The almost sure limits of the minimal position and the additive martingale in a branching random walk.

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The almost sure limits of the minimal position and the additive martingale in a branching random walk.

Yueyun Hu

To cite this version:

Yueyun Hu. The almost sure limits of the minimal position and the additive martingale in a branching random walk.. 2012. �hal-00755972v2�

The almost sure limits of the minimal position and the additive martingale in a branching random walk

Yueyun Hu¹ Universit´e Paris XIII

Summary. Consider a real-valued branching random walk in the boundary case. Using the techniques developed by A¨ıd´ekon and Shi [5], we give two integral tests which de- scribe respectively the lower limits for the minimal position and the upper limits for the associated additive martingale.

1 Introduction

Let{V(u), u∈T}be a discrete-time branching random walk on the real lineR, whereTis an Ulam-Harris tree which describes the genealogy of the particles andV(u) ∈Ris the position of the particle u. When a particleuis atn-th generation, we write|u|=nforn≥0. The branching random walkV can be described as follows: At the beginning, there is a single particle ∅located at 0. The particle∅ is also the root of T. At the generation1, the root dies and gives birth to some point process Lon R. The point process L constitutes the first generation of the branching random walk {V(u),|u| = 1}. The next generations are defined by recurrence: For each|u|=n(if suchuexists), the particleudies at the(n+ 1)-th generation and gives birth to an independent copy ofLshifted byV(u). The collection of all children of allutogether with their positions gives the(n+ 1)-th generation. The whole system may survive forever or die out after some generations.

PlainlyL=P

|u|=1δ_{V(u)}. AssumeE[L(R)]>1and that E

Z

e^−xL(dx)

= 1, E

Z

xe^−xL(dx)

= 0. (1.1)

When the hypothesis (1.1) is fulfilled, the branching random walk is called in the boundary case in the literature (see e.g. Biggins and Kyprianou [9] and [10], A¨ıd´ekon and Shi [5]). Under some integrability con- ditions, a general branching random walk can be reduced to the boundary case after a linear transformation, see Jaffuel [19] for detailed discussions. We shall assume (1.1) throughout this paper.

Denote by M_n := min_|u|=nV(u) the minimal position of the branching random walk at generationn (with conventioninf∅ ≡ ∞). Hammersly [17], Kingman [20] and Biggins [8] established the law of large numbers forM_n(for any general branching random walk), whereas the second order limits have recently attracted many attentions, see [1, 18, 12, 2] and the references therein. In particular, A¨ıd´ekon [2] proved the

1D´epartement de Math´ematiques, Universit´e Paris 13, Sorbonne Paris Cit´e, LAGA (CNRS UMR 7539), F–93430 Villetaneuse.

Research partially supported by ANR 2010 BLAN 0125 Email: yueyun@math.univ-paris13.fr

convergence in law ofM_n−³₂lognunder (1.1) and some mild conditions, which gives a discrete analog of Bramson [11]’s theorem on the branching brownian motion.

Concerning the almost sure limits of M_n, there is a phenomena of fluctuation at the logarithmic scale ([18]): Under (1.1) and some extra integrability assumption: ∃δ > 0 such that E[L(R)^1+δ] < ∞ and E R

R(e^δx+e^−(1+δ)x)L(dx)

<∞, the following almost sure limits hold:

lim sup

n→∞

M_n

logn = 3

2, P^∗-a.s., lim inf

n→∞

M_n

logn = 1

2, P^∗-a.s., where here and in the sequel,

P^∗(·) :=P(·|S),

andSdenotes the event that the whole system survives. The upper bound ³₂lognis the usual fluctuation for M_nbecause M_n− ³₂lognconverges in law ([2]). It is a natural question to ask howM_ncan approach the unusual lower bound ¹₂logn.

A¨ıd´ekon and Shi [5] proved that under (1.1) and the following integrability conditions σ² :=Eh Z

R

x²e^−xL(dx)i

<∞, (1.2)

Eh

η(log₊η)²+ηelog₊eηi

<∞, (1.3)

whereη:=R

Re^−xL(dx),ηe:=R_∞

0 x e^−xL(dx)andlog₊x:= max(0,logx), then lim inf

n→∞

M_n−1 2logn

=−∞, P^∗-a.s.

Furthermore, they asked whether there is some deterministic sequencea_n→ ∞such that

−∞<lim inf

n→∞

1 a_n

M_n−1 2logn

<0, P^∗-a.s.?

The answer is yes: we can choosea_n= log logn. Moreover, we can give an integral test to describe the lower limits ofM_n:

Theorem 1.1 Assume(1.1),(1.2)and(1.3). For any functionf ↑ ∞, P^∗

M_n−1

2logn <−f(n), i.o.

=



 0 1

⇐⇒

Z _∞ dt texp(f(t))





<∞

=∞

, (1.4)

where i.o. means infinitely often as the relevant indexn→ ∞.

As a consequence of the integral test (1.4), we have that for anyε >0,P^∗-a.s. for all largen≥n0(ω), M_n − ¹₂logn ≥ −(1 +ε) log logn whereas there exists infinitely often n such that M_n − ¹₂logn ≤

−log logn.HenceP^∗-a.s.,lim inf_{n→∞ log log}¹ _n(M_n− ₂¹logn) =−1.

The behaviors of the minimal position M_n are closely related to the so-called additive martingale (W_n)_n≥0:

W_n:= X

|u|=n

e^−V(u), n≥0, with the usual convention: P

∅ ≡ 0. By Biggins [8] and Lyons [23],W_n → 0almost surely asn → ∞. The problem to find the rate of convergence (or a Seneta-Heyde norming) for Wn arose in Biggins and Kyprianou [9] and was studied in [18]. A¨ıd´ekon and Shi [5] gave a definite result to this problem. Let

Dn:= X

|u|=n

V(u)e^−V(u), n≥1, (1.5)

be the derivative martingale (which is a martingale under the boundary condition (1.1)). It was shown in Biggins and Kyprianou [9] thatP-a.s.,D_nconverges to some nonnegative random variableD_∞. Moreover under (1.1), (1.2) and (1.3),P^∗-a.s.,D_∞>0, as shown in [9] and [2].

Theorem (A¨ıd´ekon and Shi [5]).Assume(1.1),(1.2)and(1.3). Then underP^∗,

√nW_n ^(p)→ r 2

πσ² D_∞, asn→ ∞. Moreover

lim sup

n→∞

√n W_n=∞, P^∗-a.s.

Furthermore A¨ıd´ekon and Shi conjectured that lim inf

n→∞

√n W_n= r 2

πσ² D_∞, P^∗-a.s. (1.6)

The upper limits ofW_ncan be described as follows:

Theorem 1.2 Assume(1.1),(1.2)and(1.3). For any functionf ↑ ∞,P^∗-almost surely, lim sup

n→∞

√n W_n f(n) =



 0

∞

⇐⇒

Z _∞ dt tf(t)





<∞

=∞

. (1.7)

Concerning the lower limits ofW_n, we confirm (1.6) under a stronger integrability assumption: There exists some small constantε0>0such that

E

η^1+ε0 + Z

e^−x|x|^2+ε0L(dx)

<∞. (1.8)

It is easy to see that the condition (1.8) is stronger than (1.2) and (1.3).

Proposition 1.3 Assume(1.1)and(1.8). We have lim inf

n→∞

√n W_n= r 2

πσ² D_∞, P^∗-a.s.

Combining Theorems 1.1 and 1.2, we can roughly say that the main contribution to the upper limits ofW_ncomes from the terme^−Mn. According to Madaule [25], and A¨ıd´ekon, Berestycki, Brunet and Shi [3], Arguin, Bovier and Kistler [6] (for the branching brownian motion), the branching random walk seen from the minimal position converges in law to some point process, in particular,Wne^Mn converges in law asn→ ∞, but we are not able to determine the almost sure fluctuations ofW_ne^Mn.

The whole paper uses essentially the techniques developed by A¨ıd´ekon and Shi [5]. To show Theorems 1.1 and 1.2, we firstly remark that both two theorems share the same integral test and that since Wn ≥ e^−Mn, it is enough to prove the convergence part in the integral test (1.7) and the divergence part in (1.4).

The convergence part in (1.7) will follow from an application of Doob’s maximal inequality to a certain martingale. To prove the divergence part in (1.4), we shall use the arguments in A¨ıd´ekon and Shi [5] (the proof of their Lemma 6.3) to estimate a second moment, then apply Borel-Cantelli’s lemma. We can also directly prove Theorem 1.2 without the use of the divergence part of (1.4). Finally, the proof of Proposition 1.3 relies on a result (Lemma 4.1) which is also implicitly contained in A¨ıd´ekon and Shi [5] (by following the proof of their Proposition 4.1).

The rest of this paper is organized as follows: In Section 2, we recall some known results on the branch- ing random walk (many-to-one formula, change of measure) and on a real-valued random walk. In Section 3, we prove Theorems 1.1 and 1.2, whereas the proof of Proposition 1.3 will be given in Section 4.

Throughout this paper, f(n) ∼ g(n) asn → ∞ means thatlim_n→∞^f_g(n)⁽ⁿ⁾ = 1and (c_i,1 ≤ i ≤ 36) denote some positive constants.

2 Preliminaries

2.1 Many-to-one formula for the branching random walk

In this subsection, we recall some change of measure formulas in the branching random walk, for the details we refer to [9, 13, 24, 5, 27] and the references therein.

At first let us fix some notations which will be used throughout this paper: For |u| = n, we write [∅, u]≡ {u₀ := ∅, u₁, ..., u_n−1, u_n =u}the shortest path from the root∅tousuch that|u_i|=ifor any 0≤i≤n. For anyu, v ∈T, we use the partial orderu < vifuis an ancestor ofvandu≤vifu < vor u=v. We also denote by^←v the parent ofv.

Under (1.1), there exists a centered real-valued random walk{S_n, n ≥0}such that for anyn≥1and any measurable functionf :Rⁿ→R₊,

Eh X

|u|=n

e^−V(u)f(V(u₁), ..., V(u_n))i

=E[f(S₁, ..., S_n)]. (2.1)

Moreover under (1.2),σ² =Var(S₁) =Eh P

|u|=1(V(u))²e^−V(u)i

∈(0,∞).

The renewal functionR(x)related to the random walkSis defined as follows:

R(x) :=

X∞ k=0

P

S_k≥ −x, S_k< min

0≤j≤k−1Sj

, x≥0, (2.2)

andR(x) = 0ifx <0. Moreover,

xlim→∞

R(x)

x =c_R, (2.3)

with some positive constantc_R(see Feller [16], pp.612).

Forα≥0, we define as in A¨ıd´ekon and Shi [5] two truncated processes: For anyn≥0, W_n^(α) := X

|u|=n

e^−V(u)1_{(V(u)≥−α)}, (2.4)

D^(α)_n := X

|u|=n

R_α(V(u))e^−V(u)1_{(V(u)≥−α))}, (2.5) whereV(u) := min∅≤v≤uV(v),Rα(x) :=R(α+x)andRis the renewal function defined in (2.2).

Denote by(Fn, n≥0)the natural filtration of the branching random walk. If the branching random walk starts fromV(∅) = x, then we denote its law byP_x (withP= P₀). According to Biggins and Kyprianou [9],(D_n^(α), n ≥ 0)is a(P_x,(Fn))-martingale and on some enlarged probability space (more precisely on the space of marked trees enlarged by an infinite ray(ξn, n≥0), called spine), we may construct a family of probabilities(Q^(α)x , x≥ −α)such that for anyx≥ −α, the following statements (i), (ii) and (iii) hold:

(i) For alln≥1, dQ^(α)_x

dP_x

Fn = D^(α)n

D^(α)₀ , (2.6)

Q^(α)_x ξn=uFn

= 1

D^(α)_n Rα(V(u))e^−V(u)1_{(V(u)≥−α)}, ∀|u|=n. (2.7) (ii) UnderQ^(α)_x , the process{V(ξ_n), n≥0}along the spine(ξ_n)_n≥0, is distributed as the random walk (S_n, n≥0)underPconditioned to stay in[−α,∞). Moreover for anyn≥1, x≥ −αandf :Rⁿ→R₊,

EQ^(α)_x

hf(V(ξ₁), ..., V(ξ_n))i

= 1

R_α(x)E_xh

f(S₁, ..., S_n)R_α(S_n)1_(Sn≥−α)i

. (2.8)

(iii) Let Gn := σ{u, V(u) : ^←u ∈ {ξ_k,0 ≤ k < n}}, n ≥ 0. Under Q^(α)_x and conditioned on G∞, for allu 6∈ {ξ_k, k ≥ 0} but^←u ∈ {ξ_k, k ≥ 0}the induced branching random walk(V(uv),|v| ≥ 0) are independent and are distributed asP_V(u), where{uv,|v| ≥0}denotes the subtree ofTrooted atu.

Let us mention that as a consequence of (i), the following many-to-one formula holds: For any n ≥ 1, x≥ −αandf :Rⁿ→R₊,

E_xh X

|u|=n

e^−V(u)Rα(V(u))f(V(u1), ..., V(un))1_{(V(u)≥−α)}i

=Rα(x)e^−xE

Q^(α)x

h

f(V(ξ1), ..., V(ξn))i . (2.9) 2.2 Estimates on a centered real-valued random walk

We collect here some estimates on a real-valued random walk{S_k, k≥0}, centered and with finite variance σ² >0. LetS_n:= min_0≤i≤nS_i,∀n≥0. Recall (2.2) for the renewal functionR(·).

Fact 2.1 There exists some constantc₁ >0such that for anyx≥0, P_x

S_n≥0

≤ c₁(1 +x)n^−1/2, ∀n≥1, (2.10)

P_x S_n−1 > S_n≥0

≤ c₁(1 +x)R(x)n^−3/2, ∀n≥1, (2.11) P_x

S_n≥0

∼ θ R(x)n^−1/2, asn→ ∞, (2.12)

withθ= _c¹

R

q 2

πσ². Moreover there isc₂ >0such that for anyb≥a≥0, x≥0, n≥1, P_x

S_n∈[a, b], S_n≥0

≤c₂(1 +x)(1 +b−a)(1 +b)n^−3/2, (2.13) For any0< r <1, there exists somec₃=c₃(r)>0such that for allb≥a≥0, x, y≥0, n≥1,

P_x

S_n∈[y+a, y+b], S_n≥0, min

rn≤j≤nS_j ≥y

≤c₃(1 +x)(1 +b−a)(1 +b)n^−3/2. (2.14) See Feller ([16], Theorem 1a, pp.415) for (2.10), A¨ıd´ekon and Jaffuel ([4], equation (2.8)) for (2.11), A¨ıd´ekon and Shi [5] for (2.13) and (2.14), and Kozlov [22] and Lemma 2.1 in [5] for (2.12) with the identification of the constantθ= _c¹

R

q 2 πσ².

We end this section by an estimate on the stability onxin the convergence (2.12).

Lemma 2.2 LetSbe a centered random walk with positive variance. There exists a constantc₄ >0such that for alln≥1andx≥0,

P_x(S_n≥0)

R(x)P(S_n≥0) −1 ≤c41 +x

√n .

Proof of Lemma 2.2. Denote in this proof by̺(n) := P(S_n ≥0)forn≥ 0. Letx ≥0. By considering the firstk∈[0, n]such thatS_k=S_n, we get that

P_x(S_n≥0) = P_x(S_n≥x) + Xn k=1

P_x

S_k−1> S_k ≥0, min

k<j≤nS_j ≥S_k

= ̺(n) + Xn k=1

P_x

S_k−1 > S_k≥0

̺(n−k),

by the Markov property atk. Note thatR(x) = 1 +P_∞

k=1P_x S_k−1 > S_k≥0

. It follows that P_x(S_n≥0)≤R(x)̺(n) +

Xn k=1

P_x

S_k−1 > S_k≥0

[̺(n−k)−̺(n)], (2.15) and

P_x(S_n≥0)≥R(x)̺(n)− X∞ k=n+1

P_x

S_k−1 > S_k≥0

̺(n). (2.16)

Denote respectively by I_(2.15)and I_(2.16) the sumPn

k=1 in (2.15) and the sumP_∞

k=n+1 in (2.16). Let T⁻ := inf{j ≥ 1 : S_j < 0}. By the local limit theorem (Eppel [15], see also [28], equation (22)), if the distribution ofS1is non-lattice, then

P

T⁻=k

∼ C₋

k^3/2, k→ ∞, (2.17)

with some positive constantC₋. Moreover Eppel [15] mentioned that a modification of (2.17) holds in the lattice distribution case. Then there exists some constantc₅>0such that for allk≥1,

P

T⁻=k

≤ c5

k^3/2. (2.18)

It follows that for anyk ≤n,̺(n−k)−̺(n) =P n−k < T⁻ ≤n

≤ c₅Pn

i=n−k+1i^−3/2. Then by (2.11),

I_(2.15)≤c₆(1 +x)R(x) Xn k=1

k^−3/2 Xn i=n−k+1

i^−3/2.

Elementary computations show thatPn/2

k=1k^−3/2Pn

i=n−k+1i^−3/2≤Pn/2

k=1k^−3/2×k(ⁿ₂)^−3/2 =O(_n¹) andPn

k=n/2k^−3/2Pn

i=n−k+1i^−3/2 ≤(ⁿ₂)^−3/2Pn

i=1i^−3/2×i=O(_n¹). HenceI_(2.15)≤c₇(1+x)R(x)_n¹ ≤ c₈(1 +x)R(x)√¹

n̺(n)by (2.12).

Finally again by (2.11), we get that I(2.16) ≤ c₉(1 +x)R(x)√¹

n̺(n). Then the Lemma follows from (2.15) and (2.16).

3 Proofs of Theorems 1.1 and 1.2

In view of the inequality: W_n ≥ e^−Mn, the convergence part of the integral test (1.7) yields that of (1.4), whereas the divergence part of the integral test (1.4) implies that of (1.7). We only need to show the conver- gence part in (1.7) and the divergence part in (1.4).

3.1 Proof of the convergence part in Theorem 1.2:

Lemma 3.1 Assume(1.1). For any α ≥0, there exists some constantc₁₀ =c₁₀(α) > 0such that for any 1< n≤mandλ >0, we have

P

nmax≤k≤m

√kW_k^(α)> λ

≤c10logn

√n +c101 λ

rm n. Proof of Lemma 3.1.Forn≤k≤m+ 1, define

Wf_k^(α,n):= X

|u|=k

e^−V(u)1_{(V(un)≥−α)},

where as beforeV(u_n) := min_1≤j≤nV(u_j)andu_nis the ancestor ofuatn-th generation. ThenWf_n^(α,n)= W_n^(α).

Fork ∈ [n, m], fW_k+1^(α,n) = P

|v|=k1_{(V(vn)≥−α)}P

u:^←u=ve^−V(u).The branching property implies that E

fW_k+1^(α,n)|Fk

=Wf_k^(α,n)fork∈[n, m]. By Doob’s maximal inequality, P

nmax≤k≤m

√kfW_k^(α,n)≥λ

≤

√m

λ E(fW_m^(α,n)) =

√m

λ E(W_n^(α)).

By the many-to-one formula (2.1) and the random walk estimate (2.10), E(W_n^(α)) =P

S_n≥ −α

≤ c₁₁

√n, withc₁₁:=c₁(1 +α). It follows that

P

nmax≤k≤m

√kfW_k^(α,n)≥λ

≤ c₁₁ λ

rm n.

ComparingfW_k^(α,n)andW_k^(α), we get that P

n≤k≤mmax

√kW_k^(α)> λ

≤P

n≤k≤mmin min

|u|=kV(u)<−α +c₁₁

λ rm

n. The proof of the Lemma will be finished if we can show that for alln≥2,

P

|minu|≥nV(u)<−α

≤c10

logn

√n . (3.1)

To this end, let us apply the following known result (see e.g. [27]):

P

u∈infTV(u)<−x

≤e^−x, ∀x≥0.

Then for alln≥2, P

mink≥n min

|u|=kV(u)<−α

≤ P

uinf∈TV(u)<−logn +P

mink≥n min

|u|=kV(u)<−α,inf

v∈TV(v)≥ −logn

≤ 1 n+

X∞ k=n

Eh X

|u|=k

1_{(V(u)<−α,V(un)≥−α,...,V(uk−1)≥−α, V(u)≥−logn)}i

= 1

n+ X∞ k=n

Eh

e^Sk1_{(Sk<−α,Sn≥−α,...,Sk−1≥−α, Sk≥−logn)}i

≤ 1

n+e^−αP

S_n≥ −logn ,

where the above equality is due to the many-to-one formula (2.1). Using (2.10) to bound the above proba- bility term, we get (3.1) and the Lemma.

Proof of the convergence part in Theorem 1.2: Letf be nondecreasing such thatR_{∞ dt}

tf(t) < ∞. Let n_j := 2^j for largej ≥j₀. ThenP_∞

j=j0

1

f(nj) <∞. By using Lemma 3.1, P

nj≤maxk≤nj+1

√kW_k^(α)> f(n_j)

≤c₁₀logn_j

√n_j +c₁₀

√2 f(n_j), whose sum onjconverges. The Borel-Cantelli lemma implies thatP-a.s. for all largek,√

kW_k^(α) ≤f(k).

Replacingf(k)byεf(k)with an arbitrary constantε >0, we get that lim sup

k→∞

√kW_k^(α)

f(k) = 0, P-a.s.,

for any α ≥ 0. By considering a countable α → ∞ (for instance α integer) and by using the fact that W_k^(α)=W_kon the set{inf_u∈TV(u)≥ −α}, we get the convergence part..

3.2 Proof of the divergence part in Theorem 1.1:

The following lemma is a slight modification of A¨ıd´ekon and Shi [5]’s Lemma 6.3:

Lemma 3.2 ([5]) There exist some constants K > 0andc₁₂ = c₁₂(K) > 0such that for alln≥ 2,0 ≤ λ≤ ¹₃logn,

c₁₂e^−λ≤P [²ⁿ

k=n+1

E_k^(n,λ)∩F_k^(n,λ)

6

=∅

≤ 1

c₁₂e^−λ, (3.2)

where forn < k≤2n, E_k^(n,λ) := n

u:|u|=k,1

2logn−λ≤V(u)≤ 1

2logn−λ+K, V(u_i)≥a^(n,λ)_i ,∀0≤i≤ko , F_k^(n,λ) := n

u:|u|=k, X

v∈Υ(ui+1)

(1 + (V(v)−a^(n,λ)_i )₊)e^{−(V(v)−a(n,λ)i )} ≤K e^−b(k,n)i ,∀0≤i < ko ,

where foru∈T\{∅},Υ(u) :={v:v6=u,^←v =^←u}denotes the set of brothers ofu,x₊:= max(x,0), a^(n,λ)_i :=1

2logn−λ 1₍ⁿ

2<i≤2n), 0≤i≤2n, and forn < k≤2n,

b^(k,n)_i :=i^1/121_(0≤i≤ⁿ

2)+ (k−i)^1/121₍ⁿ

2<i≤k), 0≤i≤k.

Proof of Lemma 3.2. The proof of the lower bound in (3.2) [by the second moment method] goes in the same way as that of Lemma 6.3 in A¨ıd´ekon and Shi [5] [We also keep their notations], by replacing ¹₂logn in their proof by ¹₂logn−λ. Moreover, a similar computation of the second moment will be given in the proof of Lemma 3.3. Then we omit the details.

The upper bound in (3.2) is a simple consequence of the many-to-one formula: Definings:= ¹₂logn−λ, we have that

P [²ⁿ

k=n+1

E_k^(n,λ) 6=∅

≤ X2n k=n+1

Eh X

|u|=k

1(s≤V(u)≤s+K,V(ui)≥a^(n,λ)_i ,∀i≤k)

i

= X2n k=n+1

Eh e^Sk1

(s≤Sk≤s+K,Si≥a^(n,λ)_i ,∀i≤k)

i

≤ X2n k=n+1

e^s+KP

s≤S_k≤s+K, S_i ≥a^(n,λ)_i ,∀i≤k .

By (2.14), P s ≤ S_k ≤ s +K, S_i ≥ a^(n,λ)_i ,∀i ≤ k

≤ c₁₃n^−3/2 for all n < k ≤ 2n. Hence P S2n

k=n+1E_k^(n,λ)6=∅

≤c₁₃e^−λ+Kproving the upper bound in (3.2).

Using the notations in Lemma 3.2 with the constantK, we define forn≥2and0≤λ≤ ¹₃logn, A(n, λ) :=n

∪²ⁿk=n+1 E_k^(n,λ)∩F_k^(n,λ)

6

=∅o

. (3.3)

The following estimate will be useful in the application of Borel-Cantelli’s lemma:

Lemma 3.3 There exists some constantc₁₄>0such that for anyn≥2,0≤λ≤ ¹₃lognandm≥4n,0≤ µ≤ ¹₃logm,

P

A(n, λ)∩A(m, µ)

≤c14e^−λ−µ+c14e^−µlogn

√n .

Proof of Lemma 3.3.As we mentioned before, the arguments that we use are very close to the computation of the second moment in the proof of Lemma 6.3 in [5]. The introduction of the eventsF_k^(n,λ) inA(n, λ), sometimes called a truncation argument, is necessary to control the second moment: the eventF_k^(n,λ)keeps the path(V(u_i),0≤i≤k)of a particleuinE_k^(n,λ)to stay far away from(a^(n,λ)_i ,0≤i≤k), otherwise the particleuwould give a too large expectation in the second moment. Such truncation argument was already introduced in A¨ıd´ekon [2].

Let us enter into the details of the proof of Lemma 3.3. Write for brevity s:= 1

2logn−λ, t:= 1

2logm−µ.

Similarly to (2.6) and (2.7), we may construct a new probability Qsuch that for alln ≥1, ^dQ_dP

Fn =W_n, Q ξ_n =uFn) = ^e−_W^V(u)

n ,∀|u|=n. Moreover underQ,(V(ξ_n), n ≥0)is distributed as the random walk (Sn, n ≥0) defined in Section 2, and the spine decomposition similar to (iii) in Section 2 holds underQ. We refer to [9, 13, 24, 5, 27] for details. It follows that

P

A(n, λ)∩A(m, µ)

≤ Eh 1_A(n,λ)

X2m k=m+1

X

|u|=k

1(u∈E^(m,µ)_k ∩F_k^(m,µ))

i

=

X2m k=m+1

E_Qh

1_A(n,λ)e^V(ξk)1

(ξk∈E_k^(m,µ)∩F_k^(m,µ))

i

≤ e^t+K X2m k=m+1

E_Qh

A(n, λ), ξ_k∈E_k^(m,µ)∩F_k^(m,µ)i

≤ e^t+K X2m k=m+1

X2n l=n+1

E_Qh X

|v|=l

1_(v

∈E_l^(n,λ)∩F_l^(n,λ), ξ_k∈E^(m,µ)_k ∩F_k^(m,µ))

i

=: e^t+K X2m k=m+1

X2n l=n+1

I(3.4)(k, l). (3.4)

Forn < l≤2n≤ ^m₂ < k≤2m, we may decompose the sum on|v|=las follows:

X

|v|=l

1(v∈E_l^(n,λ)∩F_l^(n,λ))= 1

(ξl∈E_l^(n,λ)∩F_l^(n,λ))+ Xl p=1

X

u∈Υ(ξp)

X

v∈T(u),|v|u=l−p

1(v∈E_l^(n,λ)∩F_l^(n,λ)),

whereT(u)denotes the subtree ofTrooted atu and|v|u = |v| − |u|the relative generation ofv ∈ T(u).

Then

I(3.4)(k, l)

= Q

ξ_l∈E^(n,λ)_l ∩F_l^(n,λ), ξ_k∈E_k^(m,µ)∩F_k^(m,µ) +

Xl p=1

E_Qh 1_(ξ

k∈E_k^(m,µ)∩F_k^(m,µ))

X

u∈Υ(ξp)

f_k,l,p(V(u))i

=: I(3.5)(k, l) + Xl p=1

J(3.5)(k, l, p), (3.5)

with

f_k,l,p(x) :=E_Qh X

v∈T(u),|v|u=l−p

1(v∈E^(n,λ)_l ∩F_l^(n,λ))

V(u) =xi

, x∈R.

In what follows, we shall at first estimateJ_(3.5)(k, l, p)thenI_(3.5)(k, l). By the branching property atu and by removing the eventF_l^(n,λ)from the indicator function inf_k,l,p(r), we get that

f_k,l,p(x) ≤ E_xh X

|v|=l−p

1(s≤V(v)≤s+K,V(vi)≥a^(n,λ)_i+p ,∀0≤i≤l−p)

i

= e^−xE_xh e^Sl−p1

(s≤Sl−p≤s+K,Si≥a^(n,λ)_i+p ,∀0≤i≤l−p)

i

≤ e^−x+s+KP_x

s≤S_l−p≤s+K, S_i ≥a^(n,λ)_i+p ,∀0≤i≤l−p

, (3.6)

where to get the above equality, we applied an obvious modification of (2.1) forE_xinstead ofE.

Let us denote by (3.6)_k,l,pthe probability term in (3.6). To estimate (3.6)_k,l,p, we distinguish as in [5]

two cases: p≤ ⁿ₂ and ⁿ₂ < p≤l. Recall thatn < l≤2n≤ ^m₂ < k≤2m. Ifp≤ ⁿ₂, (3.6)_k,l,p≤1_(x≥0)c₁₅ 1 +x

(l−p)^3/2, by using (2.14). Then for1≤p≤ ⁿ₂,

f_k,l,p(x)≤c151_(x≥0)e^s+K−x(1 +x)(l−p)^−3/2. It follows that for alln < l≤2n, m < k≤2m,

X

1≤p≤n/2

J_(3.5)(k, l, p) ≤ Xn/2 p=1

E_Qh

1(ξk∈E_k^(m,µ)∩F_k^(m,µ))

X

u∈Υ(ξp)

c₁₅1_(V(u)≥0)e^s+K−V(u)1 +V(u) (l−p)^3/2 i

≤ c₁₆e^sn^−3/2 Xn/2 p=1

E_Qh

1(ξk∈E^(m,µ)_k ∩F_k^(m,µ))

X

u∈Υ(ξp)

1_(V(u)≥0)e^−V(u)(1 +V(u))i

≤ c₁₆K e^sn^−3/2 Xn/2 p=1

E_Qh

1(ξ_k∈E_k^(m,µ)∩F_k^(m,µ))e^{−(p−1)1/12}i ,

where the last inequality is due to the definition ofξ_k∈F_k^(m,µ)[noticing thata^(m,µ)p = 0andb^(k,m)p =p^1/12 for allp≤n/2< m/2]. Then we get that

X

1≤p≤n/2

J_(3.5)(k, l, p)≤c₁₇e^sn^−3/2Q

ξ_k∈E^(m,µ)_k

≤c₁₈e^sn^−3/2m^−3/2, (3.7)

sinceQ

ξ_k∈E_k^(m,µ)

=P(t≤S_k≤t+K, S_i ≥a^(m,µ)_i ,∀0≤i≤k

≤c₁₉m^−3/2for allm < k≤2m, by using (2.14).

Now considering ⁿ₂ < p≤l,a^(n,λ)_i+p =sfor any0≤i≤l−p, hence (3.6)_k,l,p= 1_(x≥s)P_x

s≤S_l−p ≤s+K, S_l−p≥s

≤1_(x≥s)c2(1 +K)² 1 +x−s (1 +l−p)^3/2, by (2.13). It follows that

X

n 2<p≤l

J(3.5)(k, l, p) ≤ X

n 2<p≤l

E_Qh 1_(ξ

k∈E^(m,µ)_k ∩F_k^(m,µ))

X

u∈Υ(ξp)

c2(1 +K)²1_(V(u)≥s)e^s+K−V(u)1 +V(u)−s (1 +l−p)^3/2 i

.

By the definition ofξ_k∈F_k^(m,µ), for allp≤l≤2n≤ ^m₂, we have that X

u∈Υ(ξp)

1_(V(u)≥s)e^−V(u)(1 +V(u)−s)≤ X

u∈Υ(ξp)

1_(V(u)≥0)e^−V(u)(1 +V(u))≤Ke^{−(p−1)1/12}.

Then X

n 2≤p≤l

J_(3.5)(k, l, p)≤c₂₀e^se^−n1/13Q(ξ_k ∈E_k^(m,µ))≤c₂₁e^s−n1/13m^−3/2. (3.8) Combining (3.7) and (3.8), we get that

X

1≤p≤l

J(3.5)(k, l, p) ≤(c18n^−3/2+c21e^−n1/13)e^sm^−3/2. (3.9) It remains to estimateI(3.5)(k, l)forn < l≤2nandm < k≤2m[in particularl < k]. We have

I_(3.5)(k, l)

≤ Q

ξ_l∈E^(n,λ)_l , ξ_k ∈E_k^(m,µ)

= P

s≤S_l≤s+K, S_i ≥a^(n,λ)_i ,∀i≤l, t≤S_k≤t+K, S_j ≥a^(m,µ)_j ,∀j≤k

. (3.10) Let us denote by (3.10)_k,l the probability term in (3.10). Using the Markov property atl, we get that

(3.10)_k,l = Eh

1(s≤Sl≤s+K,Si≥a^(n,λ)_i ,∀0≤i≤l)P_Sl

t≤S_k−l≤t+K, S_j ≥a^(m,µ)_j+l ,∀0≤j≤k−li

≤ c₂₁ (k−l)^3/2Eh

1(s≤Sl≤s+K,Si≥a^(n,λ)_i ,∀0≤i≤l)(1 +S_l)i

(by (2.13))

≤ c₂₂(1 +s+K)(k−l)^−3/2l^−3/2.

Based on the above estimate and (3.9), we deduce from (3.4) and (3.5) that P

A(n, λ)∩A(m, µ)

≤ c₂₃e^t+K X2m k=m+1

X2n l=n+1

(1 +s+K)(k−l)^−3/2l^−3/2+e^se^−n1/13m^−3/2+e^sn^−3/2m^−3/2

≤ c₂₄e^−λ−µ+c₂₄e^−µlogn

√n ,

proving the Lemma.

Proof of the divergence part in Theorem 1.1. Letf be nondecreasing such thatR_{∞ dt}

te^f(t) =∞. Without any loss of generality we may assume that√

logt≤e^f(t) ≤(logt)²for all larget≥t₀(see e.g. [14] for a similar justification). Denote by

Bx(k) :=n

M_n+x≤ 1

2logn−f(n+k), i.o. asn→ ∞o

, x∈R, k ≥0.

Let us first prove that there exists some constantc₂₅>0such that for anyx∈Randk≥0, P

B_x(k)

≥c₂₅. (3.11)

To this end, we taken_i:= 2ⁱfori≥1,λ_i:=f(n_i+1+k)+x+Kand consider the eventA_i:=A(n_i, λ_i) in (3.3). There is some integeri₀≡i₀(x, k)≥1such that for alli≥i₀,0≤λ_i ≤ ¹₃logn_i. By Lemma 3.2,

c₁₂e^−λi ≤P(A_i)≤ 1 c12

e^−λi, ∀i≥i₀.

Note thatR_{∞ dt}

te^f(t+k) ≥R_{∞ dt}

(t+k)e^f(t+k) =∞, andRni+2

ni+1

dt

te^f(t+k) ≤(log 2)e^−f(ni+1+k)by the monotonic- ity off. HenceP

ie^−λi =∞. By Lemma 3.3, we have for anyi≥i0andj≥i+ 2, P

A_i∩A_j

≤c₁₄e^−λi−λj+c₁₄e^−λjlogn_i

√n_i ,

which implies thatPk

i,j=i0P

A_i∩A_j

≤c₁₄ Pk

i=i0e^−λi2

+ 2c₁₄ Pk

i=i0e^−λi

× P_∞

i=1log√nnii

.Using the lower boundP(A_i)≥c₁₂e^−λi and the fact thatPk

i=i0e^−λi → ∞ask→ ∞, we obtain that lim sup

k→∞

P

1≤i,j≤kP(A_i∩A_j) Pk

i=1P(A_i)2 ≤ c₁₄ c²₁₂.

By Kochen and Stone [21]’s version of the Borel-Cantelli lemma,P(Ai,i.o.i→ ∞) ≥c²₁₂/c14 =:c25

which does not depend on (x, k). Observe that{A_i,i.o.i → ∞} ⊂ B_x(k), in fact, for those isuch that A_i ≡ A(n_i, λ_i) holds, by the definition (3.3), there exits somen ∈ (n_i, n_i+1]such thatM_n ≤ ¹₂logn_i− λi+K= ¹₂logni−f(ni+1+k)−x≤¹₂logn−f(n+k)−x. Hence we get (3.11).

We have proved that for anyx ∈Randk≥ 0,P(B_x(k)) ≥c₂₅. For anyk≥0, the eventsB_x(k)are non-increasing onx. LetB_∞(k) :=∩^∞i=1B_i(k)[thenB_∞(k)is nothing but{lim inf_n→∞(M_n−¹₂logn+

f(n+k)) = −∞}]. By the monotone convergence, P(B_∞(k)) ≥c₂₅, for allk ≥ 0. Moreover, for any x∈R,P_x(B_∞(k)) =P(B_∞(k))≥c₂₅. On the other hand, if we denote byZ_k :=P

|u|=k1the number of particles in thek-th generation, then by the branching property,

P

B_∞(0)Fk

= 1_(Zk>0)

1− Y

|u|=k

(1−P_V(u)(B_∞(k)))

≥1_(Zk>0)

1−(1−c25)^Zk .

It is well-known (cf. [7], pp.8) that S = {lim_k→∞Z_k = ∞}. Then by letting k → ∞in the above inequality, we get that

1_B∞(0) = lim

k→∞P

B_∞(0)Fk

≥1_S, P-a.s.

ClearlyS^c ⊂ B_∞(0)^c by the convention on the definition ofM_non S^c. HenceS = B_∞(0), P-a.s. This proves the divergence part of Theorem 1.1.

4 Proof of Proposition 1.3

The main technical part was already done in A¨ıd´ekon and Shi [5]:

Lemma 4.1 ([5]) Assume (1.1) and (1.8). For any fixed α ≥ 0, there exist some δ = δ(ε₀) > 0 and c₂₆=c₂₆(α, δ) >0such that for alln≥2,

Var_Q(α)

√nW_n^(α) D^(α)_n

!

≤c26 n^−δ+ sup

k^1/3n ≤x≤kn

h_x+α(n−k_n) h_α(n) −1

!

, (4.1)

wherek_n:=⌊n^1/3⌋andh_x(j) :=

√jPx(S_j≥0)

R(x) forj≥1, x≥0.

Proof of Lemma 4.1. The Lemma was implicitly contained in the proof of Proposition 4.1 in [5]. In fact, in their proof of the convergence that Var_Q(α)

_√

nWn^(α)

D^(α)n

→ 0, we choosekn :=⌊n^1/3⌋in their definition ofE_n := E_n,1∩E_n,2∩E_n,3 (see the equation (4.6) in [5], Section 4). We claim that for some constant δ₁ =δ₁(ε₀)>0, there is somec₂₇=c₂₇(δ₁, α)>0such that for alln≥1,

Q^(α) E_n^c

≤ c₂₇n^−δ1, (4.2)

sup

k^1/3n ≤x≤kn

Q^(α)

E_n^cV(ξ_kn) =x

≤ c27n^−δ1. (4.3)

In fact, according to the definition ofE_n,1in [5], Q^(α)

E_n,1^c

≤Q^(α)

{V(ξ_kn)> k_n} ∪ {V(ξ_kn)< k_n^1/3}

+ sup

kn^1/3≤x≤kn

Q^(α)_x

∪ⁿ_i=0^−kn{V(ξ_i)< k_n^1/6} .

By (2.8), Q^(α)

V(ξ_kn)< k^1/3_n

= 1

R_α(0)E

1(S_kn≥−α,Skn<k^1/3n )R_α(S_kn)

≤ R_α(k^1/3n )

R_α(0) P(S_kn ≥ −α)≤c₂₈k⁻_n^1/6,