Recent common Ancestors

25 Pages·2002·0.18 MB·English

Adv.Appl.Prob. 31,1002–1026(1999)
PrintedinNorthernIreland
AppliedProbabilityTrust1999
RECENT COMMON ANCESTORS OF ALL PRESENT-DAY
INDIVIDUALS
∗
JOSEPHT.CHANG, YaleUniversity
Abstract
Previous study of the time to a common ancestor of all present-day individuals has
focused on models in which each individual has just one parent in the previous
generation. For example, ‘mitochondrial Eve’ is the most recent common ancestor
(MRCA)whenancestryisdefinedonlythroughmaternallines. InthestandardWright–
Fishermodelwithpopulationsizen,theexpectednumberofgenerationstotheMRCA
isabout2n,andthestandarddeviationofthistimeisalsoofordern. Herewestudya
two-parentanalogoftheWright–Fishermodelthatdefinesancestryusingbothparents.
Inthismodel,ifthepopulationsizenislarge,thenumberofgenerations,Tn,backto
a MRCA has a distribution that is concentrated around lgn (where lg denotes base-2
logarithm),inthesensethattheratioTn/(lgn)convergesinprobabilityto1asn→∞.
Also,continuingtotracebackfurtherintothepast,atabout1.77lgngenerationsbefore
thepresent, allpartialancestryofthecurrentpopulationends, inthefollowingsense:
withhighprobabilityforlargen,ineachgenerationatleast1.77lgngenerationsbefore
thepresent,allindividualswhohaveanydescendantsamongthepresent-dayindividuals
areactuallyancestorsofallpresent-dayindividuals.
Keywords: Coalescent; Wright–Fisher model; Galton–Watson process; genealogical
models;populationgenetics
AMS1991SubjectClassification: Primary92D25
Secondary60J85
1. Introduction
Startingwiththesetofallofuspresent-dayhumans,imaginetracingbackintimethrough
our mothers, our mothers’mothers, and so on. This is the maternal family tree of mankind,
andweareatitsleaves. Recentresearchhassuggestedthatthewomanattherootofthistree
lived roughly 100000 or 200000 years ago, perhaps inAfrica (Cann et al. (1987); Vigilant
etal. (1991)). Thiswomanhasbeendubbed‘mitochondrialEve’,sinceallpresent-dayhuman
mitochondrial DNA descended from hers. Mitochondrial Eve was undoubtedly not the only
woman alive at her time, so the name ‘Eve’ is misleading, as has been pointed out by a
number of authors; see, for example, Ayala (1995). However, this misunderstanding aside,
questionsoftheoriginsofmankindandthenatureofourrelationshipstoeachotherarestill
ofkeeninterest,andtheresearchonmitochondrialEvehasreceivedagreatdealofpublicity,
generating headlines in the popular press as well as in scientific publications. Svante Pa¨a¨bo
(1995)explains:
Received9October1997;revisionreceived19June1998.
∗
Postaladdress: StatisticsDepartment,YaleUniversity,Box208290,NewHaven,CT06520,USA.
Emailaddress:[email protected]
1002
Recentcommonancestors 1003
...therecentdateofourmitochondrialancestorisinasensethereallycontrover-
sialconclusionfromthesestudies. Everyoneagreesthatwetraceourancestryto
Homoerectus,whoemergedinAfricaandfromtherecolonizedmostofEurasia
about a million years ago or even earlier. What the mitochondrial data seem to
show,however,isthatwehaveamuchmorerecentancestor,onewholivedsome
100000or200000yearsago.
Whatcapturestheimaginationisnottheparticularchoicetotracebackthroughthematernal
line,butratheritistheideathatallofpresent-dayhumanitymayhaveacommonancestorwho
livedaslittleas100000yearsago,atimethatseemstomanytobesurprisinglyrecent. Ifwe
retain this idea while removing the restriction to the maternal line, the question becomes:
Howfarbackintimedoweneedtotracethefullgenealogyofmankindinordertofindany
individualwhoisacommonancestorofallpresent-dayindividuals? Inthispaperweaddress
thissortofquestioninasimplemathematicalmodel.
The coalescent model of Kingman (1982a) forms the basis of many of the calculations,
formal and informal, used in recent treatments of questions about mitochondrial Eve and
relatedtopics. Thecoalescentisalarge-populationlimitofanumberofthefundamentalmod-
els of population genetics, including the Wright–Fisher process. These models are haploid,
with each individual in a given generation having a single parent in the previous generation.
The Wright–Fisher model assumes ‘random mating’, in the sense that the parent of a given
individualisequallylikelytobeanyoftheindividualsinthepreviousgeneration. Thestandard
modelalsopostulatesaconstantpopulationsize,whichmaybean‘effectivepopulationsize’
whenmodelingmoregeneralsituations. Anumberofimportantpropertiesofthecoalescent
modelareusedinapplications. Forexample,themodelimpliesarelationshipbetweencoales-
cencetimesandpopulationsize: theexpectedcoalescencetime(measuredingenerations)of
alargesampleisabouttwicethepopulationsize. Hudson(1990)givesasurveyofthetheory
andapplicationsofthecoalescent.
Herewestudyanaturaltwo-parentanalogoftheWright–Fisherprocess. (Thisprocesswas
previouslyconsideredbyKa¨mmerle(1991)andMo¨hle(1994);seetheendofthissectionfor
a discussion of related work.) We assume the population size is constant at n. Generations
are discrete and non-overlapping. The genealogy is formed by this random process: in each
generation,eachindividualchoosestwoparentsatrandomfromthepreviousgeneration. The
choices are made just as in the standardWright–Fisher model—randomly and equally likely
over the n possibilities—the only difference being that here each individual chooses twice
instead of once. All choices are made independently. Thus, for example, it is possible that
whenanindividualchooseshistwoparents, hechoosesthesameindividualtwice, sothatin
factheendsupwithjustoneparent;thishappenswithprobability1/n.
Thismodelisdesignedonlyasasimplestartingpointforthought;ofcourseitisnotmeant
tobeparticularlyrealistic. Still,onemightworrythatthissimplemodelignoresconsiderations
ofsexandallowsimpossiblegenealogies. Ifthisseemsbothersome,analternativeinterpreta-
tionofthesameprocessisthateach‘individual’isactuallyacouple,andthatthepopulation
consists of n monogamous couples. Then the random choices cause no contradictions: the
husbandandwifeeachwereborntoacouplefromthepreviousgeneration. Theycouldeven
comefromthesamecoupleinthepreviousgeneration.
Our interest here is in finding individuals who are common ancestors of all present-day
individuals. Forconvenience,weusetheabbreviation‘CA’torefertoacommonancestorof
allpresent-dayindividuals,and‘MRCA’standsfor‘mostrecentcommonancestor.’
1004 J.T.CHANG
Itturnsoutthatmixingoccursextremelyrapidlyinthetwo-parentmodel,sothatCAsmay
befoundwithinanumberofgenerationsthatdependslogarithmicallyonthepopulationsize.
In particular, our first main result says that the number of generations back to an MRCA is
aboutlgn,wherelgdenoteslogarithmtobase2.
Theorem1. LetTndenotethenumberofgenerations,countingbackintimefromthepresent,
toanMRCAofallpresent-dayindividuals,inapopulationofsizen. Then
T
n P
→1 asn→∞.
lgn
Thiscontrastsdramaticallywiththeone-parentsituation. Forexampleifnis1million,then
theone-parentMRCA(‘Eve’)isexpectedtooccurabout2milliongenerationsago,whereas
atwo-parentMRCAoccurswithhighprobabilitywithinthelast20generationsorso. Also,
thevariabilityintheone-parentsituationissuchthattheactualtimetotheMRCAmayeasily
be as small as half the expected time or as large as double the expected time, say, even in
arbitrarily large populations. In contrast, the time to an MRCA for the two-parent model is
much less variable. For example, if the population is large enough, it is very unlikely that a
randomrealizationofthetwo-parentMRCAtimewilldifferfromlgnbyevenonepercent.
This paper also addresses a second related question. Imagine tracing back through the
two-parent genealogy. According to Theorem 1, after about lgn generations, we will reach
the most recent generation that contains a CA. That generation might contain just one CA,
or it might contain more than one. In any case, if we continue tracing back further through
successivegenerations,thenthetitle‘CA’becomesmuchlessofaprestigiousdistinction. For
example,bothparentsofaCAwillbeCAs,andallgrandparentsofaCAwillbeCAs,andso
on. Eventually,inagivengeneration,many(andinfactmost)oftheindividualswillbeCAs.
AtsomepointwereachagenerationinwhichsomeindividualsareCAs(havingallpresent-
dayindividualsasdescendants)andsomeare‘extinct’(havingnopresent-dayindividualsas
descendants),butnoindividualisintermediate(havingsomebutnotallpresent-dayindividuals
as descendants). That is, at this point, everyone who is not extinct is a CA. This condition
persistsforeveraswetracebackintime: everyindividualisaCAorextinct. Thenextresult
showsthatthisconditionisreachedveryrapidlyinthemodelstudiedhere.
Theorem2. Let Un denote the number of generations, counting back in time before the
present,toagenerationinwhicheachindividualiseitheraCAofallpresent-dayindividuals
or an ancestor of no present-day individual. Let γ denote the smaller of the two numbers
satisfyingtheequationγ e−γ =2e−2,andletζ =−1/(lgγ)≈0.7698. Then
U
n P
→1 asn→∞.
(1+ζ)lgn
Thus, within about 1.77lgn generations, a tiny amount of time in comparison with the
orderntimerequiredtogetaone-parentCA,everyoneinthepopulationiseitheraCAofall
present-dayindividualsorextinct.
Figure1showsasmallexampletoillustratethedefinitionsandstatements. Thepopulation
size is 5. At the bottom of the figure is generation 0, the present. Going up in the graph
correspondstogoingbackintime,sothatthetoprowisgeneration−5. Foreachindividual
I in each previous generation, we calculate the set of present-day individuals (individuals
in generation 0) that are descendants of I. For example, the set of present-day descendants
of individual #1 in generation −1 is {3,5}. The calculations propagate backward in time
Recentcommonancestors 1005
/ / S
0 0
S
S
S
/
0 S
{4,5} S
S /
0
S {4,5} {1,2,3,4}
{1,2,3,4} {2,3} /
{4,5} S 0
{2,3}
{4} {5}
{3,5} {1,2,4}
1 2 3 4 5
Figure1:Anexampleillustratingthemodel. Herethefourthindividualingeneration−2isaCA.By
generation−5,allindividualsareCAsorextinct: individuals1,4,and5areCAs,andindividuals2and
3areextinct.
according to the rule: the set of descendants of an individual I is the union of the sets of
descendantsofthechildrenofI. Forexample,thesetofpresent-daydescendantsofindividual
#4 in generation −2 is the union {3,5}∪{5}∪{1,2,4}, which is the whole population S.
Thus, individual #4 in generation −2 is a CA of the set S of all present-day individuals.
Continuing backward in time, at generation −5 we reach the stage where each individual
hasasdescendantseitherthewholepopulationS ortheemptyset∅. Thatis,eachindividual
ingeneration−5iseitheraCAorextinct,havingasdescendantseithereverybodyornobody
from the set of present-day individuals, and all generations prior to generation −5 also have
thisproperty. Intheexampleshown,T =2andU =5.
5 5
Whatisthesignificanceoftheseresults?Anapplicationtotheworldpopulationofhumans
would be an obvious misuse. For example, we would not claim that a common ancestor of
every present-day human may be found within the last lgn generations. Even if we took n
to be 5 billion, this would imply a CA just about 32 generations ago—perhaps 500 years or
so. An important source of the inapplicability of the model to this situation is the obvious
non-randomnatureofmatinginthehistoryofmankind. Forexample,parentsaremuchmore
likely to live within a few miles of their children than a thousand miles away or halfway
around the world. So the model studied here is too simple to be directly applicable to the
evolution of mankind as a whole. In such complicated situations, the results sound a note
of caution: if the logarithmic time to CAs seems patently implausible, then at least one of
the assumptions of the model, such as the random mating assumption, must be causing a
great deal of trouble. On the other hand, it would be interesting to know whether there are
simpler real-life situations in which the assumptions of the model do apply reasonably well
and the theorems provide reasonably accurate quantitative descriptions. Perhaps a relatively
homogeneouspopulationlackingdiscerniblestructures(geographicorotherwise)thatinteract
stronglywithreproductionwouldbeapromisingcandidate.
1006 J.T.CHANG
TherandomtimeanalysedinTheorem2seemsofnaturalinterestinthisprocessandmay
alsobepertinenttocertainquestionsabout‘speciestrees’or‘populationtrees’(asopposedto
‘genetrees’). Inmanycontextsthespeciestreeisconsideredtobetherealobjectofinterest,
andweusegeneticdataandgenetreestoattempttolearnaboutthespeciestree. Forexample,
forhumans,chimpanzees,andgorillas,isthe‘truespeciestree’(HC)G,(HG)C,or(CG)H?
Roughly,theconceptualframeworkofthisquestionisasfollows. Thereweretwo‘speciation
events’thatsplitasinglespeciesancestraltohumans,chimpanzees,andgorillasintothethree
separatemodernspecies. Thetree(HC)G,forexample,saysthatthefirstsuchsplitseparated
the subpopulation that eventually became modern gorillas from the remainder, which later
splittobecomemodernhumansandchimpanzees. Unfortunately,moreprecisedefinitionsof
theconceptofspeciestreethatremainusefulindifficultorunclearcasesseemhardtocome
by. One might adopt the viewpoint that the proper starting point for a definition of ‘species
tree’isthefulltwo-parentgenealogyofallpresent-dayindividuals. Givensuchadefinition,
if we knew all details of this genealogy, then we could read off an answer to the H, C, and
G question (the answer might be ‘none of the 3 choices above’—that is, the species tree is
notwelldefinedoratleastnotbifurcating). OneinterpretationofthetimeUn isasfollows.
Supposeweimagineacasewhereevolutionreallyproceededaccordingtoaneatsuccession
of ‘speciation events’. Under a certain reasonable definition of a species tree, if the times
between those speciation events exceed Un, then the species tree is guaranteed to be well
defined and coincide with the history of speciation events. This idea will be discussed more
fullyelsewhere.
Acaveattoforestallpotentialmisunderstanding: thispaperisnotaboutgenetics. Thatis,it
isnotaboutwhogetswhatgenes;itisaboutsomethingmoreprimitive,namely,theancestor–
descendantrelationship. One-parentmodelsareappropriateintracingthehistoryofasample
of nonrecombining genes or small bits of DNA; a single nucleotide descends from a single
nucleotidefromeitherthemotherorfather,butnotboth. Hereweareconsideringancestryin
themorecommon, demographicsenseoftheword, asappliedtopeople, forexample, rather
thangenes.
Previous genetics research that is somewhat related, although still very different from the
present study, considers models incorporating recombination. This type of model has been
investigated in a number of papers, including those of Hudson (1983) and Griffiths and
Marjoram(1997). ThehistoryofasampleofDNAsequencesmaybedescribedbyacollection
ofgenealogies,witheachnucleotidepositionintheDNAhavingitsownone-parentgenealogy.
The genealogies for two positions that experience no recombination between them will be
congruent, with the paths of the two genealogies going back through the same individuals,
whereas a recombination between two positions causes the genealogies of those positions to
differ. Each of the genealogies in the collection will have its own MRCA (the nucleotide at
its root), which may occur in different individuals. Each of these individuals will be a CA
in the sense considered in this paper, but the most recent of these individuals is generally
not an MRCA in our sense. Our MRCA is more recent, since the paths from ancestors
to descendants consist of all potential paths for genes to be transmitted, and may include
paths that did not happen to be taken by any genes. No previous results about these genetic
models have been similar to the results here, for example, in getting times of order logn.
This is not surprising, since the asymptotics would require an assumption that the sequence
lengths and the number of recombinations tend to infinity. This is another manifestation
of the statement that the questions we are investigating here are not fundamentally genetics
questions.
Recentcommonancestors 1007
There is some previous work on the process we study here and related processes. Two
papers of Ka¨mmerle (1989, 1991) introduce a general class of two-parent (called ‘bisexual’
in those papers) versions of the Wright–Fisher and other processes. These papers focus on
two main questions. First, they analyse the probability of extinction of a set of individuals
in the present generation, that is, the probability that the set of individuals eventually has
no descendants in some future generation. Second, in a two-parent version of the Moran
model, they study the number Rn(t) of individuals t generations ago who have at least one
descendantinthepresentgeneration. Ka¨mmerle(1989)findsthattheMarkovchain{Rn(t) :
t = 0,1,...}, suitably normalized and suitably initialized (with the initialization essentially
requiringthatthechainisstartedinsteadystate),convergesweaklyasn → ∞toadiscrete-
timeOrnstein–Uhlenbeckprocess.
Mo¨hle (1994) both generalizes and refines the results of Ka¨mmerle. In particular, Mo¨hle
providesadetailedanalysisoftheextinctionprobabilitiesinatwo-parentWright–Fishermodel
thatapproximatestheprobabilitiesuptoo(1/n). Healsoestablishesweakconvergenceina
generalclassoftwo-parentmodels,includinganOrnstein–Uhlenbecklimitforthetwo-parent
Wright–Fisherprocess. Mo¨hlealsohasanumberofotherpapersinpress,includingonethat
relaxestheassumptionofconstantpopulationsize.
Thesepreviousresultsarecomplementarytotheresultsinthispaper. Thepreviouspapers
consideredindividualswhohaveatleastonedescendantinagivenfuturegeneration. Herewe
consider CAs, who have as descendants all members of the future generation. The previous
resultsabouttheprocess{Rn(t)}applytolarget,thatis,tothebehavioroftheprocessmany
generationsbeforethepresent,withtheprocessinsteadystate. Herewefocusonthebehavior
ofarelatedprocessatsmall(i.e.recent)times,startingfarawayfromsteadystate. Weshow
that at about t = 1.77lgn generations before the present, with high probability the Rn(t)
individualswhohaveatleastonepresent-daydescendantareallinfactCAs.
2. Simulations
Table1presentsasmallsimulationstudyconsistingof25trialseachforn=500,n=1000,
n = 2000, and n = 4000. Two numbers are reported for each trial: Tn, the number of
generations back to an MRCA, and Un, the generation at which every individual is either a
CAorextinct.
InthesesimulationsthedistributionofthetimebacktoanMRCAisindeedquiteconcen-
trated around the value lgn, which is nearly 9 for n = 500, nearly 10 for n = 1000, and so
on. Thus,thesimulationresultsshowthattheasymptotic(n→∞)statementofTheorem1is
‘notsoasymptotic’,inthatitdescribesthesituationwellevenforrathersmallvaluesofn. The
behaviorpredictedbyTheorem2isalsoreflectedreasonablywellinthesimulations,although
onemighthaveguessedanumericalconstantcloserto2ratherthan1.77fromthissmallstudy.
3. Proofs
3.1. Generalideasandtools
We start with the observation that although Theorems 1 and 2 are phrased in terms of
countinggenerationsbackintimefromthepresentuntilsomeconditionobtains,theseresults
maybeprovedbycountingforwardintimefromafixedgeneration. Forexample, theevent
{Tn ≤m}requiresthataCAofallindividualsingeneration0maybefoundamonggenerations
−1,−2,...,−m. Thisisequivalenttorequiringthatifwestartwithgeneration−mandtrace
forward in time, then some individual in generation −m becomes a CA of all individuals in
somegenerationt ∈{−m+1,−m+2,...,0}.
1008 J.T.CHANG
Table1:Asmallsimulationstudy. Foreachoffourpopulationsizesn,thetwotimesTn andUn are
reportedfor25trials.
n=500 n=1000 n=2000 n=4000
1 10 18 11 19 12 24 13 24
2 9 18 11 21 12 22 13 23
3 10 21 10 20 11 23 12 24
4 9 18 11 20 12 22 13 24
5 10 19 10 20 12 21 13 24
6 9 19 11 31 12 23 13 24
7 10 19 11 20 12 21 13 24
8 10 21 10 23 12 27 13 24
9 9 19 11 20 11 24 13 24
10 9 17 11 20 12 24 13 31
11 9 19 10 21 12 26 13 23
12 9 18 11 21 11 22 13 23
13 9 19 11 21 12 21 13 25
14 9 20 11 20 12 25 13 22
15 9 19 10 20 11 24 13 24
16 10 21 10 21 11 23 13 23
17 9 19 11 19 12 24 13 23
18 9 17 10 26 12 22 13 24
19 9 19 11 21 11 23 13 25
20 10 18 10 21 12 22 13 25
21 9 19 11 19 12 22 12 25
22 10 19 11 26 12 22 13 26
23 9 19 10 20 12 24 13 27
24 10 17 11 21 12 22 13 26
25 10 19 11 23 12 23 13 25
Sowewillcountgenerationsforwardintime,andforconvenienceletusrenumbergenera-
tionssothattheinitialgenerationis‘generation0’. Thepopulationatgenerationt ≥0consists
ofnindividualsdenotedbyIt,1,It,2,...,It,n. WecanpictureIt,1,It,2,...,It,n asdotsinan
array as in Figure 1, with It,j being the jth dot in row t. The association of a number j to
individual It,j is an arbitrary labeling of the individuals within generation t. Assigned only
as a means of referring to individuals, the labels have no significance in the model, which
does not order the individuals within a generation. Let µt,1,νt,1,µt,2,νt,2,...,µt,n,νt,n be
independent and uniformly distributed on the set {1,...,n}. We interpret µt,j and νt,j as
labels of the parents of individual It,j; that is, the parents of It,j are It−1,µt,j and It−1,νt,j.
DefiningasequenceofrandomsetsGi,Gi,... recursivelybyGi ={i}and
0 1 0
Git ={j ≤n:µt,j ∈Git−1orνt,j ∈Git−1},
Git isthesetoflabelsofthedescendantsofI0,i ingenerationt. LetGit denotethecardinality
ofGit. TheconditionalprobabilitythatindividualIt+1,j hasatleastoneparentamongtheGit
membersofGit is
P({µt+1,j ∈Git}∪{νt+1,j ∈Git}|Git)=(Git/n)+(Git/n)−(Git/n)(Git/n).
Recentcommonancestors 1009
Theprocess{Git :t =0,1,...}isaMarkovchainwithtransitionprobabilities
(cid:1) (cid:2) (cid:3) (cid:4)
(Git+1 |Git)∼Bin n,2Gnit − Gnit 2 , (1)
whereBin(n,p)denotesthebinomialdistributionforthenumberofsuccessesinnindepen-
denttrialseachhavingsuccessprobabilityp.
Throughout the proof, {Gt} will denote a Markov chain with transition probabilities as in
(1), although in different parts of the proof we will consider different possible initial values
G . For example, taking G = 1 corresponds to following the descendants of a particular
0 0
individualingeneration0. Intheearlystagesoftheprocess,whileGt remainssmallrelative
ton,inviewof(1)theconditionaldistributionofGt+1givenGt isnearlyPoisson(2Gt),thatis,
thePoissondistributionwithmean2Gt. Inotherwords,while{Gt}remainssmall,itevolves
nearly as a Galton–Watson branching process {Yt} with offspring distribution Poisson(2).
Ka¨mmerle (1991) gave a formal statement of a result of this nature. A special case of his
result says that for fixed u, the joint distribution of (G0,G1,...,Gu) converges to that of
(Y0,Y1,...,Yu)asn→∞. Forourpurposes,wewillusethefollowingresultthatallowsus
toapproximateprobabilitiesfortheGprocessbythosefortheY processuptoahigherorder
ofaccuracyandoverlongerintervalsoftimethatmayhaverandomlengths.
Lemma3. Let Y ,Y ,... denote a Galton–Watson branching process with offspring dis-
0 1
tribution Poisson(2). Suppose that Y0 = G0 = 1. Define τbY = inf{t : Yt ≥ b} and
τ0Yb =inf{t :Yt =0orYt ≥b},withcorrespondingdefinitionsforτbGandτ0Gb. Asn→∞,if
mandbsatisfymb2 =o(n),then
P{τG >m}=P{τY >m}(1+o(1)) (2)
b b
and
P{τG >m}=P{τY >m}(1+o(1)). (3)
0b 0b
Proof.Astraightforwardcalculationboundsthelikelihoodratio
L(y |x):= P{Gt+1 =y |Gt =x}
P{Yt+1 =y |Yt =x}
P{Bin(n,(2x/n)−(x2/n2))=y}
= ≤e2x(1−(2x/n)+(x2/n2))n−y,
P{Poisson(2x)=y}
sothat (cid:2) (cid:3)
2x x2
logL(y |x)≤2x+(n−y) − + ≤(x2+2xy)/n.
n n2
ThisholdswheneverthedenominatorP{Yt+1 = y | Yt = x}ispositive,thatis,forallx > 0
andy ≥ 0,andalsoforx = y = 0. Thus,forallsuchpairsofx andy satisfyingx < band
y <b,wehave
logL(y |x)≤3b2/n.
Asimilarcalculationgivesthelowerbound
logL(y |x)≥−5b2/(2n)[1+O(b/n)],
1010 J.T.CHANG
so that logL(y | x) ≥ −3b2/n for sufficiently large n. So if x1,...,xm are all less than b,
then
P{G1 =x1,...,Gm =xm}=P{G1 =x1 |G0 =1}···P{Gm =xm |Gm−1 =xm−1}
=P{Y1 =x1,...,Ym =xm}L(x1 |1)···L(xm |xm−1)
≤P{Y1 =x1,...,Ym =xm}e3mb2/n
and
P{G1 =x1,...,Gm =xm}≥P{Y1 =x1,...,Ym =xm}e−3mb2/n.
Thus,
(cid:5) (cid:5)
P{τbG >m}= ··· P{G1 =x1,...,Gm =xm}
0≤(cid:5)x1<b 0≤(cid:5)xm<b
≤ ··· P{Y1 =x1,...,Ym =xm}e3mb2/n
0≤x1<b 0≤xm<b
=P{τY >m}e3mb2/n (4)
b
and, similarly, P{τG > m} ≥ P{τY > m}e−3mb2/n, so that, by the assumption that mb2 =
b b
o(n), we obtain P{τG > m} = P{τY > m}(1+o(1)). This proves (2). The proof of (3)
b b
uses the same reasoning, with the summations in (4) ranging over 0 < xt < b rather than
0≤xt <b.
The previous result will be useful because the Poisson Galton–Watson process is simple
andwellunderstood. Thenextlemmarecordsafewwellknownitemsforfuturereference.
Lemma4. Let Y ,Y ,... denote a Galton–Watson process with offspring distribution
0 1
Poisson(2). Definethemomentgeneratingfunctionψ(z) = E(zY1) = e−2+2z. Theextinction
probability ρ = P{Yt = 0 forsomet} ≈ 0.20319 is the smaller of the two solutions of
ψ(ρ) = ρ, and ρ = γ/2, where γ is as defined in Theorem 2. The t-fold composition
ψt =ψ ◦···◦ψ satisfiesψt(z)↑ρ forall0≤z≤ρ.
Therelationρ =γ/2isconfirmedbycomparingthedefinitionsofρandγ. Despitethesimple
relationship,wewillkeepthetwodifferentlettersinournotationforconceptualclarity.
Defininggt =Gt/n,wehave
E(gt+1 |gt)=2gt −gt2 =gt(2−gt). (5)
That is, if the fraction of descendants of a given individual is currently gt, it is expected to
multiplybyafactorof2−gt inthenextgeneration. Forexample, intheearlystagesofthe
processwhenthefractiongt issmall,itnearlydoublesinexpectationinthenextgeneration.
Forverysmallgt (oftheorder1/n,forexample)therandomvariabilityislarge;forexample,
theprocesscouldeasilygoextinct. ThisiswhenitismostusefultoapproximatetheGprocess
bythePoisson(2)Galton–Watsonprocess.
On the other hand, for larger values of gt, the multiplication factor gt+1/gt, although
expected to be somewhat smaller, has much less variability. The deviations of this factor
from its expected value are bounded probabilistically by large deviations inequalities for the
binomialdistribution. WewillusethefollowinginequalityofBernstein(1946)asabasictool.
Recentcommonancestors 1011
Lemma5. (Bernstein’sinequality.) IfX ∼Bin(n,p)andr >0,then
(cid:6) (cid:7)
−r2
P{X ≥np+r}≤exp . (6)
2np(1−p)+(2/3)r
Since n − X ∼ Bin(n,1 − p), the right-hand side of (6) is also an upper bound for the
probabilityP{X ≤np−r}.
3.2. ProofofTheorem1
Outline.The proof will be divided into several parts. We start from generation 0 and trace
forwardintime.
Stage 1: By the end of Stage 1, we identify an individual I in generation 0 who has a
numberofdescendantsthatissmallcomparedton,butlargeenoughsothatI isunlikelyever
to become extinct. In particular, we look for a generation t such that some individual I in
generation0hasatleastlg2(n)descendantsingenerationt. Withprobabilityapproaching1,
thishappensintimeo(lgn),negligiblecomparedwithlgn; thisisshownbyusingLemma3
to approximate our process by a Poisson Galton–Watson process. The rest of the proof will
show that with probability approaching 1, individual I becomes a CA within (1+()(lgn)
generations,where( isanarbitrarypositivenumber.
Stage2: Letβ ∈ (0,1). Stage2followsthedescendantsofI untilreachingageneration
containingatleastnβ descendants. Inviewof(5),sincenβ isasmallfractionofnforlargen,
throughoutStage2thenumberofdescendantsinagenerationisexpectedtobenearlydouble
thenumberofdescendantsinthepreviousgeneration. Andlg2(n)islargeenoughsothatthe
multiplicationfactorwillbeveryclosetoitsexpectedvalue,withhighprobability. SoStage2
shouldnottakemuchmorethanaboutlg(nβ)=βlg(n)generations.
Stage 3: This stage brings the count of descendants of I up from nβ to (1/2)n. Since
the fraction of descendants during Stage 3 stays below 1, the expected multiplication factor
2
is at least 2 − 1 = 3. Again, this multiplication factor is very reliable, so that with high
2 2
probability Stage 3 takes no more than about log3/2{(n/2)/(nβ)} generations. We can make
thisanarbitrarilysmallfractionoflgnbychoosingβ closeenoughto1.
Stage 4: Now we switch to looking at the fraction Bt of individuals in a generation who
arenot descendantsofindividualI. Thisfractionisexpectedtosquareeachgeneration. This
causesBt todecreaseveryquickly. Fixingα ∈(1,2),weshowthatStage4,whichtakesthe
2 3
fractionBt from 1 downton−α,takesonlyorderlglgntime.
2
Stage 5: This completes the process, ending when the B process hits 0, and individual I
hasbecomeaCA.Weshowthatthistakesjustonegenerationwithhighprobability.
Upper bound: Combining the results of Stages 1 through 5 gives the probabilistic upper
boundlimn→∞P{Tn ≤(1+()lgn}=1.
Lowerbound: Hereweshowthatlimn→∞P{Tn ≥(1−()lgn}=1. Thisisdonebyusing
Bernstein’sinequalitytoproveanassertionofthefollowingform: forpositiver andδ,once
theprocessofdescendantsofanygivenindividualreachesapowernr ofn,itisveryunlikely
toincreasebyafactorofmorethan2+δ inageneration,whereasitwouldhavetodosoin
ordertohaveTn <(1−()lgn.
3.2.1. Stage1.Herewewillshowthatwithhighprobability,withinanumberofgenerations
negligiblecomparedtolgn,wecanfindagenerationwithatleastlg2nindividualswhoshare
a common ancestor. For simplicity we give a crude argument that circumvents the need to