Dynamics of the time to the most recent common ancestor in a large branching population

25 Pages·2010·0.29 MB·English

TheAnnalsofAppliedProbability
2010,Vol.20,No.1,1–25
DOI:10.1214/09-AAP616
©InstituteofMathematicalStatistics,2010
DYNAMICS OF THE TIME TO THE MOST RECENT COMMON
ANCESTOR IN A LARGE BRANCHING POPULATION
BY STEVEN N. EVANS1 AND PETER L. RALPH2
UniversityofCaliforniaatBerkeleyandUniversityofCaliforniaatDavis
If we follow an asexually reproducing population through time, then
theamountoftimethathaspassedsincethemostrecentcommonancestor
(MRCA)ofallcurrentindividualslivedwillchangeastimeprogresses.The
resulting“MRCAage”processhasbeenstudiedpreviouslywhenthepopu-
lationhasaconstantlargesizeandevolvesviathediffusionlimitofstandard
Wright–Fisherdynamics.Foranypopulationmodel,thesamplepathsofthe
MRCAageprocessaremadeupofperiodsoflinearupwarddriftwithslope
+1punctuatedbydownwardjumps.WebuildotherMarkovprocessesthat
havesuchpathsfromPoissonpointprocessesonR++×R++withintensity
measuresoftheformλ⊗μwhereλisLebesguemeasure,andμ(the“family
lifetimemeasure”)isanarbitrary,absolutelycontinuousmeasuresatisfying
μ((0,∞))=∞andμ((x,∞))<∞forallx>0.Specialcasesofthiscon-
structiondescribethetimeevolutionoftheMRCAagein(1+β)-stablecon-
tinuousstatebranchingprocessesconditionedonnonextinction—aparticular
caseofwhich,β=1,isFeller’scontinuousstatebranchingprocesscondi-
tioned on nonextinction. As well as the continuous time process, we also
consider the discrete time Markov chain that records the value of the con-
tinuousprocessjustbeforeandafteritssuccessivejumps.Wefindtransition
probabilitiesforboththecontinuousanddiscretetimeprocesses,determine
whentheseprocessesaretransientandrecurrentandcomputestationarydis-
tributionswhentheyexist.Moreover,weintroduceanewfamilyofMarkov
processesthatstandsinarelationwithrespecttothegeneral(1+β)-stable
continuousstatebranchingprocessanditsconditionedversionthatissimilar
totheonebetweenthefamilyofBessel-squareddiffusionsandtheuncondi-
tionedandconditionedFellercontinuousstatebranchingprocess.
1. Introduction. Anyasexuallyreproducingpopulationhasauniquemostre-
centcommonancestor,fromwhomtheentirepopulationisdescended.Insexually
reproducingspecies,thesameistrueforeachnonrecombiningpieceofDNA.For
instance, our “mitochondrial Eve,” from whom all modern-day humans inherited
ReceivedDecember2008;revisedApril2009.
1SupportedinpartbyNSFGrantsDMS-04-05778andDMS-09-07630.
2SupportedinpartbyaVIGREgrantawardedtotheDepartmentofStatistics,UniversityofCali-
forniaatBerkeley.
AMS2000subjectclassifications.92D10,60J80,60G55,60G18.
Keywordsandphrases.Genealogy, most recent common ancestor, MRCA, continuous state
branching, Poisson point process, Poisson cut-out, transience, recurrence, stationary distribution,
duality,Bessel,self-similar,piecewisedeterministic.
1
2 S.N.EVANSANDP.L.RALPH
theirmitochondrialDNA,isestimatedtohavelivedaround180,000yearsago[19]
whileour“Y-chromosomalAdam”isestimatedtohavelivedaround50,000years
ago[33].TherehavealsobeeneffortstoestimatethetimesincetheMRCAlived
(whichwewillalsocallthe“ageoftheMRCA”)inpopulationsofotherorganisms,
particularly pathogens [32, 34]. These studies, using sophisticated models incor-
porating ofdemographic history, arefocused onestimatingtheageoftheMRCA
atasinglepointintime(thepresent).
As time progresses into the future, eventually the mitochondrial lineages of all
but one of the daughters of the current mitochondrial Eve will die out, at which
pointthenewmitochondrialEvewillhavelivedsomewhatlaterintime.Theageof
theMRCAisthusadynamicallyevolvingprocessthatexhibitsperiodsofupward
lineargrowthseparatedbydownwardjumps.
Recently, [26] and [29] independently investigated the MRCA age process for
thediffusionlimitoftheclassicalWright–Fishermodel.TheWright–Fishermodel
isperhapsthemostcommonlyusedmodelinpopulationdynamics:eachindividual
in a fixed size population independently gives birth to an identically distributed
random number of individuals (with finite variance), and after the new offspring
are produced, some are chosen at random to survive so that the total population
sizeremainsconstant.Thediffusionlimitarisesbylettingthepopulationsizegoto
infinityandtakingthetimebetweengenerationstobeproportionaltothereciprocal
ofthepopulationsize.
In this paper, we investigate the MRCA age process for a parametric family
of population models in a setting in which the population size varies with time,
and, by suitable choice of parameters, allows control over the extent to which
rareindividuals can have large numbers of offspring that survive to maturity. The
model for the dynamics of the population size is based on the critical (1+β)-
stablecontinuousstatebranchingprocessfor0<β ≤1.Theseprocessesariseas
scalinglimitsofGalton–Watsonbranchingprocessesasfollows.
(n)
Write Z for the number of individuals alive in a critical continuous time
t
Galton–Watson branching process with branching rate λ and offspring distribu-
tionγ.Thedistributionγ hasmean1(andthus,theprocessis“critical”).Suppose
thatifW isarandomvariablewithdistributionγ,thentherandomwalkwithsteps
distributed as the random variable (W −1) falls into the domain of attraction of
a stable process of index 1+β ∈(1,2]. The case β =1 corresponds to γ having
finite variance and the random walk converging to Brownian motion after rescal-
ing. Set X(n) =n−1/βZ(n) and suppose that X(n) →x as n→∞. Then, up to
t t 0
a time-rescaling depending on λ and the scaling of the stable process above, the
processes X(n) converge to a Markov process X that is a critical (1+β)-stable
continuous state branching process, and whose distribution is determined by the
Laplacetransform
(cid:2) (cid:3)
θx
(1.1) E[e−θXt|X =x]=exp − .
0 (1+θβt)1/β
DYNAMICSOFTHEMRCA 3
Ifβ =1,thisisFeller’scriticalcontinuousstatebranchingprocess[17,22].(Note
that time here is scaled by a factor of 2 relative to some other authors so that the
generatorofour“Fellercontinuousstatebranchingprocess”isx ∂2 .)
∂x2
Let τ =inf{t >0:X =0} denote the extinction time of X (it is not hard to
t
showthatX =0forallt ≥τ).Takingθ →∞in(1.1)gives
t
(cid:2) (cid:3)
x
P{τ >t|X =x}=1−exp − ,
0 t1/β
so X dies outalmostsurely.However,itispossible tocondition X tolive forever
inthefollowingsense:
1
lim E[f(X )|X =x,τ >T]= E[f(X )X |X =x].
t 0 t t 0
T→∞ x
(cid:8) (cid:8)(cid:8)
Thus if P (x ,dx ) are the transition probabilities of X, then there is a Markov
t
processY withtransitionprobabilities
1
Q (y(cid:8),dy(cid:8)(cid:8))= P (y(cid:8),y(cid:8)(cid:8))y(cid:8)(cid:8).
t y(cid:8) t
The process Y is the critical (1+β)-stable continuous state branching process X
conditioned on nonextinction. The distribution of Y is determined by its Laplace
t
transform
(cid:2) (cid:3)
yθ
(1.2) E[exp(−θY )|Y =y]=exp − (1+tθβ)−(β+1)/β
t 0 (tθβ +1)1/β
(see Section 6). Moreover, it is possible to start the process Y from the initial
stateY =0,andtheformula(1.2)continuestoholdfory=0.Thesuper-process
0
generalizationofthisconstructionwasconsideredforβ =1in[11,14,15]andfor
generalβ in[16].
For β =1, the conditioned process Y can be described informally as a single
“immortalparticle”constantlythrowingoffinfinitesimallysmallmasseswitheach
massthenevolvingaccordingtothedynamicsoftheunconditionedprocess.These
infinitesimalmassescanbeinterpretedasthesingleprogenitorsoffamilieswhose
lineagesplitsfromtheimmortalparticleatthebirthtimeoftheprogenitorandare
eventually doomed to extinction. Most such families die immediately, but a rare
few live for a noninfinitesimal amount of time. More formally, there is a σ-finite
measureν onthespaceofcontinuouspositiveexcursionpaths
E0:={u∈C(R+,R+):u0=0&∃γ >0s.t.ut >0⇔0<t <γ},
suchthatif(cid:10)isaPoissonpointprocessonR+×E0 withintensityλ⊗ν whereλ
is Lebesgue measure, and (X¯t)t≥0 is an independent copy of X begun at X¯0=y,
thentheprocess
(cid:2) (cid:3)
(cid:4)
(1.3) X¯t + u(t−s)∨0
(s,u)∈(cid:10) t≥0
4 S.N.EVANSANDP.L.RALPH
has the same distribution as (Yt)t≥0 begun at Y0 =y (see [15]). A point (s,u)∈
(cid:10) corresponds to a family that grows to nonnegligible size; the time s records
the moment the family splits off from the immortal particle, and the value u of
r
the trajectory u gives the size of the family r units of time after it split off. The
family becomes extinct after the period of time γ(u):=inf{r >0:u(t)=0,∀t >
r}. The σ-finite measure ν may be identified explicitly, but it suffices to remark
here that it is Markovian with transition probabilities the same as those of the
unconditionedprocessX—inotherwords,ν arisesfromafamilyofentrancelaws
¯
for the semigroup of X. The process (Xt)t≥0 records the population mass due
to descendants of individuals other than the immortal particle who are present at
time0.
AnanalogousdescriptionoftheconditionedprocessY forthecaseβ ∈(0,1)is
presentedin[16].Thereisagainasingleimmortallineage,butnowfamiliessplit
offfromthatlineagewithanoninfinitesimalinitialsize,reflectingtheheavy-tailed
offspring distributions underlying these models. More precisely, a decomposition
similar to (1.3) holds, but the Poisson point process (cid:10) is now on R+×E where
E ={u∈D(R+,R+):∃γ >0s.t.ut >0⇔0≤t <γ}, the set of càdlàg paths
starting above zero that eventually hit zero. The nondecreasing process (Mt)t≥0
where
(cid:4)
M := u
t 0
(s,u)∈(cid:10)∩[0,t]×E
isthetotaloftheinitialfamilysizesthatsplitofffromtheimmortalparticleinthe
timeinterval[0,t].Itisastablesubordinatorofindexβ.
Suppose now that β ∈ (0,1] is arbitrary. Take Y = 0 so that X¯ ≡ 0 in the
0 t
decomposition (1.3) and all “individuals” belong to families that split off from
the immortal particle at times s ≥0. Extend the definition of γ(u) given above
for u ∈ E to u ∈ E in the obvious way. The individuals, besides the immortal
0
particlealiveattime t >0,belongtofamiliesthatcorrespondtothesubsetA :=
t
{(s,u) ∈ (cid:10):0 ≤ t −s < γ(u)} of the random set (cid:10). At time t, the amount of
timesincethemostrecentcommonancestoroftheentirepopulationlivedisA :=
t
sup{t −s:(s,u)∈At}. As depicted in Figure 1, the MRCA age process (At)t≥0
hassaw-toothsamplepathsthatdriftupwithslope1untilthecurrentoldestfamily
is extinguished, at which time they jump downward to the age of the next-oldest
family.
It is not necessary to know the Poisson point process (cid:10) in order to construct
the MRCA age process (At)t≥0. Clearly, it is enough to know the point process
(cid:11) on R+ ×R++ given by (cid:11):={(s,γ(u)):(s,u)∈(cid:10)}. Indeed, if we define the
left-leaningwedgewithapexat(t,x)tobetheset
(1.4) (cid:15)(t,x):={(u,v)∈R2:u<t &u+v>t +x},
then
A =t −inf{s:∃x>0s.t.(s,x)∈(cid:11)∩(cid:15)(t,0)}
t
DYNAMICSOFTHEMRCA 5
FIG.1. Thepointsof(cid:11)aremarked“x”;thesamplepathoftheprocessAisdrawnwithasolidline;
theoldestextantfamilyattimet isrepresentedbythepointin(cid:11)labeled“O”andtheleft-leaning
wedge(cid:15)(t,0)isthedarklyshadedregionwithapexat(t,0).AttimeT,thefamilyrepresentedbyO
willdieoutandthefamilyrepresentedbyNwillcontaintheMRCAbecauseOisontheboundaryof
(thelightlyshaded)(cid:15)(T,0).IfthecoordinatesofO are(u,y)andthecoordinatesofN are(v,z),
thenthesizeofthejumpthatAmakesatT isAT−AT−=u−v=(T−v)−(T−u)=(T−v)−y.
(seeFigure1).
Note that (cid:11) is a Poisson point process with intensity λ⊗μ where μ is the
push-forward of ν by γ; that is, μ((t,∞))=ν({u:γ(u)>t}). We will show in
Section5thatμ((t,∞))=(1+β)/(βt).
With these observations in mind, we see that if (cid:11) is now any Poisson point
processonR+×R++ withintensityλ⊗μwhereμisanymeasureonR++ with
μ(R++)=∞ and 0<μ((x,∞))<∞ for all x >0, then the construction that
built(At)t≥0 fromtheparticularpointprocess(cid:11)consideredabovewillstillapply
andproduceanR+-valuedprocesswithsaw-toothsamplepaths.Wearetherefore
ledtothefollowinggeneraldefinition.
DEFINITION 1.1. Let(cid:11)beaPoissonpointprocessonR+×R++ withinten-
sity measure λ⊗μ where λ is Lebesgue measure and μ is a σ-finite measure on
R++ withμ(R++)=∞andμ((x,∞))<∞forallx>0.Define(At)t∈R+ by
A :=t −inf{s≥0:∃x>0s.t.(s,x)∈(cid:11)∩(cid:15)(t,0)},
t
where(cid:15)(t,0)isdefinedby(1.4),andA =0if(cid:11)∩(cid:15)(t,0)isempty.
t
We will suppose from now on that we are in this general situation unless we
specify otherwise. We will continue to use terminology appropriate for the ge-
nealogical setting and refer to (At)t≥0 as the MRCA age process and μ as the
lifetimemeasure.Wewillassumeforconveniencethatthemeasureμisabsolutely
continuous with a density m with respect to Lebesgue measure that is positive
Lebesguealmosteverywhere.Itisstraightforwardtoremovetheseassumptions.
6 S.N.EVANSANDP.L.RALPH
The strong Markov property of the Poisson point processes (cid:11) implies that
(At)t≥0 is a time-homogeneous strong Markov process. In particular, there is a
family of probability distributions (Px)x∈R+ on the space of R+-valued càdlàg
paths with Px interpreted in the usual way as the “distribution of (At)t≥0 started
from A =x.” More concretely, the probability measure Px is the distribution of
0
the process (Axt)t≥0 defined as follows. Let (cid:11)x be a point process on [−x,∞)×
R++ that has the distribution of the random point set {(t −x,y):(t,y)∈(cid:11)}∪
{(−x,Z)}whereZ isanindependentrandomvariablethatisdefinedonthesame
probability space ((cid:12),F,P) as the point process (cid:11), takes values in the interval
(x,∞)andhasdistribution
P{Z≤z}=μ((x,z])/μ((x,∞)).
Thenwecandefine
Ax :=t −inf{s≥−x:∃y>0s.t.(s,y)∈(cid:11)x ∩(cid:15)(t,0)}, t ≥0,
t
whereweadopttheconventionthatAx :=0ifthesetaboveisempty[seetheproof
t
ofpart(a)ofTheorem1.1belowforfurtherinformation].Fromnowon,whenwe
speakoftheprocess(At)t≥0 wewillbereferringeithertotheprocessconstructed
asinDefinition1.1fromthePoissonprocess(cid:11)definedonsomeabstractprobabil-
ity space ((cid:12),F,P) or to the canonical process on the space of càdlàg R+-valued
pathsequippedwiththefamilyofprobabilitymeasures(Px)x≥0.Thisshouldcause
noconfusion.
Wenoteinpassingthattheanalogueof(At)t≥0 intheconstantpopulationsize
Wright–FishersettingisnotMarkov(seeRemark4.1.3of[26]).
Weprovethefollowingpropertiesoftheprocess(At)t≥0 inSection2.
THEOREM 1.1. (a) The transition probabilities of the time-homogeneous
Markovprocess(At)t≥0 haveanabsolutelycontinuouspart
(cid:2) (cid:5) (cid:3)
μ((x,x+t]) x+t
Px{A ∈dy}= exp − μ((z,∞))dz μ((y,∞))dy,
t μ((x,∞)) y
fory<x+t andasingleatom
μ((x+t,∞))
Px{A =x+t}= .
t μ((x,∞))
(b) Thetotalrateatwhichtheprocess(At)t≥0 jumpsfromstatex>0is
m(x)
,
μ((x,∞))
and when the process jumps from state x >0, the distribution of the state to
whichitjumpsisabsolutelycontinuouswithdensity
(cid:2) (cid:5) (cid:3)
x
y(cid:17)→exp − μ((z,∞))dz μ((y,∞)), 0<y<x.
y
DYNAMICSOFTHEMRCA 7
(c) The probability P0{∃t > 0:At = 0} that the process (At)t≥0 returns to the
statezeroispositiveifandonlyif
(cid:5) (cid:2)(cid:5) (cid:3)
1 1
exp μ((y,∞))dy dx<∞.
0 x
(d) If
(cid:5) (cid:2) (cid:5) (cid:3)
∞ x
exp − μ((y,∞))dy dx=∞,
1 1
then for each x >0 the set {t ≥0:A =x} is Px-almost surely unbounded.
t
Otherwise,limt→∞At =∞,Px-almostsurely,forallx≥0.
(e) Astationarydistributionπ existsfortheprocess(At)t≥0 ifandonlyif
(cid:5)
∞
μ((z,∞))dz<∞,
1
inwhichcaseitisunique,and
(cid:2) (cid:5) (cid:3)
∞
π(dx)=μ((x,∞))exp − μ((z,∞))dz dx.
x
(f) If(At)t≥0 hasastationarydistributionπ,then
(cid:2) (cid:5) (cid:3)
∞ μ([x,x+t))
d (Px{A ∈·},π)≤1−exp − μ((y,∞))dy × ,
TV t t+x μ([x,∞))
where d denotes the total variation distance. In particular, the distribution
TV
ofA underPx convergestoπ intotalvariationast →∞.
t
Specializing Theorem 1.1 to the case when A is the MRCA age process of the
conditioned critical (1+β)-stable continuous state branching process gives parts
(a)to(d)ofthefollowingresult.Part(e)followsfromanobservationthataspace–
timerescalingofthisMRCAageprocessisatime-homogeneousMarkovprocess
that arises from another Poisson process by the general MRCA age construction
ofDefinition1.1.TheproofisinSection5.
COROLLARY 1.1. Suppose that A is the MRCA age process associated with
thecritical(1+β)-stablecontinuousstatebranchingprocess.
(a) The transition probabilities of the process A have an absolutely continuous
part
(1+β)ty1/β
Px{A ∈dy}= dy, 0<y<x+t,
t β(x+t)2+1/β
andasingleatom
x
Px{A =x+t}= .
t x+t
8 S.N.EVANSANDP.L.RALPH
(b) The total rate at which the process A jumps from the state x >0 is 1/x, and
when it jumps from state x >0, the distribution of the state to which it jumps
isabsolutelycontinuouswithdensity
y1/β
(1+1/β) , 0<y<x.
x1+1/β
(c) TheprobabilityP0{∃t >0:A =0}thattheprocessAreturnstothestatezero
t
is0.
(d) Foreachx≥0,limt→∞At =∞,Px-almostsurely.
(e) Theprocess
(e−tAet)t∈R
indexed by the whole real line is a time-homogeneous Markov process under
Px for any x ≥0, and it is stationary when x =0. Moreover, A /t converges
t
in distribution to the Beta(1+1/β,1) distribution as t →∞ under Px for
any x ≥0, and A /t has the Beta(1+1/β,1) distribution for all t >0 when
t
x=0.
Note that the sample paths of (At)t≥0 have local “peaks” immediately before
jumps and local “troughs” immediately after. We investigate the discrete time
MarkovchainofsuccessivepairsofpeaksandtroughsinSection4.Wealsocon-
siderthejumpheightsandinter-jumpintervalsanddescribeaninterestingduality
betweenthesesequencesinSection3.
Finally,recallthattheBessel-squaredprocessindimensionγ,whereγ isanar-
bitrarynonnegativerealnumber,istheR+-valueddiffusionprocesswithinfinites-
imalgenerator 2xd2/dx2+γ d/dx.When γ isapositiveinteger,suchaprocess
has the same distribution as the square of the Euclidean norm of a Brownian mo-
tion in Rγ. Feller’s critical continuous state branching process is thus the zero-
dimensionalBessel-squaredprocess,moduloachoiceofscaleintimeorspace.It
wasshowninExample3.5of[28]thatfor 0≤γ <2,theBessel-squaredprocess
withdimension γ conditionedonneverhittingzeroistheBessel-squaredprocess
with dimension 4−γ. Thus for β = 1, the conditioned process Y is the four-
dimensional Bessel-squared process. We introduce a new family of processes in
Section 6 that are also indexed by a nonnegative real parameter and play the role
oftheBessel-squaredfamilyforvaluesofβ otherthan1.Theseprocesseswillbe
studiedfurtherinaforthcomingpaper.
We end this introduction by commenting on the connections with previous
work. First, we may think of each point (s,x)∈(cid:11) as a “job” that enters a queue
with infinitely many servers at time s and requires an amount of time x to com-
plete.Wethushaveaclassical M/G/∞ queue[31],except weareassumingthat
the total arrival rate of jobs of all kinds is infinite. With this interpretation, the
quantity A is how long the oldest job at time t has been in the queue. Properties
t
DYNAMICSOFTHEMRCA 9
ofsuch M/G/∞ queues withinfinitearrivalrateshavebeenstudied(see,forex-
ample,[10]),buttheageoftheoldestjobdoesnotappeartohavebeenstudiedin
thiscontext.
Second,notethattheprocess(At)t≥0isanexampleofapiecewisedeterministic
Markov process: it consists of deterministic flows punctuated by random jumps.
Such processes were introduced in [5] and studied further in [6] (see also [20],
where the nomenclature jumping Markov processes is used). The general proper-
tiesofsuchprocesseshavebeenstudiedfurtherin,forexample,[3,4,7].
Last, piecewise deterministic Markov processes like (At)t≥0 that have periods
oflinearincreaseinterspersedwithrandomjumpshavebeenusedtomodelmany
phenomena, such as stress in an earthquake zone [2], congestion in a data trans-
missionnetwork[8]andgrowth-collapse[1].Theyalsohaveappearedinthestudy
oftheadditivecoalescent[12]andR-tree-valuedMarkovprocesses[13].
2. ProofofTheorem1.1. (a)Supposethat (At)t≥0 isconstructedfrom (cid:11) as
in Definition 1.1. For s ≥x consider the conditional distribution of As+t given
A = x. The condition A = x is equivalent to the requirements that there is a
s s
point (s −x,Z) in (cid:11) for some Z >x and that, furthermore, there are no points
of (cid:11) in the left-leaning wedge (cid:15)(s −x,x). The conditional probability of the
event {Z >z}, given A =x, is μ((z,∞))/μ((x,∞)) for z ≥x. If Z >x +t,
s
thenAs+t =x+t.Otherwise,As+t <x+t.Thesecondclaimofpart(a)follows
immediately.
Now consider P{As+t ∈ dy|As = x} for y < x +t. This case is depicted in
Figure2.Byconstruction,As+t =y ifandonlyifthereisapoint(s+t−y,W)∈
(cid:11) forsome W >y,andtherearenopointsof (cid:11) in (cid:15)(s+t −y,y).Fromabove,
theconditionA =x requirestheretobenopointsof(cid:11)inthewedge(cid:15)(s−x,x)
s
FIG.2. ThecomputationofP{As+t ∈dy|As=x}.
10 S.N.EVANSANDP.L.RALPH
(thelightlyshadedregioninFigure2).Therefore,ifAs =x,thenAs+t =y ifand
onlyifZ≤x+t,therearenopointsof(cid:11)inthedarklyshadedregionofFigure2
andthereisapointof(cid:11)oftheform(s+t −y,w)withw>y.
Now,theconditionalprobabilityoftheeven{Z≤x+t}givenA =x is
s
μ((x,x+t])
.
μ((x,∞))
Theprobabilitythatnopointsof(cid:11)areinthedarklyshadedregionis
(cid:2) (cid:5) (cid:3)
x+t
exp − μ((u,∞))du .
y
The probability that (cid:11) has a point in the infinitesimal region [s +t −y,s +t −
y +dy]×(y,∞) is μ((y,∞))dy. Multiplying these three probabilities together
givesthefirstclaimofpart(a).
(b)Bothclaimsfollowreadilybydifferentiatingtheformulaein(a)at t =0.It
isalsopossibletoarguedirectlyfromtherepresentationintermsof(cid:11).
(c)Supposethat(At)t≥0 isconstructedfrom(cid:11)asinDefinition1.1.Notethat
(cid:6) (cid:7)
{t >0:At =0}=R++ (t,t +x).
(t,x)∈(cid:11)
Thequestionofwhensucha“Poissoncutout”randomsetisalmostsurelyempty
wasaskedin[23],andthenecessaryandsufficientconditionpresentedinpart(c)
issimplytheonefoundin[30](seealso[18]).
(d)Forx>0,therandomset{t ≥0:A =x}isadiscreteregenerativesetunder
t
Px thatisnotalmostsurelyequalto{0}.Hencethissetisarenewalprocesswithan
inter-arrival time distribution that possibly places some mass at infinity; in which
case, the number of arrivals is almost surely finite with a geometric distribution,
andthesetisalmostsurelybounded.
Suppose that the set {t ≥0:A =x} is Px-almost surely unbounded for some
t
x >0. Let 0=T <T <··· be the successive visits to x. It is clear that Px{∃t ∈
0 1
[0,T ]:A = y} > 0 for any choice of y > 0 and hence, by the strong Markov
1 t
property, the set {t ≥ 0:A = y} is also Px-almost surely unbounded. Another
t
application of the strong Markov property establishes that the set {t ≥0:A =y}
t
is Py-almost surely unbounded. Thus the set {t ≥0:A =x} is either unbounded
t
Px-almostsurelyforallx>0orboundedPx-almostsurelyforallx>0.
Recall that away from the set {t ≥0:A =0} the sample paths of A are piece-
t
wiselinearwithslope1.Itfollowsfromthecoareaformula(see,e.g.,Section3.8
of[24])that
(cid:5) (cid:5)
∞ ∞
f(A )dt = f(y)#{t >0:A =y}dy
t t
0 0
foraBorelfunctionf :R+→R+.HencebyFubini’stheorem,
(cid:5) (cid:5) (cid:5)
∞ ∞ ∞ Px{A ∈dy}
f(y)Ex[#{t >0:A =y}]dy= f(y) t dtdy
t
dy
0 0 0