor g t+1 /g t, although expected to be somewhat smaller, has much less variability. The deviatios of this factor from its expected value are bouded probabilistically by large deviatios iequalities for the biomial distributio. We will use the followig iequality of Berstei (1946) as a basic tool. Lemma 5.[Berstei s iequality] If X Bi(, p) ad r > 0, the { r 2 } P {X p + r} exp. (6) 2p(1 p)+(2/3)r Sice X Bi(, 1 p), the right side of (6) is also a upper boud for the probability P {X p r}. 3.2 Proof of Theorem 1 Outlie. The proof will be divided ito several parts. We start from geeratio 0 ad trace forward i time. Stage 1: By the ed of stage 1, we idetify a idividual I i geeratio 0 who has a umber of descedats that is small compared to, but large eough so that I is ulikely ever to become extict. I particular, we look for a geeratio t such that some idividual I i 10

11 geeratio 0 has at least lg 2 () descedats i geeratio t. With probability approachig 1, this happes i time o(lg ), egligible compared with lg ; this is show by usig Lemma 3 to approximate our process by a Poisso Galto-Watso process. The rest of the proof will show that with probability approachig 1, idividual I becomes a CA withi (1 + ɛ)(lg ) geeratios, where ɛ is a arbitrary positive umber. Stage 2: Let β (0, 1). Stage 2 follows the descedats of I util reachig a geeratio cotaiig at least β descedats. I view of (5), sice β is a small fractio of for large, throughout Stage 2 the umber of descedats i a geeratio is expected to be early double the umber of descedats i the previous geeratio. Ad lg 2 () is large eough so that the multiplicatio factor will be very close to its expected value, with high probability. So stage 2 should ot take much more tha about lg( β )=β lg() geeratios. Stage 3: This stage brigs the cout of descedats of I up from β to (1/2). Sice the fractio of descedats durig stage 3 stays below 1/2, the expected multiplicatio factor is at least 2 1/2 = 3/2. Agai, this multiplicatio factor is very reliable, so that with high probability stage 3 takes o more tha about log 3/2 {(/2)/( β )} geeratios. We ca make this a arbitrarily small fractio of lg by choosig β close eough to 1. Stage 4: Now we switch to lookig at the fractio B t of idividuals i a geeratio who are ot descedats of idividual I. This fractio is expected to square each geeratio. This causes B t to decrease very quickly. Fixig α (1/2, 2/3), we show that stage 4, which takes the fractio B t from 1/2 dow to α, takes oly order lg lg time. Stage 5: This completes the process, edig whe the B process hits 0, ad idividual I has become a CA. We show that this takes just oe geeratio with high probability. Upper boud: Combiig the results of Stages 1 through 5 gives the probabilistic upper boud lim P {T (1 + ɛ)lg} =1. Lower boud: Here we show that lim P {T (1 ɛ)lg} = 1. This is doe by usig Berstei s iequality to prove a assertio of the followig form: For positive r ad δ, oce the process of descedats of ay give idividual reaches a power r of, it is very ulikely to icrease by a factor of more tha 2 + δ i a geeratio, whereas it would have to do so i order to have T < (1 ɛ)lg. Stage 1. Here we will show that with high probability, withi a umber of geeratios egligible compared to lg, we ca fid a geeratio with at least lg 2 idividuals who share a commo acestor. For simplicity we give a crude argumet that circumvets the eed to cosider ay depedece amog the processes {G i t : t 0} startig from differet idividuals I 0,i. This could also be doe alog the lies of the argumet i Lemma 19 below, where we eed to cofrot this depedece. Lemma 6. Defie τ b = if{t : G t b}. Assumig that G 0 =1, lim if P {τ lg 2 3lglg} > 0. Proof. Let b ad m deote lg 2 ad 3lglg, respectively. Let {Y t } be a Galto-Watso process with offsprig distributio Poisso(2), ad defie M t = Y t 2 t. The process {M t } is a oegative martigale that coverges almost surely to a limit M, say, with 11

12 P {M =0} = ρ<1. Note that P {τb Y >m} P {Y m <b} = P {M m <b2 m }. Therefore, usig Fatou s lemma ad the assumptio that b2 m 0, lim sup P {τb Y >m} lim sup P {M m <b2 m } P(lim sup{m m <b2 m }) = P {M m <b2 m ifiitely ofte} P {M =0} = ρ<1. By Lemma 3, P {τ b >m} = P {τ Y b >m}(1 + o(1)) as. Therefore, lim sup P {τ b >m} lim sup P {τ Y b >m} ρ<1. So lim if P {τ b m} 1 ρ>0. Propositio 7. Let G i t deote the umber of descedats i geeratio t of idividual I 0,i (the ith idividual i geeratio 0), ad let G t = max 1 i {G i t}. Defie τb G = if{t : G t b}. The τ G lg 2 = o P (lg ). Proof. We use a geometric trials argumet. Let m = 3lglg, ad choose a sequece {k } with k ad k m = o(lg ). Perform a sequece of k trials as follows. For the first trial, start with idividual I 0,1, ad follow his progey for m geeratios. We say the trial is a success if I 0,1 has at least lg 2 descedats i geeratio m ; by Lemma 6this happes with probability at least c, say, where c>0. If the trial is a failure, start a ew trial, followig the progey of idividual I m,1 for m more geeratios. Ad so o. We stop at the first success, havig foud a idividual with at least lg 2 descedats. The probability that this sequece of trials fails to termiate by geeratio k m is at most (1 c) k, which teds to 0. Thus, with probability tedig to 1, there is a κ {0,...,k 1} such that idividual I κm,1 has at least lg 2 descedats i geeratio (κ +1)m. Let I deote ay acestor of I κm,1 i geeratio 0. We will show i the remaider of the proof that for each ɛ>0, with probability tedig to 1 as, idividual I becomes a CA withi (1 + ɛ)lg geeratios. Stage 2. The followig simple cosequece of Berstei s iequality will be a coveiet tool. Lemma 8. If δ 3/4 ad G t δ/20, the P {G t+1 (2 δ)g t G t } exp( δ 2 G t /5). The ext result shows that the probability that Stage 2 takes more tha lg geeratios approaches 0 as. I fact, we show that this probability is o(1/); this will be used i the proof of Theorem 2. Propositio 9. Assume that G 0 lg 2, ad let 0 <β<1. Defie T 2 = if{t : G t β }. The P {T 2 > lg } = o(1/) as. Proof. Take 0 <δ<3/4 such that lg(2 δ) >β, ad defie ( ) β b() = log 2 δ lg 2. 12

13 Note that b() β lg lg, lg(2 δ) at least for 3, so that P {T 2 > lg } P {T 2 >b()}. We will show that P {T 2 >b()} = o(1/). The iequality T 2 >b() implies that G t+1 < (2 δ)g t for some 0 t b() 1. The first such t must also satisfy G t lg 2. Thus, b() 1 P {T 2 >b()} P {G t+1 < (2 δ)g t,g t lg 2, T 2 >b()} b() 1 t=0 t=0 P { G t+1 < (2 δ)g t, lg 2 G t β}. However, β δ/20 for sufficietly large. Therefore, o the evet {lg 2 G t β },we may apply Lemma 8 to obtai ( ) P {G t+1 < (2 δ)g t G t } exp δ2 5 lg2 = (δ2 /5)(lg e)(lg ). Thus, P {T 2 >b()} b() (δ2 /5)(lg e)(lg ) = o(1/) as. Stage 3. This stage starts i a geeratio i which the umber of descedats of I is just over β ad eds whe the umber of descedats i a geeratio reaches (1/2). Defiig g t = G t /, wehavee(g t+1 g t )=g t (2 g t ). The idea is that if g t 1/2, the i the ext geeratio g t is expected to multiply by a factor of 2 g t 3/2. So with high probability, throughout stage 3, at each geeratio the umber of descedats will multiply by at least 2, say, sice 2 < 3/2. So to get from β to (1/2), we should eed at most log 2 (1/2)1 β = 2[(1 β)lg 1] geeratios. Propositio 10. Assume G 0 β, ad defie T 3 = if{t : G t (1/2)}. The P {T 3 > 2(1 β)lg} = o(1/) as. Proof. The proof is similar to that of Propositio 9. For β G t /2, a straightforward calculatio usig Berstei s iequality gives P {G t+1 2G t G t } exp(.001g t ) exp(.001 β ). Note that log 2 {(/2)/β } = 2(1 β)lg 2. So if T 3 > 2(1 β)lg, the we must have G t+1 2G t for some t<2(1 β)lg satisfyig β G t /2. Thus, P {T 3 > 2(1 β)lg} 2(1 β)(lg ) exp(.001 β )=o(1/) as. 13

14 Stage 4. Let B t deote 1 G t /, the fractio of idividuals i geeratio t who are ot descedats of the chose idividual I. The (B t+1 B t,b t 1,...) 1 Bi(, B2 t ), (7) sice a idividual is ot a descedat of I whe both of his parets fail to be descedats of I. Fix α (1/2, 2/3). Stage 4 takes the B t process from 1/2 dowto α. The idea is this. Sice E(B t+1 B t )=Bt 2, we expect B t to square each geeratio. We will show that the probability P {B t+1 B 3/2 t } is small throughout stage 4 (ote 3/2 < 2). This will be good eough, sice if B t+1 <B 3/2 t holds throughout stage 4, the stage 4 is completed i order lg lg time. Propositio 11. Cosider a process B 0,B 1,... satisfyig (7), ad suppose B 0 1/2. Let α (1/2, 2/3) ad defie T 4 = if{t : B t α }. The P {T 4 2lglg} = o(1/) as. Proof. By Berstei s iequality, P {B t+1 B 3/2 t B t } = P {Bi(, Bt 2 ) B 3/2 t B t } { 2 Bt 3 (1 B 1/2 t ) 2 } exp 2Bt 2(1 B2 t )+(2/3)B3/2 t (1 B 1/2 t ) { B t (1 B 1/2 t ) 2 } = exp 2(1 Bt 2)+(2/3)B 1/2 t (1 B 1/2. t ) If α B t 1/2, the (1 B 1/2 t ) , ad B t (1 B 1/2 t ) 2 2(1 Bt 2)+(2/3)B 1/2 t (1 B 1/2 t ) α 2+(2/3) (the last iequality holdig for 6 2/α ), so that P {B t+1 B 3/2 t { B t } exp (3/2)α}. α/ (3/2)α For 2, if B 0 1/2 ad B t+1 B 3/2 t B 2lglg 1 α. Therefore, for t =0, 1,..., 2lglg 1, the {T 4 > 2lglg } {B 2lglg > α } 2lglg 1 t=0 {B t+1 >B 3/2 t, α <B t 1/2}, so that P {T 4 > 2lglg } 2lglg 1 t=0 2lglg exp P {B t+1 B 3/2 t, α <B t 1/2} { (3/2)α} = o(1/). 14

15 Stage 5. This stage starts with the {B t } process below α ad eds whe it hits 0. We show that with high probability this takes just oe geeratio. Propositio 12. Suppose B 0 α. The P {B 1 =0} 1 as. Proof. Sice B 1 (1/)Bi(, B0 2 ) ad 2α >1, we have P {B 1 =0} =(1 B0 2) (1 2α ) 1. Upper boud. Propositio 13. For each ɛ>0, P {T > (1 + ɛ)lg} 0 as. Proof. Defie T 1 to be the time at which stage 1 eds. The we kow that T 1 is fiite with probability 1, ad, for arbitrary positive ξ, P {T 1 >ξlg } 0as. At the ed of stage 1 we have foud a idividual I, say, i geeratio 0 who has at least lg 2 () descedats i geeratio T 1. Let G t deote the umber of descedats of I i geeratio t, ad let τ(b) deote if{t : G t b}. Our previous results have show that Thus, P {τ( β ) T 1 > lg } = o(1/), P {τ(/2) τ( β ) > 2(1 β)lg τ( β ) < } = o(1/), P {τ( 1 α ) τ(/2) > 2lglg τ(/2) < } = o(1/), P {τ() τ( 1 α ) > 1 τ( 1 α ) < } = o(1). P {T >ξlg +lg + 2(1 β)lg + 2 lg lg +1} P {T 1 >ξlg } + P {T 1 <,τ( β ) T 1 > lg } + P {τ( β ) <,τ(/2) τ( β ) > 2(1 β)lg} + P {τ(/2) <,τ( 1 α ) τ(/2) > 2lglg} + P {τ( 1 α ) <,τ() τ( 1 α ) > 1} = o(1) + o(1/) + o(1/) + o(1/) + o(1) = o(1). Give ɛ>0, takig ξ ad β such that ξ + 2(1 β) <ɛ, we see that P {T > (1 + ɛ)lg} 0. Lower boud. We will use Berstei s iequality i the followig form. Lemma 14. For δ 3/2, P {G t+1 (2 + δ)g t G t } exp[ δ 2 G t /5]. Propositio 15. For each ɛ>0, P {T < (1 ɛ)lg} 0. 15

16 Proof. Fix ɛ (0, 1). Proceedig forward i time from geeratio 0, we wat to show that the probability that oe of the idividuals i geeratio 0 becomes a CA before geeratio (1 ɛ)lg teds to 1 as. Defie G 0 = 1 ad (G t+1 G t,...,g 0 ) Bi(, 2G t / (G t /) 2 ). Here we thik of G t as the umber of descedats of idividual I 0,1 i geeratio t. Fix r (0,ɛ) so that 2 (1 r)/(1 ɛ) (2, 3.5). Let { G t } evolve like {G t } except that it is trucated (or reflected ) below at the value r. That is, ( G t+1 G t,..., G 0 ) max Bi, 2 G t ( ) 2 Gt, r. Defiig τ G = if{t : G t = } ad τ G = if{t : G t = }, obviously P {τ G u} P {τ G u} for all u. Sice G 0 = r,ifτ G (1 ɛ)lg, the we must have G t+1 G t 2 (1 r)/(1 ɛ) for some t< (1 ɛ)lg. Defiig δ =2 (1 r)/(1 ɛ) 2 (0, 3/2), by Lemma 14 the probability of this is at most (1 ɛ)lg exp( δ 2 r /5), which is o(1/) as. Thus, we have show that the probability that idividual I 0,1 has become a CA by geeratio (1 ɛ)lg is o(1/). So the evet that at least oe of the idividuals i geeratio 0 becomes a CA by geeratio (1 ɛ)lg is a uio of such evets of probability o(1/), ad hece has probability that teds to 0 as. 3.3 Proof of Theorem 2 Idea. The idea of the proof is as follows. Defie t = (ζ ɛ)lg ad u = (ζ + ɛ)lg. For each i =1,...,, the process {G i t : t =0, 1,...} follows the descedats of idividual I 0,i.We are waitig util all of the processes {G 1 t },...,{G t } have reached either 0 or (some will reach 0 ad some will reach ). The key igrediet of the argumet is this assertio: With high probability, there are may i s such that G i t [1, lg 2 ()] ad there is o i such that G i u [1, lg 2 ()]. This follows from Lemma 3 together with a aalysis of the Galto-Watso process with offsprig distributio Poisso(2). For a upper boud, cosider the situatio at time u. Some of the processes have become extict ad reached 0, ad we are just waitig for the other, oextict processes to reach 0 or. The key assertio says that with high probability, all of the oextict processes have reached values above lg 2 (). This level is high eough so that with high probability these processes will all icrease predictably ad reach withi (1 + ɛ)lg additioal geeratios; this was show i the proof of Theorem 1. So with high probability, U u +(1+ɛ)lg. For a lower boud, the key assertio states that with high probability may of the processes are i the iterval [1, lg 2 ()] at time t.itisvery ulikely that all of these will go extict. Furthermore, sice these processes are startig from at most lg 2 () at time t, with high probability it will take more tha (1 ɛ)lg additioal geeratios for ay of them to reach. SoU >t +(1 ɛ)lg with high probability. A brachig process result. Lemma 16. Let {Y t } be a Galto-Watso process whose offsprig distributio is Poisso with mea 2, startig at Y 0 =1. Defie γ as i Theorem 2, ad let b 1,b 2,... be positive itegers 16

17 satisfyig lg(b t )=o(t) as t. The 1 lim t t lg P {1 Y t b t } =lg(γ) Proof. We use a umber of results from chapter 1 of Athreya ad Ney (1972). First, the Mootoe Ratio Lemma says that for each k there is a λ k < such that Also, Λ(s) := P {Y t = k} P {Y t =1} λ k as t. λ k s k < for all s (0, 1). Fially, usig the otatio ad facts collected i Lemma 4, we have k=1 P {Y t =1} = ψ t(0) = ψ [ψ t 1 (0)]ψ t 1(0) = ψ [ψ t 1 (0)]P {Y t 1 =1}, so that P {Y t =1} P {Y t 1 =1} ψ (ρ) =2ρ = γ. I particular, (1/t)lgP{Y t =1} lg γ. For s (0, 1), b t P {Y t = k} P {Y t =1} λ k k=1 k=1 b t b t P {Y t =1}s bt λ k s k k=1 P {Y t =1}s bt Λ(s). (8) If we take s close to 1 (e.g. s =1 b 1 t, say), the the term s bt will remai bouded ad preset o difficulty. So we would like to kow how Λ(s) grows as s 1. Defie ϕ to be the iverse fuctio ψ 1, ad ϕ k to be the k-fold compositio ϕ ϕ. By equatio (6) o page 12 of Athreya ad Ney (1972), for each s (ρ, 1), Λ(ϕ(s)) = γ 1 [Λ(s) Λ(e 2 )] γ 1 Λ(s). Therefore, sice ρ<1/2 <ϕ(1/2) <ϕ 2 (1/2) < < 1, Λ(ϕ k (1/2)) γ k Λ(1/2). However, sice ψ (1) = 2, we may choose a umber Φ so that ϕ k (1/2) 1 (1.9) k Φ ad, therefore, Λ(1 (1.9) k Φ) γ k Λ(1/2) 17

18 hold for all sufficietly large k. From this, it follows that Λ(1 y) Λ(1/2)(y/Φ) (lgγ)/(lg1.9) y 2lgγ holds for all sufficietly small positive y. Now substitutig s =1 b 1 t i (8), there is a fiite costat C such that b t k=1 P {Y t = k} Cγ t Λ(1 b 1 t ) Cγ t b 2lg(1/γ) t. Thus, as log as b t grows subgeometrically, that is, lg(b t )=o(t), we have lim t (1/t)lgP {1 Y t b t } lg(γ). Combiig this with the fact that lim t (1/t)lgP {Y t =1} =lg(γ) completes the proof. Upper boud. Lemma 17. Let I 0,i deote idividual i i geeratio 0. Defie G i t to be the umber descedats of I 0,i i geeratio t; i particular, G i 0 =1for all i =1,...,. Also defie τ i 0,b = if{t : G i t =0or G i t b}, ad let The P (A ) 0 as. A = {τ i 0,lg 2 > (ζ + ɛ)lg}. (9) i=1 Proof. Defie τ Y 0b = if{t : Y t =0orY t b}. Sice {τ Y 0b >t} {1 Y t <b}, Lemma 16ad (3) give lim (1/t)lgP {τ i 0b >t} lg(γ) if lg(b) =o(t) ad tb 2 = o(). (10) Lettig ɛ>0 ad applyig (10) to t =(ζ + ɛ)(lg ) ad b =lg 2 () gives lg P {τ i 0,lg 2 > (ζ + ɛ)(lg )} (lg(γ)+δ)(ζ + ɛ)(lg ) () for all δ ad all sufficietly large. Takig δ sufficietly small, from the defiitio of ζ we see that P {τ i 0,lg 2 > (ζ + ɛ)(lg )} = o(1/) as, () so that P (A )=o(1). We have show that, with high probability, all idividuals i geeratio 0 have either o descedats or more tha lg 2 () descedats i geeratio (ζ + ɛ)lg for ɛ>0. Next we will show that for ay give ɛ>0, with high probability, each idividual havig more tha lg 2 () descedats i geeratio (ζ + ɛ)lg becomes a CA withi (1 + ɛ)lg additioal geeratios. Most of the work required to prove this has already bee doe i the proof of 18

19 Theorem 1; the extra igrediet is the followig lemma, which takes a closer look at stage 5. We retai the defiitio B t =1 (G t /) from above. Lemma 18. Let α (1/2, 2/3) ad take k(α) > 1/(2α 1). Suppose that B 0 α ad defie T 5 = if{t : B t =0}. The P {T 5 >k(α)} = o(1/) as. Proof. Sice (B t+1 B t ) 1 Bi(, B2 t ), o the evet {B t α } we have P {B t+1 > 0 B t } =1 (1 B 2 t ) 1 (1 2B 2 t )=2B 2 t 2 1 2α, where the first iequality holds for sufficietly large (sice α>1/2 implies that B 2 t is arbitrarily small for sufficietly large ). I particular, P {0 <B t+1 α 0 <B t α } 2 1 2α for sufficietly large. (11) Next, by Berstei s iequality, o the evet {B t α }, P {B t+1 > α B t } = P { 1 Bi(, B2 t ) >Bt 2 +( α Bt 2 ) B t } [ 2 ( α Bt 2 ) 2 ] exp 2Bt 2(1 B2 t )+ 2 3 ( α Bt 2 [ ] ) exp 2 2α 2 1 2α α Sice the expoet is asymptotic to (3/2) 1 α, clearly P {B t+1 > α B t } exp[ 1 α ] o {B t α } for sufficietly large. Assumig that B 0 α, Therefore, sice we obtai k k 1 {B t > α } {B t α,b t+1 > α }. t=0 t=0 P {B t α,b t+1 > α } = E [ {B t α }P {B t+1 > α B t } ] exp[ 1 α ], Thus, usig (11) ad (12),. [ k ] P {B t > α } k exp[ 1 α ]. (12) t=0 ( k ) ( k ) P {T 5 >k} P {B t > α } + P {0 <B t α } t=0 t=0 k exp[ 1 α ]+(2 1 2α ) k. 19

20 Applyig this to k = k(α) > 1/(2α 1) gives the desired result. Proof of the upper boud i Theorem 2. Let U deote the time at which everyoe from geeratio 0 has become a CA or extict. Recall the defiitio of A from (9), ad let τ i (b) = if{t : G i t b}. Sice {U > (1 + ζ +2ɛ)lg} A [A c {U > (1 + ζ +2ɛ)lg}] A {τ i (lg 2 ) (ζ + ɛ)lg, τ i () > (1 + ζ +2ɛ)lg}, i=1 to show that P {U > (1 + ζ +2ɛ)lg} = o(1), by Lemma 17 it suffices to show that P {τ 1 (lg 2 ) (ζ + ɛ)lg, τ 1 () > (1 + ζ +2ɛ)lg} = o(1/). To see this, observe that the results of Stages 2 through 4 from the proof of Theorem 1 show that ad Lemma 18 gives Cosequetly, P {τ 1 ( β ) τ 1 (lg 2 ) > lg τ 1 (lg 2 ) < } = o(1/), P {τ 1 (/2) τ 1 ( β ) > 2(1 β)lg τ 1 ( β ) < } = o(1/), P {τ 1 ( 1 α ) τ 1 (/2) > 2lglg τ 1 (/2) < } = o(1/), P {τ 1 () τ 1 ( 1 α ) >k(α) τ 1 ( 1 α ) < } = o(1/). P {τ 1 () τ 1 (lg 2 ) > lg + 2(1 β)lg + 2 lg lg + k(α) τ 1 (lg 2 ) < } = o(1/). Choosig β sufficietly close to 1, we see that for ay give ɛ>0, Thus, as desired. P {τ 1 () τ 1 (lg 2 ) > (1 + ɛ)lg τ 1 (lg 2 ) < } = o(1/). P {τ 1 (lg 2 ) (ζ + ɛ)lg, τ 1 () > (1 + ζ +2ɛ)lg} P {τ 1 (lg 2 ) <,τ 1 () τ 1 (lg 2 ) > (1 + ɛ)lg} = o(1/), Lower boud. The proof goes as follows. First we show that at time t = (ζ ɛ)lg, there are may idividuals i who have G i t [1, lg 2 ]. The probability that all of these idividuals evetually become extict is egligibly small. I the probable evet that ot all of these idividuals become extict, the time U must wait for at least oe of them to become a CA. From the previous results we kow that this will take a additioal (1 ɛ)lg geeratios. Here is some otatio that will be used throughout the proof. Let t deote (ζ ɛ)lg. For 1 i, defie J i to be the evet {G i t [1, lg 2 ] for all t t }; we will also deote by J i 20

21 the idicator radom variable correspodig to this evet. Thus, J i = 1 meas that idividual i i geeratio 0 does ot become extict by time t ad that the umber of descedats of this idividual also remais relatively small (o more tha lg 2 ()) up to time t. At time t these idividuals still have a chace to become CA s, but they have ot yet made much progress toward doig so. The umber of such idividuals is N = i=1 J i. The ext lemma shows that there is little depedece betwee the umbers of descedats of differet idividuals i the early stages of the process. The lemma gives a upper boud o a probability; a similar lower boud may be obtaied, but it is ot eeded i the remaider of the proof. Lemma 19. P (J 1 J 2 ) [P (J 1 )] 2 (1 + o(1)) as. Proof. Cosider idividuals I 0,1 ad I 0,2, that is, idividuals 1 ad 2 i geeratio 0. Let A t deote the umber of idividuals i geeratio t who are descedats of I 0,1 but ot of I 0,2. Let C t deote the umber of idividuals i geeratio t who are descedats of I 0,2 but ot of I 0,1. Let B t deote the umber of idividuals i geeratio t who are descedats of both I 0,1 ad I 0,2. This otatio is local to this proof; i particular, B t has a differet meaig here tha it did i the proof of Theorem 1. So G 1 t = A t + B t ad G 2 t = C t + B t. Lettig H t =(A t,b t,c t ), the process {H t } is a Markov chai. For coveiece we use the otatio P H (a t,b t,c t ) for P {A t = a t,b t = b t,c t = c t }, P H (a t+1,b t+1,c t+1 a t,b t,c t ) for P {A t+1 = a t+1,b t+1 = b t+1,c t+1 = c t+1 A t = a t,b t = b t,c t = c t }, ad so o. We begi by observig that P (J 1 J 2 ) P (J 1 J 2 {B t = 0 for all t t }). (13) This is easy to see ituitively: If A t ad C t are both bouded by lg 2 () ad B t = 0, the the coditioal probability that B t+1 > 0 is at most 2A t C t / = O(lg 4 ()/). This suggests that for each s t, give the evet J 1 J 2, the coditioal probability that B t is positive for the first time at t = s is O(lg 4 ()/). Addig these probabilities over all s t = O(lg ) would the give P (B t > 0 for some t t J 1 J 2 )=O(lg 5 ()/). This is correct, ad the relatio [ ( lg 5 )] () P (J 1 J 2 )=P(J 1 J 2 {B t = 0 for all t t }) 1+O follows from a rather tedious calculatio whose details we omit. The calculatio bouds ratios of biomial probabilities i similar way to a argumet that is give later i this proof. Lettig L deote the iterval [1, lg 2 ], we wat a upper boud o the probability P (J 1 J 2 {B t = 0 for all t t }) = P H (a 1, 0,c 1,a 2, 0,c 2,...,a t, 0,c t ) a 1 L a t L c 1 L c t L = P H (a 1, 0,c 1 ) P H (a 2, 0,c 2 a 1, 0,c 1 ) a 1,c 1 L a 2,c 2 L P H (a t, 0,c t a t 1, 0,c t 1). a t,c t L 21

22 Defiig ad we may write α s = 2a s a s(a s +2c s ) 2, β s = 2a sc s 2, γ s = 2c s c s(c s +2a s ) 2, P H (a t, 0,c t a t 1, 0,c t 1 )=P {Bi(, α t 1 )=a t }P {Bi( a t, P {Bi( a t c t, β t 1 1 α t 1 γ t 1 )=0}. γ t 1 1 α t 1 )=c t } We wat to compare this to the aalogous probability for two idepedet {G t } processes, that is, to P G (a t a t 1 )P G (c t c t 1 )=P {Bi(, α t 1 + β t 1 )=a t }P {Bi(, γ t 1 + β t 1 )=c t }. The ratio is the product of three terms: P H (a t, 0,c t a t 1, 0,c t 1 ) P G (a t a t 1 )P G (c t c t 1 ) (14) P {Bi(, α t 1 )=a t } P {Bi(, α t 1 + β t 1 )=a t }, (15) γ P {Bi( a t, t 1 1 α t 1 )=c t } P {Bi(, γ t 1 + β t 1 )=c t }, (16) ad β t 1 P {Bi( a t c t, )=0}. (17) 1 α t 1 γ t 1 We boud the third term (17) simply by 1; i fact it is close to 1. The term (15) is αt 1 at (1 α ( ) t 1) at (α t 1 + β t 1 ) at (1 α t 1 β t 1 ) β t 1 1+ at 1 α t 1 β t 1 ( lg 4 ) () =1+O, sice β t 1 β t 1 2lg4 () 1 α t 1 β t 1 2 for a t 1,c t 1 lg 2 (). By a similar calculatio, (16) is also 1 + O we obtai ( P H (a t, 0,c t a t 1, 0,c t 1 ) lg 4 ) P G (a t a t 1 )P G (c t c t 1 ) =1+O (). 22 ( ) 1 lg 4 (). Multiplyig,

23 Thus, P (J 1 J 2 {B t = 0 for all t t }) = P H (a 1, 0,c 1 ) P H (a 2, 0,c 2 a 1, 0,c 1 ) a 1,c 1 L a 2,c 2 L P H (a t, 0,c t a t 1, 0,c t 1) a t,c t L P G (a 1 1)P G (c 1 1) P G (a 2 a 1 )P G (c 2 c 1 ) a 1,c 1 L a 2,c 2 L [ ( lg 4 )] t () P G (a t a t 1)P G (c t c t 1) 1+O a t,c t L = P G (a 1 1)P G (a 2 a 1 ) P G (a t a t 1) a t,...,a t L P G (c 1 1)P G (c 2 a 1 ) P G (c t a t 1) c t,...,c t L [ ( [ ] 2 lg = P {G 1 5 )] () t L for all t t } 1+O ( lg =[P (J 1 )] [1+O 2 5 )] (). This completes the proof. [ 1+O ( lg 4 () )] t Lemma 20. N i probability as. Proof. We will show that the mea ad stadard deviatio of N satisfy EN ad SD(N )=o(e(n )). To see that EN = P J 1, begi with (3), which gives P (J 1 ) P {Y t [1, lg 2 ()] for all t t }. This last probability is very close to P {Y t [1, lg 2 ()]}. Ideed, the differece P {Y t [1, lg 2 ()]} P{Y t [1, lg 2 ()] for all t t } = P {Y t [1, lg 2 ()],Y t > lg 2 () for some t<t }, (18) is the probability that the Y process exceeds lg 2 () some time before t but the decreases to be below lg 2 () at time t. Sice the Berstei iequality applied to the Poisso distributio gives P {Y t+1 Y t Y t } exp[ (3/14)Y t ] exp[ (3/14) lg 2 ()] o {Y t > lg 2 ()}, the differece (18) is bouded by t exp[ (3/14) lg 2 ()] = o(1/). By Lemma 16, lg P {Y t [1, lg 2 ()]} (ζ ɛ)lg 1 t lg P {Y t [1, lg 2 ()]} lg γ = 1 ζ. 23

24 which implies that P {Y t [1, lg 2 ()]}. Thus, P (J 1 ) P {Y t [1, lg 2 ()] for all t t } = [P {Y t [1, lg 2 ()]} + o(1/)]. Fially, to see that SD(N )=o(e(n )), we apply Lemma 19 to obtai Var(N )=E(N) 2 (EN ) 2 = P J 1 + ( 1)P (J 1 J 2 ) (P J 1 ) 2 P J 1 + ( 1)[P (J 1 )] 2 (1 + o(1)) (P J 1 ) 2 = o( 2 (PJ 1 ) 2 )=o((en ) 2 ). Proof of the lower boud i Theorem 2. Defiig W = {i : G i t [1, lg 2 ()]}, wehave P {U t +(1 ɛ)lg()} P {W = } + P {evetual extictio for all i W } +P {G i t = for some i W +(1 ɛ)lg() }. (19) P The cardiality of the set W is N. Sice N, clearly the probability that all idividuals {I 0,i : i W } evetually become extict coverges to 0; this is a easy cosequece of results about extictio probabilities of Kämmerle (1991) or Möhle (1994). So it remais to show that the last probability i (19) teds to 0. To see this, takig i W, observe that for the evet {G i t = } to occur the +(1 ɛ)lg() {Gi t} process must go from below lg 2 () at time t to at time t +(1 ɛ)lg(). That is, the process must go from below lg 2 () to withi a time spa of at most (1 ɛ)lg() geeratios. However, by the proof of Propositio 15, we kow that this has probability o(1/), so that, takig the uio over i W gives a total probability of o(1). 4 Discussio A motivatio behid this study was the iterest surroudig the idea of all of makid havig a recet commo acestor. I thikig about a mathematical treatmet of that idea, it seemed atural to remove the restrictio to the materal lie ad cosider a two-paret model. We have see that CA s occur very recetly i the two-paret model studied here. The most recet CA occurs, with high probability, about lg geeratios ago. Withi 1.77 lg geeratios, with high probability, all idividuals who are ot extict are CA s. These results describe the behavior of populatios satisfyig certai assumptios of radom matig ad so o. If our world really satisfied such assumptios, the athropological excitemet about the recetess of mitochodrial Eve would be misplaced: I oly a tiy fractio of the time back to mitochodrial Eve, commo acestors of makid would aboud, ad i fact a radomly chose idividual would be a CA with probability about 0.8. If we wish to uderstad aalogous questios i more complicated models that could better address pheomea such as the evolutio of makid, further study is required. For example, the absece of geographic structure is a key feature limitig the applicability of the model studied here to such situatios. 24