www.imstat.org/aihp 2014, Vol. 50, No. 1, 84–94
DOI:10.1214/12-AIHP504
© Association des Publications de l’Institut Henri Poincaré, 2014
A branching-selection process related to censored Galton–Walton processes
Olivier Couronné and Lucas Gerin
Université Paris Ouest Nanterre La Défense, Equipe Modal’X, 200 avenue de la République, 92000 Nanterre, France.
E-mail:ocouronn@u-paris10.fr
Received 2 November 2011; revised 4 June 2012; accepted 19 June 2012
Abstract. We obtain the asymptotics for the speed of a particular case of a particle system with branching and selection introduced by Bérard and Gouéré [Comm. Math. Phys.298(2010) 323–342]. The proof is based on a connection with a supercritical Galton–
Watson processcensoredat a certain level.
Résumé. Nous étudions un cas particulier de système de particules avec branchement et sélection introduit par Bérard et Gouéré [Comm. Math. Phys.298(2010) 323–342]. Nous obtenons l’asympotique pour la vitesse, en remarquant un lien avec un processus de Galton–Watson surcritique censuré à un certain niveau.
MSC:60J80
Keywords:Galton–Watson process; Censored branching process; Branching and selection
1. Models and results
1.1. The censored Galton–Watson process
For a probability distributionX on non-negative integers, the Galton–Watson process withoffspringX is the process defined byZa0=aand
Zka+1=
Zka
i=1
Xk,i,
where theXk,i’s are i.i.d. copies ofX (Zak+1=0 ifZak =0). We write(Zk)=(Zk1). We deal here with supercritical Galton–Watson processes (i.e. whenE(X) >1) and to avoid trivialities we imposeX(0) >0. In this case, it is well- known (we refer to [1] for basic facts on branching processes) thatZkdies with a certain probability 0< q <1 which is the unique solution in(0,1)of
f (x):=
i≥0
X(i)xi=x.
The first step of this article will be to study a kind of constrained Galton–Watson process, in which the constraint is a
“roof” at a given height that prevents the process from exploding.
Definition 1. Given N a positive integer and a probability distributionX on integers,the Galton–Watson process censored at levelN≥2with offspringX is the process(XkN)k≥0defined byXN0 =N and,fork≥0,
XNk+1=
min
N,XNk i=1Xk,i
ifXNk >0,
0 otherwise,
where theXk,i’s are i.i.d.copies ofX.
The process(XNk )k≥0is a finite state Markov process which is typically stuck for a long time onN, but eventually dies. LetUN be thesurvival timeof the Galton–Watson process:
UN=min
k≥0, XkN=0 .
Let us describe heuristically the asymptotic behavior ofUN. When the censored process dies, it happens very sud- denly: in the uncensored underlying process, the progenies of theN particles pass away almost simultaneously in a few generations. This latter event occurs with a probability close toqN, and therefore the censored process is expected to survive a time close to a geometric random variable with parameterqN.
We state this in the following theorem, which may be derived from results of [5].
Theorem 1. The following convergence holds in law:
UNqN N→∞→ E(1), (1)
whereE(1)stands for the exponential distribution with parameter one.The convergence also holds in mean:
E(UN)∼(1/q)N asN→ ∞. (2)
The convergence in law (1) is a consequence of ([5], Th. 1 (3), (4)) and the asymptotic (2) follows from ([5], Th. 1 (2)). The results of [5] are actually much accurate. As we will only need these simple estimates, and for the sake of completeness, we provide in the two following sections an elementary and more concise proof of Theorem1.
We also mention [6], in which the Galton–Watson process is censored by a function depending on time; the author obtained a criterion for the degeneracy of the process.
1.2. A connection with a branching-selection process
The present authors considered Theorem1when trying to find an asymptotic of the speed of a very particular case of a certain branching particle system with selection, studied by Bérard and Gouéré ([2], Section 7, Theorem 5). The branching-selectionprocess we will study is a generalization of their Bernoulli branching-selection process and can be described as follows. For a distribution(X,X)onN2such that the expectation ofXis strictly greater than 1 and X+X≥1 a.s., the particle system is the discrete-time particle system ofNparticles moving onZand starting from the origin such that, at each time unit,
1. Each of theN particles is replaced byX particles just on the right of that particle and byXparticles on the same position, independently of each other;
2. Among all these new particles, we keep only theNrightmost particles.
It is convenient to see the process of the locations of theN particles as a finite point measure. We setY0N=N δ0
and write YkN=
≥0
δYkN(),
whereYkN()is the number of particles at timekat the position; by construction we have at any time
≥0YkN()= N. We write
maxYkN:=max
≥0;YkN() >0
for the position of the rightmost particle at timek. We have maxYkN≤kby construction. SinceE(X) >1, and since YkN+1≥YkN by the property ofX, it is likely forN large that maxYkN is close tok. Remark also that in this process, all the particles at timekare on{maxYkN−1,maxYkN}after the selection.
Bérard and Gouéré in fact studied the particle system defined as follows. At every time unit each particle is dupli- cated and moves one step forward with probabilityα∈(12,1)and stays put with probability 1−α. With our notations, this corresponds to the case whereX is a Bernoulli variableB(2, α)withα > 12, andX=2−X (we will denote their model as theBernoulli branching-selection process). They proved that(maxYkN)/kconverges almost surely to a constantvN(α), and noticed thatvN(α)lies between(1−exp(−c1(α)N ))and(1−exp(−c2(α)N ))for some positive constantsc1(α)andc2(α). The stress of their paper was a large class of distributions for the walk performed by the particles, and the case of the Bernoulli random walk withα >1/2 was in fact just mentioned as a degenerate case.
The existence of the asymptotic speedvN(α)would apply as well in our case, but not their proof for the bounds on this speed.
The key point for our result is that the process of the number of particles that are at the rightmost possible position (YkN(k))k has the same law as the censored Galton–Watson process(XNk )k, when the offspring distribution isX. Theorem 2. For the branching-selection process with the distribution(X,X)onN2such thatE(X) >1andX + X≥1a.s.,we have
Nlim→∞
1−vN(α) qN =1,
whereqis the extinction probability of a Galton–Watson process whose offspring isX. In particular, for the Bernoulli branching-selection model studied in [2] withα >12,
Nlim→∞
1−vN(α) (qα)N =1, where
qα=1−2α(1−α)−√
1−4α(1−α) 2α2
is the extinction probability of a Galton–Watson process whose offspring is the Binomial distribution with parameters (2, α).
Remark that the exact law ofXhas no influence, and we could have always taken X=1X=0.
UsingXallows us essentially to have a true generalization of the Bernoulli branching-selection process.
The article is organized as follows. The proof of Theorem1is decomposed over Sections2and3. A key ingredient in Section3is to compare our censored process at the last time it reaches the levelN, and the classical Galton–Watson process starting at N and conditioned to die. The main contribution is the application to the branching-selection particle system and is given in Section4.
2. Preliminaries
We will repeatedly use the following left-tail bound for a sum of copies ofX:
Lemma 1. LetX1, . . . , XNbe i.i.d.copies of lawX.There existb, c >0such that,whenNis large enough, P
X1+ · · · +XN≤N (1+b)
≤exp(−cN ).
To prove this, we chooseM such thatE[min{X1, M}]>1. We then can apply ([4], III, Chap. 4) to the bounded random variables min{Xi, M}and we get the desired bound. This implies in particular that
P
XNk+1< N|XNk =N
≤exp(−cN ). (3)
Note that throughout the paper,cstands for a generic positive constant, which might differ at each appearance. We introduce the last time for which our process is equal toN. This variable will turn out to be equivalent toUN and more simple to approximate by a geometric variable.
Definition 2. Let VN=max
k≥0, XkN=N ,
and letqNbe the probability that(XNk)kdoes not ever hitN after time zero:
qN=P
XNk < N,∀k >0
=P(VN=0).
We deduce from (3) that qN≤P
X1N< N
≤exp(−cN ). (4)
We will compareqN with the probability that the classical Galton–Watson process(ZNk )starting fromNparticles dies out. By independence of the progeny of theN particles, we haveP((ZkN)dies out)=qN. By conditioning over the first passage over some leveln, we get
P
(Zk)dies out|∃k≥0 such thatZk≥n
≤qn. (5)
3. Proof of Theorem1
The main tool of the proof is the following lemma, which says that the Galton–Watson process(ZkN), conditioned to die, roughly behaves likeXNk after its last passage inN. Recall thatqstands for the probability thatZdies whileqN
is the probability thatXNdoes not ever hitN.
Lemma 2. There existsc >0such that forN large enough qN
qN −1
≤exp(−cN ).
Proof. By Lemma1, we can takeb >0 andc >0 such that P
ZN1 < (1+b)N
≤exp(−cN ).
The point is that if(ZkN)k dies out, then with high probability it does not hitN after timek=0. LetR=inf{k≥ 1, ZkN≥N}(with the convention thatR= +∞isZndoes not hitNafter time zero),
P
R <∞;ZNdies out
≤P
Z1N< (1+b)N;R <∞;ZNdies out +P
ZN1 ≥(1+b)N;ZNdies out
≤P
Z1N< (1+b)N;R <∞;ZNdies out
+q(1+b)N
≤
≥N
P
ZNR =;R <+∞;ZN1 < (1+b)N;ZNdies out
+q(1+b)N
≤
≥N
P
ZNd.o.|ZRN=;R <+∞;Z1N< (1+b)N P
ZR=;R <+∞;Z1N< (1+b)N
+q(1+b)N
≤
≥N
P
ZNd.o.|ZRN=;R <+∞
P
ZN1 < (1+b)N
+q(1+b)N
≤
≥N
qe−cN+q(1+b)N≤ 1
1−q exp(−cN )qN+q(1+b)N. Finally, forN large enough
P
ZNdies out
−P(VN=0)=P
ZN dies out, R <∞
≤qNexp
−cN
, (6)
which finishes the proof.
By constructionUN≥VN, we first prove thatUNis close toVN, in the following sense.
Lemma 3. The sequence(1+UNV
N)N≥1converges to1in probability.
Proof. By definition we haveUN≥VN+1. Forε >0, P
UN
1+VN −1≤ −ε
≤P
UN−(VN+1) >√ εN
+P
UN< 1
√εN
.
We study the first term using a standard technique in branching processes theory:
P Zk1
dies;ZK>0
=q−fK(0),
wherefK is both theKth iterate off and the generating function ofZK. Our assumptions imply that|q−fK(0)| ≤ exp(−cK)for somec >0 ([1], Th. 1, Chap. I.11).
By the Markov property, the process(XNk+)≥0conditioned toXkN=N has the same law as(XN)≥0, it follows that
P
UN−(VN+1) > K
=P(UN> K+1|VN=0).
Recall (6):
P(VN=0)≥P
ZN dies out
−2qNexp
−cN
≥qN/2 forN large enough. We then write
P(UN> K+1|VN=0)=P
ZKN+1>0|ZkN< N for eachk≥1
≤P
ZKN+1>0|
ZkN
dies out
× P((ZNk )dies out) P(ZkN< Nfor eachk≥1)
≤NP
ZK1+1>0|
ZkN
dies out
× qN qN/2
≤2Nexp(−cK)
q , (7)
where we finally used ([1], Sec. I.11). We obtain P(UN−VN−1>√
εN )≤CNexp(−c√ εN )
for someCandc >0 and for anyN≥1 andε >0. The second term is handled thanks to (4) as follows. ForN >1/ε, P
UN<
1
√εN
≤P
VN<
1
√εN
≤P
∃v≤ 1
√εN
, VN=v
≤P
∃v≤ 1
√εN
, XvN=N andXvN+1< N
≤ 1
√εNexp(−cN )
for a certainc >0. LettingN go to infinity finishes the proof.
We introduce another process and an associated time whose properly renormalized law will converge and which is equivalent toVN. LetAk be thekth passage inN of the censored process:A0=0 and fork≥0,Ak+1=inf{ >
Ak, XN=N}, with the usual convention that inf(∅)= +∞. LetT be the survival time ofA:
T =sup{k≥0, Ak<∞}.
It is clear by contruction thatT has the law ofG(qN)−1, whereG(r)is a geometric of parameterr. By the Markov property of our branching process, the variablesAk+1−Ak are independent and identically distributed. With high probability,Ak+1=Ak+1, we can be more precise:
Lemma 4. There existsc >0such that for allN P(Ak+1> Ak+1)≤exp(−cN ).
There existsc, C >0such that for allK≥1,for allNlarge enough, P(Ak+K < Ak+1<∞)≤Cexp
−c K logN −cN
.
Proof. The first assertion is just (3). The second one is a consequence of the following inequality: for any 0< i < N P
0< Zki < Nfor allk≤K
≤Cexp
−c K logN
. (8)
To prove so, take 1< μ< μ≤ +∞. The sequenceZk/μktends to infinity with positive probability (see [1], Th. 3, Chap. I.10). Hence we have positive constantsδ, βsuch that, for any integersk, n,
P
Znk> δμk
≥P
Zk1> δμk
≥β. (9)
Now, setm= log(N/δ)/log(μ). We assume thatNis large enough, such thatm≥1. We first assume thatK > m.
{0< Zk< Nfor allk≤K} ⊂ {Zm< N} ∩ {Zm>0 andZ2m< N}
∩ {Z2m>0 andZ3m< N} ∩ · · · ∩ {ZK/mm−m>0 andZK/mm< N}. Thanks to (9), the probability of the right-hand side is smaller than(1−β)K/m. IfK≤mthenK/logN≤2/logμ for N large enough. By choosingC large enough, the right-hand side in (8) is greater than one, and the claimed inequality also holds.
To finish the proof of the lemma, it suffices to remark that forK≥1, P(Ak+K < Ak+1<∞)≤P(Ak+1< Ak+1<∞)P
0< ZN < N∀≤K
≤Cexp(−cN )exp
−c K logN
.
Lemma 5. The sequence(11++VTN)N≥1converges to one in probability.
Proof. By construction we haveT ≤VN. In order to bound the difference, we first notice that,T being aG(qN)−1, we have for large enoughNthe bound
P(T <1/√
qN)≤(1/√
qN+1)P(T =0)≤(1/√
qN+1)qN≤2√
qN≤3qN/2, (10)
where we finally used Lemma2. We now write
VN(law)= T
k=0
Ak(law)= T+ T
k=0
Ak,
where(Ak)is a sequence of i.i.d. random variables with the same law asA1−1 conditioned byA1<∞, and the sequence is independent fromT.
Now takeε >0. By Lemma4, forNlarge enough we haveE(A1)≤ε/2.
P
VN+1−(T+1)≥ε(T +1)
=P T
k=0
Ak≥ε(T +1)
≤P T
k=0
Ak−E(Ak)≥(T+1)ε/2
≤P T
k=0
Ak−E(Ak)≥ε(T+1)/2;T ≥1/√ qN
+P(T <1/√ qN)
≤
t≥1/√ qN
P t
k=0
Ak−E(Ak)≥ε(t+1)/2
+P(T <1/√ qN)
≤
t≥1/2√
qN
P t
k=0
Ak−E(Ak)≥ε(t+1)/2
+P(T <1/√ qN),
where we finally used Lemma2. The second term on the right-hand side goes to zero thanks to (10). Let us now handle the sum. Using Lemma4we have
P(Ak≥K)=P(K≤A1<∞)P(A1<∞)−1≤2 exp(−cK/logN )
for a certainc >0. From such a tail, we deduce (see [4], Sec. III.4) that there existsc >0 such that for allt,εandN, P
t
k=0
Ak−E(Ak)≥εt /2
≤exp
−cε t logN
,
and the sum goes to zero.
End of proof of Theorem1. We now write UNqN= UN
VN+1 qN qN
qN(T+1)VN+1 T+1 , where, whenNgoes to infinity,
UN/(VN+1)→1 (in prob.) by Lemma3, qN/qN→1 by Lemma2,
qN/(T +1)→E(1) (in law) sinceT is geometric with parameterqN, (VN+1)/(T +1)→1 (in prob.) by Lemma5.
This proves the convergence in distribution. For the convergence in mean, recall that one may write E[VN] =E
T +
T
k=0
Ak
=E[T]
1+E[A1]
=qN
1+o(1) ,
thanks to Lemmas2and4. SinceE[UN] =E[VN] +E[UN−VN], we finally obtain with (7)
E[UN] ∼E[VN] ∼(1/q)N, (11)
which will be useful in the next section.
4. Application: Branching-selection particle system
We now describe more precisely the connection with the branching-selection process defined in the Introduction, in order to exploit the results of the previous section. Recall thatYkN()is the number of particles at timekat the position and thatvN:=limk→∞(maxYkN)/k(a.s.).
The connection with the censored Galton–Watson process in is the following lemma, whose proof is immediate by construction of the process.
Lemma 6. In the branching-selection process with a pair (X,X)of laws onN,the number of particles that are at the rightmost possible position has the law of a censored Galton–Watson process,when the offspring distribution isX:
YkN(k)
k≥0 law=
XkN
k≥0.
In particular, if we defineV1as the last time at which theN particles are at the rightmost possible location:
V1=max
k;YkN(k)=N ,
then V1 law= VN, whereVN, defined in the previous section, is the last time at which the censored Galton–Watson hitsN.
We now are ready to estimatevN. Lemma 7. For all integerN≥1,
vN≥1− 1 E(VN)+1,
where the progeny distribution definingVNisX.
Proof. The proof is inspired by that of Proposition 4 in [2]. The main idea is to design adominatedprocessY˜kmoving slower than thetrueprocessYk.
Let us skip the exponent N in order to lighten the notations, and consider an i.i.d. family (X,k,i, X,k,i )≥0,k≥0,i∈[1,N]of integer variables with the same law as(X,X).
We sample the particle system(Yk)by means of these variables: the populationYkat timekbeing given, we let
Tk+1=∞
=0 Yk()
i=1
X,k,iδ+1+X,k,i δ
be the locations of the particles after branching. The populationYk+1is then obtained by keeping only theNrightmost particles amongTk+1.
We modifyY in the following way. At timeV1+1, let continue the process as if all the particles were at position V1at timeV1+1. More precisely, set(Yk(1))=(Yk)and introduce a process(Yk(2))k≥V1 starting from
Y(2)
V1+1=N δV1.
Then, fork≥V1+1,Yk(2)+1is sampled fromYk(2)exactly asYk+1is sampled fromYkN(with the same(X,k,i,X,k,i )’s).
The main point is that, at timeV1+1, theNparticles ofY(1)
V1+1are at positionsV1orV1+1 while theN particles ofY(2)
V1+1are at positionV1. Hence, the point measureY(1)
V1+1dominates Y(2)
V1+1, and this domination will continue throughout the process, since the same(X,X)’s are used to generateY(1)andY(2).
Now, letV2 be such thatV1+1+V2is the last time k at which theN particles are at the rightmost possible location (which is positionk−1) forY(2):
V2=max
k;Yk(2)(k−1)=N
− V1+1
.
The random timeV2has the same law asV1and, as a function of{X,k,i,X,k,i ; > V1}, is independent ofV1. We define recursively similar processesY(3), Y(4), . . .and random variablesV3, V4, . . .. Setting also
Γ0=0, Γi+1=Γi+Vi+1, theΓi’s are renewals.
Let us introduce a last process(Y˜k)k≥0as follows: ifk∈ [Γi−1, Γi), then Y˜k=Yk(i).
One can see in Fig.1the first steps of a realization of the process(Yk), and in Fig.2the corresponding modificated process(Y˜k). By construction,YkdominatesY˜k for eachk, and in particular maxYk≥maxYk. Let us also note that at each renewalΓi the quantity maxYk is decreased by one. SetIkbe the renewal process associated to the renewals Γi, that isIkis the only integerisuch thatΓi ≤k < Γi+1.On the one hand we have
Γi+1−Γilaw=V1+1,
so by applying the renewal theorem (see for instance [3], Chap. 3.4) to the renewalsΓ1, Γ2, . . .we get
klim→∞
1
kE(Ik)→ 1 E[V1] +1. On the second hand we have
maxYk=k−Ik.
Fig. 1. A realization of thetrueprocessYwithN=3 particles, for the particular case of the Bernoulli branching-selection process. At each step are represented the 6 particles (after replication) and the 3 leftmost particles are striked. Here, we see thatV1=3, since from timek=5 there can be no more particles at the rightmost possible position, and because at time 4 only one particle is on the rightmost possible position.
Fig. 2. A realization of the associateddominatedprocess: at timeVN1+1=4, we let the process restart withallparticles at position 3.
Hence we get 1
kE[maxYk] ≥1
kE[maxYk]k→∞→ 1− 1 E[V1] +1.
To conclude, recall that thanks to the connection with the censored Galton–Watson processE[V1] =E[VN].
Lemma 8. For all integerN≥1, vN≤1− 1
E(UN).
Proof. The proof is an adaptation of the previous one, replacingV1byU1, which is the first time at which there is no more particle at the rightmost possible position:
U1=min
k;YkN(k)=0 .
(On the example drawn in Fig.1, one hasU1=5.) Then,U1 has the same law as theUN defined in Section2. In a similar manner to the previous proof, we let the process restart at time U1as if all the particles were at position
U1−1. The end of the proof is similar.
We now combine these two estimates ofvNwith the results of the previous section in order to prove Theorem2:
1
E(UN)≤1−vN≤ 1 E(VN)+1
and both sides, thanks to (11), are equivalent toqN. Acknowledgments
We warmly thank J.-B. Gouéré for many helpful comments on this work. We are also very thankful to the two referees and the associate editor for recommending numerous improvements.
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