Necessary and sufficient conditions for the convergence of the consistent maximal displacement of the branching random walk

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1 Necessary and sufficient conditions for the convergence of the consistent maximal displacement of the branching random walk Bastien Mallein January 4, 08 Abstract Consider a supercritical branching random walk on R. he consistent maximal displacement is the smallest of the distances between the traectories of individuals at the nth generation and the boundary of the process. It has been proved see Fang and Zeitouni [], Faraud, Hu and Shi [] that the consistent maximal displacement grows almost surely at rate λ n /3 for an explicit λ. We obtain here ecessary and sufficient condition for this asymptotic behaviour to hold. Introduction A branching random walk on R is a process defined as follows. It starts with one individual located at 0 at time 0. Its children are positioned in R according to a point process of law L, and form the first generation of the process. hen for any n N, each individual in the n-th generation makes children around its current position according to an independent point process with law L. We write for the genealogical tree of the population. For any u we denote by V u the position of the individual u and by u the generation to which u belongs. he random marked tree, V is the branching random walk with reproduction law L. We assume the Galton-Watson tree is supercritical, i.e. E >,. u = and we write S = {# = } for the survival event, which happens with positive probability under assumption.. We also assume the branching random walk, V is in the boundary case in the sense of [8]: E V u e = and E V u V ue = 0.. u = u = LAGA, Université Paris 3 Villetaneuse.

2 Under these assumptions, Biggins [7] proved that n min u =n V u converges to 0 almost surely on S. Any branching random walk with mild integrability assumption can be normalized to be in the boundary case, see e.g. Bérard and Gouéré [6]. We also assume that σ := E V u V u e <..3 u = Let n N. For any u such that u = n and k n we denote by u k the ancestor of u alive at generation k. he consistent maximal displacement of the branching random walk is the quantity defined as L n := min u =n max k n V u k. It corresponds loosely to the maximal distance between the lower boundary of the branching random walk and the traectory of the individual that stayed as close as possible to it. he asymptotic behaviour of L n has been studied by Fang and Zeitouni [] and by Fauraud, Hu and Shi []. Under stronger integrability assumptions, they proved that L n behaves as λ n /3 almost surely for some explicit λ > 0. he main result of this article is ecessary and sufficient condition for this asymptotic behaviour to hold. Roberts [] computed the second order of the asymptotic behaviour of L n for a similar model, the branching Brownian motion. his On /3 asymptotic behaviour for the consistent maximal displacement is non-obvious, as it is different of the asymptotic behaviour of the minimum of the branching random walk, M n := min u =n V u. Indeed, it was proved by Addario-Berry and Reed [] and by Hu and Shi [3] that under some additional assumptions, the minimal displacement satisfies lim M n log n = 3 in probability. hus, the consistent maximal displacement grows much faster than the minimal displacement does. Note that it was proven in [0] that the traectory yielding to the minimal position at time n, when rescaled in time by n and in space by n /, converges toward a Brownian excursion. herefore, the maximal distance from 0 of this traectory is of order n /, much larger than what is expected for the consistent maximal displacement. As a result, one conclude that particles realizing the minimal displacement and particles realizing the consistent maximal displacement form distinct sets. We introduce the integrability assumption lim e V u { } = 0..4 log v = e V v x x E u = Observe that this assumption is weaker than the classical integrability assumptions [, Equation.4] for branching random walks. hese stronger conditions are necessary and sufficient to obtain the regularity of the asymptotic behaviour of many quantities associated to the extremal particles in the branching random walk, such as the minimal displacement M n [], or the derivative martingale [9]. x

3 heorem.. We assume that.,. and.3 hold. hen.4 is a necessary and sufficient condition for lim L n n /3 = 3π σ /3 a.s. on S. In the rest of the article, we denote by N the set of positive integers, Z + the set of non-negative integers, R + the set of non-negative real numbers. For any x R +, we write [x] = [0, x] Z + the set of non-negative integers that are smaller or equal to x. he rest of the article is organised as follows. In Section we introduce the spinal decomposition of the branching random walk and the Mogul skiĭ s small deviations estimate. hese results are used to obtain a law of large numbers for L n in Section 3, which is then used to obtain its almost sure asymptotic behaviour. Preliminary results In this section, we first introduce the spinal decomposition, that links additive moments of the branching random walk with random walk estimates. Using this result, to study the consistent maximal displacement, good estimates of the probability for random walks to stay in a small interval will be needed. We introduce them in a second time, by expanding Mogul skiĭ results [0] on small deviations for random walk traectories.. Spinal decomposition of the branching random walk For n Z +, we write W n = u =n e V u and F n = σ V u, u [n]. Under assumption., W n is on-negative F n -martingale. We introduce the probability P such that for any n Z +, P Fn = W n P Fn. he spinal decomposition consists in an alternative description of P as a branching random walk with a distinguished individual with a different reproduction law. It generalizes a similar construction for Galton-Watson processes, that can be found in [7]. his result has been proved by Lyons in [6]. Let be a tree, a spine of is a sequence w = w n Z+ such that for any n Z + and k [n] we have w n = n and w n k = w k. We write L for the law of the point process V u, u = under the law P. We now define the law P of a branching random walk with spine, V, w. It starts with a unique individual w 0 located at 0 at time 0. Its children are positioned according to a point process of law L. he individual w is then chosen at random among these children u with probability proportional to e V u. Similarly at each generation n, every individual u makes children independently, according to law L if u w n, or L otherwise; then w n+ is chosen at random among the children v of w n with probability proportional to e V v. Proposition. Spinal decomposition, Lyons [6]. Assuming. and.3, for any n Z +, we have P Fn = P Fn, for any u = n, P w n = u F n = e V u /W n, and V w n, n Z + is a centred random walk with variance σ. 3

4 he spinal decomposition is widely used in branching random walk literature. In particular, it implies the so-called many-to-one lemma, introduced for the first time by Kahane and Peyrière [, 5]. For any n N, for any measurable non-negative function f : R n+ R, we have E fv u, n = E fv u, n u =n W n u =n = Ê e u =n V u W n e V u u =n = Ê e V wn fv w, n. fv u, n In other words, to compute the mean of an additive functional of the branching random walk, it is enough to compute the mean of a similar functional for the sole random walk V w n, n 0, up to an exponential tilting. Remark.. Note that.4 can be rewritten, using the spinal decomposition, as lim x P e V v e x = 0.. x v = In other words, this assumption translates into a condition on the tail of the distribution of the progeny of the spine particle.. Small deviations estimate for enriched random walk he spinal decomposition for the branching random walk allows to simplify branching random walk computations by focusing only on the spine particle. For example, to compute the average number of particles satisfying a given property, it is often enough to consider the probability for the spine to satisfy this property, under the size-biased law. In the rest of the article, we often have to deal with the fact that the progeny of the spine particle is usually correlated with its displacement. As a result, we develop in this section tight estimates on the small deviations for enriched random walks, that we use as toy-models for the spine of a branching random walk. An enriched random walk can be constructed as follows. Let X n, ξ n n N be a sequence of i.i.d. vectors in R such that EX n = 0 and EX n = σ 0,,. We denote by S n = S 0 +X + +X n. For any z R, P z is the probability such that P z S 0 = z =. We simply write P for P 0. Setting ξ 0 = 0, an enriched random walk is the process S n, ξ n, n Z +, that takes values in R. Enriched random walks appear under other names in the literature. For example, the process S n + ξ n, n 0 is often called a perturbed random walk. his process has been studied in the perpetuity literature see e.g. [3, 4]. We study in this section the probability for an enriched random walk to stay during n units of time in an interval of width on /, generalizing the Mogul skiĭ small deviations estimate [0]. 4

5 heorem.3. Let R N + such that lim = and = on /. We assume. and that lim x x Pξ > x = ϱ [0, ]. For any continuous functions f < g such that f0 < 0 < g0, for all λ > 0 and t > 0, lim = lim t = 0 a n n log sup S P zan [f/n, g/n], ξ z [f0,g0] n log P S [f/n, g/n], ξ λ, [tn] π σ ϱt ds gs fs λ. a n λ, [tn] he rest of the section is devoted to the proof of heorem.3, using the same techniques as in [9, Lemma.6]. he first step relies on the following observation. Lemma.4. For n N and t R +, we introduce the functions S n t = S nt n / and P n t = nt = {ξ >n / }. We assume., and that there exists an increasing sequence n k N N and ϱ [0, such that lim n kp ξ > n k / = ϱ..3 k hen lim k S nk, P nk = σb, P in law, for the Skorokhod topology on the space of càdlàg functions, where B is a Brownian motion and P an independent Poisson process with parameter ϱ. his lemma shows that the events { S [f/n, g/n], [n]} and {ξ λ, [n]} become asymptotically independent, which is an heuristic explanation for heorem.3. Proof. o lighten the notation, in this proof, every asymptotic behaviour written as n is implicitly considered along the subsequence n k that is given as a hypothesis. We observe that Lemma.4 is straightforward if ϱ = 0. Indeed, P n is an increasing function, and for any t R +, we have PP n t = 0 = P ξ > n / nt as n. herefore, lim P n = 0 in probability. Moreover, lim S n = σb by Donsker s theorem. We conclude by Slutsky s theorem that S n, P n converges toward σb, 0 in law, for the Skorokhod topology. We now assume that ϱ > 0. Let t R +, for any λ, µ R, we compute nt E exp iλs n t + iµp n t = E exp i λ X n / + iµ {ξ>n / } nt = Φλn / + e iµ Pξ > n / Ψ n λ, where Φs = Ee isx and Ψ n s = Eexpis X ξ n / > n /. Note that E X ξ > n / E X {ξ>n = / }. n / npξ > n / 5

6 hus by dominated convergence, we have lim E X ξ > n / = X n / ϱ lim E {ξ>n / } = 0. X herefore, conditioned to {ξ n / > n / } converges to 0 in L, hence in law. his yields lim Ψ n λ = for all λ R. Moreover, by. and.3 respectively, we have Φλn / λ σ n and Pξ > n / ϱ n as n. As a result, we have lim E expiλs n t + iµp n t = exp t λ σ tϱe iµ which proves that lim S n t, P n t = σb t, P t for all t > 0. Using this result and the independent of the increments, we obtain the finitedimensional distributions of S n, P n toward σb, P. By [3, heorem 5.], the finite-dimensional distribution convergence of a random walk in R implies the convergence in law for the Skorokhod topology of the process toward the associated Lévy process, concluding the proof. We use this convergence in law to prove the following result. Lemma.5. Let R N + be a sequence such that lim = and = on /. Under assumption., if there exist an increasing sequence n k N N and ϱ [0, ] such that lim k k Pξ > k = ϱ, then for any continuous functions f < g such that f0 < 0 < g0, for all t > 0, a n lim k k n k log P ξ k [f/n k, g/n k ], k, [tn k ] S = t 0 π σ ds ϱt..4 gs fs Proof. Again, to simplify the notation, every asymptotic behaviour as n is implicitly understood as along the subsequence n k. First note that if ϱ =, then lim a npξ > =. As a result, lim sup n log P S [f/n, g/n], ξ, [tn] a n lim sup a n n log Pξ tn log lim sup Pξ > ta n =, concluding the proof in this case. In the rest of the proof, we assume that ϱ <. We first prove that.4 holds when functions f and g are constant. More precisely, given t > 0, a < 0 < b and c < d such that a c and d b, and writing I = [a, b], we first prove that a n n log P S tn [c, d], S I, ξ, [tn] tπ σ b a tϱ..5 o prove this result, we will divide the time interval [0, n] into On/a n intervals of width Oa n. On each of these time intervals, the random walk traectory, 6

7 properly normalized, converges toward a Brownian motion. We finally use Brownian motion estimates to bound the probability for the random walk to leave the interval I in any of these time intervals. Let 0 < ɛ < d c 8 and N. For any n N, we set r n = a n and denote by K n = tn /r n. For k [K n ], we introduce the times m n,k = kr n and set m n,kn+ = tn. Applying the Markov property at times m n,kn, m n,kn,... m n,, and considering only traectories that are at each time m n,k within distance ɛ from 0, we have P Sn [c, d], S I, ξ, [tn] πn Kn π n,.6 where we have set S π n = inf P rn S h h ɛa ɛ, I, ξ, [r n ] n S n π n = inf P h h ɛa [c, d], S I, ξ, [ n ] n and n = tn K n r n. We now study the asymptotic behaviour of π n as n. We introduce the rescaled traectories, defined for s [0, ] by S s n = a n S a n s and P s n = a s n = {ξ>}. We observe that the probability π n can be rewritten as π n = inf P h P n = 0, S n ɛ, Sn s I, s h ɛ = inf P P n = 0, S n + h ɛ, S s n + h I, s, h ɛ using the fact that the law of S n, P n under P han and S n + h, P n under P 0 are the same. Moreover, note that for all h [0, ɛ], we have P P n = 0, S n [ ɛ h, ɛ h], S s n [a h, b h], s P P n = 0, S n [ ɛ, 0], S n [a, b ɛ], s, as [ ɛ, 0] [ ɛ h, ɛ h] and [a, b ɛ] [a h, b h] for all h [0, ɛ]. Similarly, for all h [0, ɛ], we have P P n = 0, S n [ ɛ h, ɛ h], S s n [a h, b h], s P P n = 0, S n [0, ɛ], S n [a + ɛ, b], s. Rewriting the infimum over the interval [ ɛ, ɛ] as the minimum of the infimum over the intervals [ ɛ, 0] and [0, ɛ], we obtain π n min P P n = 0, S n + δ ɛ, 0, S s n + δ a, b ɛ, s [0, ]. δ { ɛ,0} s s 7

8 By Lemma.4, we know that S n, P n converges toward σb, P, where B is a Brownian motion and P an independent Poisson process with intensity ϱ. herefore, by Portmanteau theorem, we have π n min P σb + δ ɛ, 0, P = 0, σb s a, b ɛ, s [0, ]. δ { ɛ,0} Using similar estimates, we obtain that π n > 0. Note that K n tn as n. herefore,.6 yields a n a n n log P Sn [c, d], S I, ξ, [tn] a n n K n log π n + log π n t log π n. hus, using the above estimates and the independence between P and B, we obtain that for all > 0, n log P Sn [c, d], S I, ξ, [tn] a n ϱt + t log min P σb + δ ɛ, 0, σb s + δ a, b ɛ, s [0, ]. δ { ɛ,0} Using [4, Chapter.7, Problem 8], we know that for all δ ɛ, we have lim log P σb + δ ɛ, 0, σb s + δ a, b ɛ, s [0, ].7 π σ = b a ɛ. herefore, letting in.7, then ɛ 0 yields.5. In a second time, we prove a uniform version of.5. Let a < x < x < b, we choose 0 < ɛ < min{d c,b x } 8 and set N = x x ɛ. We note that inf P S n ha h [x,x n ] [c, d], S I, ξ, [tn] min inf P S n [N] h [x+ɛ,x++ɛ] + h [c, d], S + h I, ξ, [tn] min P S n x+ɛ [N] [c, d ɛ], S [a, b ɛ], ξ, [tn], using the same techniques as the ones used to bound π n. hus, applying.5 to let n, then letting ɛ 0 we obtain a n n inf log P Sn ha h [x,x n ] [c, d], S [a, b], ξ, [tn] tπ σ tϱ..8 b a Finally, using.8, we can tackle general continuous functions. Let f < g be two continuous functions such that f0 < 0 < g0. We introduce a continuous 8

9 function h such that f < h < g and h0 = 0. Let K N, for k [K] we set I k = [ k ] [0, ], as well as K, k+ K f k = sup s I k fs, g k = inf s I k gs and h k = hk/k. We choose some large K N and some small ɛ > 0. Applying the Markov property at times k n/k for k [K], we obtain, for all n N large enough S P [f/n, g/n], ξ, [tn] tk inf P S n/k S s s h k ɛ h k+ ɛ, [f k, g k ], ξ, [ n/k ]. k=0 herefore, using.8, we obtain n log P S [f/n, g/n], ξ, [tn] a n π σ + Kgk f k ϱ tk K. tk Letting K, we obtain the lower bound of.4. he upper bound for this equation is obtained in a very similar way, replacing inf by sup, by and min by max. More precisely, one can prove an analogue of.8, by dividing the time interval [0, n] into On/a n intervals of length Oa n, using the Markov property and replacing inf h ɛan by sup h [aan,b]. hen, approximating functions f and g by staircase functions and using again the Markov property, the analogue of.8 yields the upper bound of.4. Finally, to prove heorem.3, we observe that under its assumptions, for any λ > 0 we have lim a npξ > λ = ϱ λ. Using Lemma.5 with the increasing sequence n, the proof is concluded. 3 ail of the consistent maximal displacement We denote by the ancestor of the branching random walk. For any u, we write πu for the parent of u, Ωu for the set of children of u, ξu = log V u V v e { ξπu if u and ξu = 0 if u =. v Ωu Note that by., we have P ξ > x e x for any x R +. We set k=0 3π λ σ /3 =. 3. In this section, we obtain upper and lower bounds for the left tail of L n, ultimately obtaining the following large deviation estimate for the consistent maximal displacement. 9

10 heorem 3.. Under assumptions.,.,.3 and.4, we have for all λ [0, λ ] lim n log PL /3 n λn /3 = λ λ. 3. In a first time, we prove the upper bound in heorem 3., for which the assumption.4 is not necessary. Lemma 3.. We assume.,. and.3. For any λ 0, λ, we have lim sup n log P L /3 n λn /3 λ λ. Proof. Let λ 0, λ, we set f : t λ λ t /3. For n N, write Y n = {V u<f u /nn /3 } {V u [f/nn /3,λn /3 ], [ u ]}. u n As f0 < 0 and f λ, we have by Markov inequality PL n λn /3 PY n EY n. 3.3 Using [8], we can compute the asymptotic behaviour of EY n. More precisely, in that article, the branching random walks that are considered are such that, V is in the boundary case. herefore, [8, Lemma 3.] yields t lim sup n /3 π σ log EY n sup ft t [0,] 0 λ fs ds sup ft π σ t [0,] λ 3 t /3 sup ft λ t /3 = λ λ, t [0,] as λ = 3π σ λ by definition 3.. As a result, we conclude from 3.3 that lim sup n /3 log EY n λ λ. We now assume that.4 does not hold, and use Lemma.5 instead of the usual Mogul skii estimate to improve the upper bound in the asymptotic left tail of L n along a well-chosen subsequence. In particular, this shows that.4 is necessary for heorem 3. to hold. Lemma 3.3. We assume.,.,.3 and lim sup x x Pξ > x > 0. here exists λ > λ such that n log P L /3 n λn /3 < 0. Proof. Let ϱ 0, ] and x k R N be such that lim x n = and lim k x k Pξ > x k = ϱ. We set R = 3λ and n k = x k /R 3. We note 0

11 that k n k /3 Pξ > Rn k /3 ϱ R. Up to modifying ϱ and extracting a subsequence from n k, we may assume without loss of generality that lim n k /3 Pξ > Rn k /3 = ϱ. 3.4 k Let λ 0, λ and f be a continuous increasing function that satisfies f0 λ, 0 and f = λ. For k [n], we set I n k = [ fk/nn /3, λn /3] and we denote by { } G n = u : u n, V u I n, ξu Rn /3, [ u ]. For k [n] with k > 0, we introduce the quantities X n k = n {πu Gn,V u<fk/nn /3 } and Y k = u =k u =k We observe that P L n λn /3 =P u = n : [n], V u λn /3 n n P X n + Y n E = { u G n, ξu>rn /3 }. = X n + Y n. 3.5 Using the spinal decomposition we have E X n V u e k =Ê e V u W {V u<fk/nn /3 k } {πu G n} u =k =Ê e V wk {V wk <fk/nn /3 } {w k G n} e fk/nn/3 P wk G n. Similarly, as for any u = k, ξu is independent of F k, and by., we have Ee ξu = for all u. herefore E Y n = E P ξu > Rn /3 k u =k {u Gn} e Rn/3 Ê e V wk {wk G n} e λ Rn/3 P wk G n. Consequently, as λ R < λ < ft for any t [0, ], 3.5 becomes P L n λn /3 n e fk/nn/3 P wk G n. k= Let A N, for any a [A], we write m a = na/a. As f is increasing, for any a [A ] and k m a, m a+ ] N, we have e fk/nn/3 P wk G n e fa+/an/3 P wma G n.

12 Moreover, by the spinal decomposition, V w, ξw, Z + is an enriched random walk under law P. By 3.4, we use Lemma.5 to obtain for any a [A ] with a 0, lim k n k log P w /3 ma G nk = a/a 0 π σ λ fs ds ϱa A. his limit also trivially holds for a = 0 with the convention 0. = 0. herefore, letting n along the subsequence n k, 3.5 yields n log P L /3 n λn /3 n A log e fa+/an/3 P wma G n/3 n A a=0 a/a π σ max fa + /A a [A] λ fs ds ϱa. A hen, letting A, we obtain n log P L /3 n λn /3 t π σ sup ft ds ϱt. t [0,] 0 λ fs 3.6 Note that if ϱ =, we can choose f : t λ / + λ t and λ = 3λ /. In that case, 3.6 allows to conclude the proof. hus, in the rest of the proof, we assume that ϱ <. For all λ λ and µ 0, we denote by f λ,µ the solution of t [0,, y t = π σ λ yt + µ, with y = λ Note that f λ,µ is continuous with respect to λ, µ, decreasing in µ, and that t [0, ], f λ,0 t = λ λ t /3 hus for a given ϱ > 0, there exists λ > λ close enough to λ such that f λ,ϱ 0 < 0. Applying 3.6 with this choice of λ and f = f λ,ϱ yields which concludes the proof. n log P L /3 n λn /3 f λ,ϱ 0 < 0, he two previous lemmas allow to bound from below the consistent maximal displacement, showing in particular that with high probability L n λ ɛn /3 for all ɛ > 0 and n large enough. o bound L n from above, we prove that with high probability there exists an individual staying below λn /3 for n units of time, as soon as λ is large enough, using a second moment computation. 0 Lemma 3.4. We assume.,.,.3 and.4. For any λ 0, λ, we have n log P L /3 n λn /3 λ λ.

13 Proof. Let λ 0, λ, δ > 0, and f : t [0, ] λ λ + δ t /3. We denote by I n = [ f/nn /3, λn /3] for all [n]. We set Z n = { } V u I n,ξu δn /3, [n]. u =n We compute the first two moments of Z n to bound from below PZ n > 0. Using the spinal decomposition, we have EZ n = Ê e V wn { } V w I n,ξw δn /3, [n] e fn/3 P V w I n, ξw δn /3, [n]. As lim n/3 Pξw > δn /3 = 0 by.4, heorem.3 yields n log EZ n f π σ /3 0 ds λ fs λ λ + δ / Similarly, to compute the second moment we observe that EZn = Ê V u e Z n e V u { V u I n u =n W n = Ê Z n e V wn { V w I n e λn/3 Ê Z n { V w I n,ξw δn /3, [n],ξw δn /3, [n] Under the law P, Z n can be decomposed as follows Z n = { V w I n,ξw δn /3, [n] where Z n u = v =n,v>u { V u I n n } + k=0,ξu δn /3, [n],ξu δn /3, [n] } }. } u Ωw k,u w k+ Z n u, G = σ w n, Ωw n, V u, u Ωw n, n N. }. We denote by Observe that conditionally on G, for any u Ωw k such that u w k+, the subtree of the descendants of u has the law of a branching random walk starting from V u. herefore, writing P x for the law of, V + x, for any k [n ] and u Ωw k such that u w k+, we have Ê Z n u G E V u { } V v I n k++,ξv δn/3, [n k ] v =n k E V u { }. V v I n k++, [n k ] v =n k 3

14 Applying spinal decomposition, for any x R and p [n], E x v =n p { } = Ê e V wn p { } V v I n p+, [n p] V w +x I n p+, [n p] e x P λn/3 V w + x I n p+, [n p]. Let A N, for any a [A] we set m a = na/a and Ψ n a,a = sup P V w + y I n m, [n m a+ a]. y I n ma Using the previous equation, for any a [A ] and k [n ] such that m a k < m a+, we have Ê Z n u G e λn/3 Ψ n V u e u Ωw k u w k+ a+,a u Ωw k e λn/3 V w k +ξw k+ Ψ n a+,a. As V w k fm a /nn /3 using the fact that f is increasing, we obtain EZn e λn/3 P V w I n, [n] A + e λ+δn/3 nψ n e fma/nn/3 a+,a P a=0 herefore, applying heorem.3 and the fact that Ψ n V w I n, [n]. A,A lim sup n log /3 EZ n λ + δ π σ ds 0 λ fs + max fa/a π σ a [A ] Letting A, we obtain a+/a =, we have ds λ fs. lim sup n log /3 EZ n λ λ + δ /3 + δ + λ δ / By Cauchy-Schwarz inequality, we have P L n λn /3 P Z n > 0 EZ n EZn. hus, using 3.7 and 3.8, we obtain n log P L /3 n λn /3 λ λ + δ /3 δ λ δ /3. Letting δ 0 allows to conclude. 4

15 Proof of heorem 3.. Using Lemmas 3. and 3., we observe that assuming that.4, we have lim which concludes the proof. n /3 log PL n λn /3 = λ λ, Proof of heorem.. We now extend heorem 3. to obtain the almost sure asymptotic behaviour of L n. We first assume that.4 does not hold. By Lemma 3.3, there exists λ > λ and an increasing sequence n k N N such that PL nk n /3 λ <. k= λ a.s. here- herefore, by Borel-Cantelli lemma, we have k fore, we conclude that k lim sup > λ a.s, n/3 L n L nk n /3 k proving that.4 is necessary for the convergence in heorem. to hold. We assume in the rest of the proof that.4 holds. By Lemma 3., for any λ < λ, we have n= PL n n /3 λ <. As a result, we have L n n /3 λ a.s. by taking λ λ. We now bound L n from above. By Lemma 3.4, for any δ > 0, we have n log PL /3 n λ n /3 > δ. We work in the rest of the proof conditionally on the survival event S. We write for the subtree of consisting of individuals having an infinite line of descent. By [5, Chapter, heorem.], is a supercritical Galton-Watson process that never dies out. Applying [8, Lemma.4] to the branching random walk, V, there exists a > 0 and ϱ > such that the event Ap = {# { u = p : p, V u pa} ϱ p } is verified a.s. for p N large enough. Let η > 0, we set p = ηn /3. Applying the Markov property at time p, we have P L n+p λ + aηn /3 Ap P L n λ n /3 ϱ p. Using the Borel-Cantelli lemma, we conclude that lim sup a.s. on S. We let η 0 to conclude the proof. L n n /3 λ + aη Note the integrability hypothesis.4 is weaker than [8, Assumption.3], but is enough to make the proof of [8, Lemma.4] hold. 5

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