The near-critical Gibbs measure of the branching random walk
Michel Pain
TL;DR
This paper investigates the near-critical behavior of the Gibbs measure in a supercritical branching random walk, establishing convergence results for the partition function and particle trajectories as the inverse temperature approaches criticality.
Contribution
It extends previous results by analyzing the near-critical regime, proving convergence of the rescaled partition function and describing the limiting behavior of particle trajectories.
Findings
Rescaled partition function converges to a multiple of the derivative martingale limit.
Particle trajectories under the near-critical Gibbs measure interpolate between Brownian meander and excursion.
Continuous families of limiting processes are identified from meander to excursion or Brownian motion.
Abstract
Consider the supercritical branching random walk on the real line in the boundary case and the associated Gibbs measure on the generation, which is also the polymer measure on a disordered tree with inverse temperature . The convergence of the partition function , after rescaling, towards a nontrivial limit has been proved by A\"{\i}d\'ekon and Shi in the critical case and by Madaule when . We study here the near-critical case, where , and prove the convergence of , after rescaling, towards a constant multiple of the limit of the derivative martingale. Moreover, trajectories of particles chosen according to the Gibbs measure have been studied by Madaule in the critical case, with convergence towards the Brownian meander, and by Chen, Madaule and Mallein in the strong…
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Taxonomy
TopicsStochastic processes and statistical mechanics · Theoretical and Computational Physics · Random Matrices and Applications
The near-critical Gibbs measure of the branching random walk
Michel Pain111DMA, École Normale Supérieure and LPMA, Sorbonne Universités.
Abstract
Consider the supercritical branching random walk on the real line in the boundary case and the associated Gibbs measure on the th generation, which is also the polymer measure on a disordered tree with inverse temperature . The convergence of the partition function , after rescaling, towards a nontrivial limit has been proved by Aïdékon and Shi [3] in the critical case and by Madaule [34] when . We study here the near-critical case, where , and prove the convergence of , after rescaling, towards a constant multiple of the limit of the derivative martingale. Moreover, trajectories of particles chosen according to the Gibbs measure have been studied by Madaule [35] in the critical case, with convergence towards the Brownian meander, and by Chen, Madaule and Mallein [22] in the strong disorder regime, with convergence towards the normalized Brownian excursion. We prove here the convergence for trajectories of particles chosen according to the near-critical Gibbs measure and display continuous families of processes from the meander to the excursion or to the Brownian motion.
MSC 2010: 60J80, 60F05, 60F17.
Keywords: Branching random walk, additive martingale, trajectories, phase transition.
1 Introduction and main results
1.1 Definitions and assumptions
The branching random walk on the real line is a natural extension of the Galton-Watson process, with addition of a position to each individual, and is defined as follows. Initially, there is a single particle at the origin, forming the 0th generation. It gives birth to children, scattered in according to some point process and forming the 1st generation. Then, each particle of the 1st generation produces its own children disposed around its position according to the law of independently of others: this set of children forms the 2nd generation. The system goes on indefinitely, unless there is no particle at some generation.
The genealogical tree of the branching random walk, denoted by , is a Galton-Watson tree (where an individual can have an infinity of children). For , we denote by \mathopen{}\mathclose{{}\left\lvert z}\right\rvert the generation of the particle and by its position in . We denote by the log-Laplace transform of : we set, for each ,
[TABLE]
noting that has the same law as (V(z),\mathopen{}\mathclose{{}\left\lvert z}\right\rvert=1).
Throughout the paper, we assume the following integrability conditions on the reproduction law . First of all, we need to assume that the Galton-Watson tree is supercritical, that is
[TABLE]
so that the survival event has positive probability and, thus, we can introduce the new probability . Moreover, we work in the boundary case (Biggins and Kyprianou [12]) by assuming
[TABLE]
which means and . See the arXiv version of Jaffuel [30] for discussion on the cases where the branching random walk can be reduced to assumption (1.2). We assume also that
[TABLE]
where we set, for , and
[TABLE]
with . The first part of (1.3) gives . We will say that is -lattice if is the largest real number such that the support of is contained by and, then, is called the span of . In this paper, we work in both lattice and nonlattice cases, but we will need sometime to distinguish these cases. Finally, we set two last assumptions that are not supposed to hold in the whole paper, but only in specific cases of the results. The following assumption,
[TABLE]
means that is finite on and, thus, analytic on and, using (1.2) and (1.3), we have as by a Taylor expansion. The second one,
[TABLE]
comes from Madaule [35] and is probably not optimal for the results where it is used. Under assumption (1.6), is finite on and, therefore, analytic on a neighbourhood of 1, so we can improve the Taylor expansion: as .
For and , we set
[TABLE]
as soon as , which holds a.s. for under assumption (1.2). Then, is a random probability measure on \{z\in\mathbb{T}:\mathopen{}\mathclose{{}\left\lvert z}\right\rvert=n\}, which is called the Gibbs measure of parameter on the th generation of the branching random walk. It is also the law of a directed polymer on the tree in a random environment, introduced by Derrida and Spohn [24] as a mean field limit for directed polymer on a lattice as dimension goes to infinity: with this terminology, is the energy of the path leading from the root to particle , is the inverse temperature and is the partition function. In the case , we can define as the uniform measure on the random set \{\mathopen{}\mathclose{{}\left\lvert x}\right\rvert=n:V(x)=\min_{\mathopen{}\mathclose{{}\left\lvert z}\right\rvert=n}V(z)\}.
According to Derrida and Spohn [24], there is a critical parameter for the directed polymer on a disordered tree (with our setting , see Subsection 1.2 for more details) and our aim in this paper is to study the near-critical case, where depends on and tends from above and below to as . The near-critical case has been recently studied for the directed polymer on the lattice in dimension and by Alberts, Khanin and Quastel [4] and Caravenna, Sun and Zygouras [17, 18], with the emergence of the so-called intermediate disorder regime. For the polymer on a tree, some work near criticality has been done by Alberts and Ortgiese [5] and Madaule [35], mostly on the partition function.
Before stating our results, we recall some well-known properties of the branching random walk, that hold under assumptions (1.1), (1.2) and (1.3). First, the sequence
[TABLE]
is a martingale, called the derivative martingale, and Biggins and Kyprianou [11] (under slightly stronger assumptions) and Aïdékon [1] showed that we have
[TABLE]
Moreover, Chen [20] proved that these assumptions are optimal for the nontriviality of . Furthermore, Aïdékon [1] also showed that, in the nonlattice case, \min_{\mathopen{}\mathclose{{}\left\lvert z}\right\rvert=n}V(z)-\frac{3}{2}\log n converges in law under and described the limit as a random shift (depending on ) of a Gumbel distribution. In the lattice case, we do not have this convergence, but the tightness still holds (see Equation (4.20) of Chen [21] or Mallein [36]): for each , it exists such that
[TABLE]
for large enough.
1.2 The partition function
The process for some fixed has been intensively studied because, if is finite, then the renormalized process is a nonnegative martingale, called additive martingale, and, therefore, converges a.s. to some limit . Kahane and Peyrière [31], Biggins [7] and Lyons [33] have determined when this limit is nontrivial: under the additional assumption that the expectation is finite, we have the following dichotomy
[TABLE]
With the terminology of polymers’ literature (see [23]), the region is thus called the weak disorder regime and the region the strong disorder regime.
In the strong disorder regime , it is natural to seek a proper renormalization of , so that it converges towards a nontrivial limit. This question has already been answered when does not depend on . In the critical case , Aïdékon and Shi [3] proved that we have
[TABLE]
and that the particles that contribute mainly to are those of order . On the other hand, Theorem 2.3 of Madaule [34] shows (in the nonlattice case) that, in the case , we have
[TABLE]
where is a random variable independent of , whose law have been described by Barral, Rhodes and Vargas [6]. Moreover, the particles that contribute mainly to are in with some large constant and, thus, are close to the lowest particle at time (see (1.8)).
Since there is a discontinuity in the size of the partition function as between and , and also as between exponential and polynomial size, we try to understand this transition by considering the near-critical case where as . Some work has been done to this end by Alberts and Ortgiese [5], who considered the case where for and proved, under stronger assumptions, that
[TABLE]
Moreover, the behavior of the limit of the additive martingale has been studied near criticality by Madaule [35], who showed under assumption (1.6) the following convergence
[TABLE]
and, although this is not exactly our setting where and simultaneously, this result and its proof are useful in this paper. The following theorem improves Alberts and Ortgiese’s result, showing convergence to a nontrivial limit of after rescaling, for every sequence .
Theorem 1.1**.**
Assume (1.1), (1.2), (1.3) and that as . Let denotes the Brownian meander of length 1.
- (i)
If and as , then we have
[TABLE] 2. (ii)
If and as , then we have
[TABLE] 3. (iii)
If (1.5) holds, and as , then we have
[TABLE] 4. (iv)
If (1.6) holds, and as , then we have
[TABLE]
Note that, in case (i), the size of can be a as soon as : this possibility does not appear in Alberts and Ortgiese’s result.
1.3 Trajectory of particles under the Gibbs measure
The second main result of this paper concerns the trajectory of particles chosen according to the Gibbs measure. We first need to introduce some additional notation. For a particle at the th generation and , we denote by its ancestor at the th generation and we set
[TABLE]
the rescaled trajectory of ’s lineage. We work here in the set of the càdlàg functions from to , with the Skorokhod distance (see Section A.2). For and , we denote by the image measure of the Gibbs measure by , that is the random measure on such that, for each , we have
[TABLE]
where denotes the set of continuous bounded functions from to .
Convergence of has already been studied in the strong disorder regime when does not depend on , under assumptions (1.1), (1.2) and (1.3). In the critical case , Theorem 1.2 of Madaule [35] shows222Actually, Madaule [35] considers the linear interpolation of the trajectory, instead of , and the convergence on , instead of . But the convergence (1.14) follows from Madaule’s result. that, for all ,
[TABLE]
On the other hand, in the case , Chen, Madaule and Mallein [22] proved (in the nonlattice case) that, under , we have, for all uniformly continuous ,
[TABLE]
where is a sequence of i.i.d. normalized Brownian excursions and follows an independent Poisson-Dirichlet distribution with parameter . The convergence in (1.15) is believed to hold for all . Moreover, Chen [21] considered the case and showed that, for all , we have \mathbb{E}^{*}[\mu_{n,\infty}(F)]\to\mathbb{E}\mathopen{}\mathclose{{}\left[F(\mathfrak{e})}\right] as , where denotes the normalized Brownian excursion.
Finally, in the weak disorder regime , if there is some such that , we have the following convergence, with , for all ,
[TABLE]
in -probability333 To our knowledge, this result has not been proved yet (except when only depends on the final position, see [8]), so a proof is given in Section C of the appendix.. It means that the trajectory is a straight line of slope at first order and around which Brownian fluctuations occur at second order.
Our aim is to prove the convergence for trajectories of particles chosen according to the Gibbs measure in the near-critical case, in order to explain how happens the transition between the Brownian excursion, the Brownian meander and the straight line with Brownian fluctuations.
Theorem 1.2**.**
Assume (1.1), (1.2), (1.3) and that as . Let denotes the normalized Brownian excursion, the Brownian meander of length 1 and the Brownian motion.
- (i)
If and as , then we have, for all ,
[TABLE] 2. (ii)
If and as , then we have, for all ,
[TABLE] 3. (iii)
If (1.5) holds, and as , then we have, for all ,
[TABLE] 4. (iv)
If (1.6) holds, and as , then we have, for all , in -probability,
[TABLE]
We now state a corollary of this theorem, concerning the location of the mass of the Gibbs measure. Note that, for the terminology of the polymers’ literature, this position is the typical energy of the polymer in the near-critical case.
Corollary 1.3**.**
Assume (1.1), (1.2), (1.3) and that as .
- (i)
If and as , then, for all , it exists such that for large enough
[TABLE] 2. (ii)
If and as , then, for all , it exists such that for large enough
[TABLE] 3. (iii)
*If (1.5) holds, and as , then the same property as in case (ii) holds. * 4. (iv)
If (1.6) holds, and as , then, for all , it exists such that for large enough
[TABLE]
Proof.
In cases (ii), (iii) and (iv), it is a direct consequence of Theorem 1.2. For case (i), the assertion will be proved at the end of Subsection 5.2. ∎
1.4 Genealogy under the Gibbs measure
We state here a direct consequence of Theorem 1.2 concerning the overlap in the branching random walk, introduced by Derrida and Spohn [24] in the context of polymers on trees. We set, for ,
[TABLE]
that is the generation of the most recent common ancestor of and . For some couple of particles chosen according to , we are interested in the overlap between and defined by \mathopen{}\mathclose{{}\left\lvert x\wedge y}\right\rvert/n. Thus, we set, for a Borel set ,
[TABLE]
so that is a random probability measure on .
Madaule [35] gives the following consequence of (1.14) in the case :
[TABLE]
For the other extremal case , one can prove in the nonlattice case only444For example, using the stopping line \mathcal{Z}[A]\coloneqq\{z\in\mathbb{T}:V(z)\geq A>\max_{k<\mathopen{}\mathclose{{}\left\lvert z}\right\rvert}V(z_{k})\} for large , it is possible to show that, with high probability for large, all particles under have the same ancestor in . Then (1.17) follows from Lemma 3.3 of Chen [21]. In the lattice case, it is clear that (1.17) cannot hold. that,
[TABLE]
The transition between this two cases appears with case , with which Chen, Madaule and Mallein [22] deal, but their result (1.15) is only proved for and, thus, the convergence in law of cannot be obtained as a corollary. However, Mallein [37] shows that, under , we have
[TABLE]
where and follows a Poisson-Dirichlet distribution with parameter . It confirms a conjecture of Derrida and Spohn [24].
In the near critical case, we can state the following consequence of Theorem 1.2.
Corollary 1.4**.**
In each case of Theorem 1.1, under the same assumptions, we have
[TABLE]
Proof.
We give the proof for the case (i), but it is exactly the same for other cases (only the limiting trajectory changes). First note that, for all , we have, with and independent normalized Brownian excursions,
[TABLE]
in -probability and therefore in , because is bounded. Indeed, by Theorem 1.2, (1.18) holds when is of the form for some and, the general case follows. Then, we consider some and the closed set of . We have
[TABLE]
using (1.18) and the Portmanteau theorem. ∎
1.5 Comments on the results
Theorem 1.1 fully describes the transition from the size when to the size when and Corollary 1.3 shows the transition in the location of the mass of the Gibbs measure from when , to when (this follows from (1.16)). Secondly, Theorem 1.2 describes the transition between the Brownian excursion in case (i) and the straight line with Brownian fluctuations in case (iv). Note that, since a.s., it is natural that the limiting trajectory stays nonnegative on in cases (i) to (iii). In case (iv), this constraint disappears in the limit due to the drift. Indeed, staying above a constant for a random walk with drift approximately needs effort until times of order , so it disappears in the trajectory after scaling by . However, this effort has a cost of order , which explains the presence of this factor in the size of the partition function.
In cases (ii) and (iii), when is of order or larger, the perturbation is too small to change radically the behavior of the Gibbs measure in regards to the critical case : the size of the partition function is still , the typical energy is of order and the limiting trajectory has a density w.r.t. the Brownian meander. Therefore, cases (ii) and (iii) are called the critical window by Alberts and Ortgiese [5]. It brings to light a family of laws on defined by for all , including the law of the Brownian meander of length 1 for and such that as , where denotes the law of . But there is no convergence as , because the trajectory is sent to infinity. However, we can consider another family defined by , so that, in case (iii), we have
[TABLE]
This family includes also the Brownian meander’s law and we have as .
As opposed to this, cases (i) and (iv) are called the near-critical window and highlight some new behaviors. In case (i), the transition between the critical size and the strong disorder size appears. The factor starts to behave differently than when and this is also the region where the particles mainly contributing to are not simply those in : the fact that the lowest particle at time is around plays a role only in this region (see Lemma 5.2). Nevertheless, in the near-critical regime, the lowest particle at time never has a positive weight in in the limit, unlike in the case . Since the particles mainly contributing to are far below , the endpoint of the limiting trajectory has to be 0 and, therefore, the excursion appears.
In case (iv), since tends to slowly enough, we find the same asymptotic behavior for the partition function as in (1.13) when we first take and then . For the limiting trajectory, the result is also similar to (1.16) in the case . Moreover, when is not too small, the different results in case (iv) can be rewritten only in terms of : if , the size of the partition function is and, if , the location of the mass is in . But, on the contrary, if is too small, there is a break of universality.
Finally, we stress that there is no discontinuity between the different cases of the results. Indeed, using that and as , all cases of Theorem 1.1 (requiring in case (iv)) can be written
[TABLE]
noting that as soon as . For Theorem 1.2, the continuity between the different cases follows from the convergences and as .
1.6 Organization of the paper
Sections of this paper correspond to the different cases in the results: case (i) is treated in Section 5, cases (ii) and (iii) in Section 3 and case (iv) in Section 4. The behavior in the critical window (cases (ii) and (iii)) is a direct consequence of the analogue results (1.12) and (1.14) in the critical case (apart from some technical details). The near-critical window in the weak disorder regime (case (iv)) needs slightly more work, but relies mainly on Madaule’s [35] results and on inequalities techniques (see [9]). Finally, the main part of this paper is dedicated to the proof of case (ii), which follows some ideas of Aïdékon and Shi [3], with change of measure and spine decomposition techniques. One main difference with the previous literature on the branching random walk is that we need here to consider particles that are far below but also far above the lowest particle. Note that we prove in Subsection 5.6 a new version of the so-called peeling lemma (see Shi [40]) with a more general setting than what is needed for the aim of this paper and that could be of independent interest.
On the other hand, Section 2 regroups some well-known results on the branching random walk and on classical random walk. Some new results are stated in this section and proved in the appendix. Note that none of the results of Section 2 are needed for the proof of cases (ii) and (iii) and only a few of them for case (iv). The appendix contains some other technical results.
Throughout the paper, the ’s denote positive constants, we set and, for , . For two sequences and of positive real numbers, we say that as if , that u_{n}=O\mathopen{}\mathopen{}\mathclose{{}\left(v_{n}}\right) as if , and that u_{n}=o\mathopen{}\mathopen{}\mathclose{{}\left(v_{n}}\right) or if . For a metric space, let be the set of bounded continuous functions from and be its subset of uniformly continuous functions. For , we set \mathopen{}\mathclose{{}\left\lVert F}\right\rVert\coloneqq\sup_{x\in S}\mathopen{}\mathclose{{}\left\lvert f(x)}\right\rvert. For , we will denote by a modulus of continuity for function : is a continuous bounded nondecreasing function from such that and , \mathopen{}\mathclose{{}\left\lvert F(x)-F(y)}\right\rvert\leq\omega_{F}(d(x,y)).
2 Preliminary results
In this section, we state some preliminary results that are mostly needed in Section 5. In Subsection 2.1, we present some well-known tools to study the branching random walk and the next subsections contain results concerning one-dimensional random walk.
2.1 Many-to-one lemma and changes of probabilities
For , let denote a probability measure under which is the branching random walk starting from , and the associated expectation (for brevity we will write and instead of and ). We define a random walk associated to the branching random walk: under , a.s. and the law of the increments is given by
[TABLE]
for all measurable . This random walk is well-defined and centred thanks to assumption (1.2). Moreover, by assumption (1.3), we have . Then, by induction, one gets the following result (see Biggins and Kyprianou [10]). It is also a corollary of the forthcoming Proposition 2.2.
Lemma 2.1** (Many-to-one lemma).**
*For all , and all measurable function , we have *
[TABLE]
We now state some well-known change of probabilities and spinal decomposition results. This method dates back at least to Kahane and Peyrière [31], Rouault [38] and Chauvin and Rouault [19]. See also Biggins and Kyprianou [11] for spinal decomposition in more general type of branching structures and Shi [40] for a survey on this topic. Let denote the -algebra generated by (V(z),\mathopen{}\mathclose{{}\left\lvert z}\right\rvert\leq n) and .
We first introduce Lyons’ change of measure [33]: since is a nonnegative martingale of mean under , we can define a new probability measure on , by letting for all ,
[TABLE]
We will denote by the associated expectation and we will write and instead of and . Let be a point process on which has the law of (V(z),\mathopen{}\mathclose{{}\left\lvert z}\right\rvert=1) under .
Lyons [33] proved the following description for the branching random walk under , with a decomposition along a spine which is a marked ray in the the genealogical tree (in order to be mathematically rigorous, one should enlarge the probability space and work on a product space, see Lyons [33]). The system starts with one particle at position , forming the [math]th generation. For each , individuals of the th generation give birth independently of each other and of the foregoing. Individuals other than generate offspring around their position according to the law of and breeds according to the law of . Then, is chosen independently among ’s children, with probability proportional to for each child . Moreover, Lyons showed the following result concerning the spine under .
Proposition 2.2**.**
Let .
- (i)
For each and \mathopen{}\mathclose{{}\left\lvert z}\right\rvert=n, we have
[TABLE] 2. (ii)
The process under has the same law as under .
Now, we present another change of measure, that was first introduced by Biggins and Kyprianou [11]. For this, we need to define the renewal function in the first strict descending ladder height process of the random walk . For ,
[TABLE]
where is the first strict descending ladder height process: we set , and, for , \tau_{k}\coloneqq\inf\mathopen{}\mathclose{{}\left\{n>\tau_{k-1}:S_{n}<S_{\tau_{k-1}}}\right\} and . Then, we introduce the truncated derivative martingale: for and ,
[TABLE]
where, for , and, for \mathopen{}\mathclose{{}\left\lvert z}\right\rvert=n, . Fix now some . For , under , is a nonnegative martingale of mean and therefore we can define another probability measure by
[TABLE]
We will denote by the associated expectation and write and instead of and . For , let be a point process on with the law of (V(z),\mathopen{}\mathclose{{}\left\lvert z}\right\rvert=1) under .
Biggins and Kyprianou [11] proved the following spinal decomposition description for the branching random walk under , where the spine is denoted by . The description is similar to the previous one, but here have offspring according to and is chosen among these children, with probability proportional to for each child . Moreover, we get the following analogue of Proposition 2.2.
Proposition 2.3**.**
Let and .
- (i)
For each and \mathopen{}\mathclose{{}\left\lvert z}\right\rvert=n, we have
[TABLE] 2. (ii)
The process under has the same law as under conditioned to stay in : for all and all measurable function ,
[TABLE]
2.2 One-dimensional random walks
Up to the end of this section, we consider a centred random walk with finite variance . In this subsection, we state various results concerning this one-dimensional random walk and the associated renewal function . For , we set .
Recall that is the renewal function associated to the first strict descending ladder height process . Since and , by Feller [27, Theorem XVIII.5.1 (5.2)], we have . Thus, it follows from Feller’s [27, p. 360] renewal theorem that it exists a constant such that
[TABLE]
and so there exist also such that, for all ,
[TABLE]
We are interested in the behavior of random walks staying above a barrier. First, we recall the following estimate for the probability of a random walk to stay above : by Kozlov [32, Theorem A], it exists a constant such that for all ,
[TABLE]
and it exists such that, for all and , we have the uniform bound
[TABLE]
Constants and will appear all along the paper and they are related by the following equation, from Aïdékon and Shi [3, Lemma 2.1],
[TABLE]
The following result states the convergence in law of conditioned to stay above towards a Rayleigh distribution, with uniformity in the position of the barrier : by Aïdékon and Jaffuel [2, Lemma 2.2], if is a sequence of positive numbers such that as , then we have, for all continuous bounded function ,
[TABLE]
uniformly in .
From Lemmas 2.2 and 2.4 of Aïdékon and Shi [3], we have the following inequalities, sometimes called ballot theorems: it exists such that, for all , and ,
[TABLE]
and, for , it exists such that for all , , and , we have
[TABLE]
where we added a second barrier between times and . From the previous results, it follows (see Aïdékon [1, Lemma B.2]) that it exists such that, for all ,
[TABLE]
Finally, we state a last result that is used for the proof of the peeling lemma in Subsection 5.6 and is proved in Subsection B.2.
Lemma 2.4**.**
*Let be an increasing sequence of positive numbers such that we have . We set, for , and , *
[TABLE]
For any and , it exists such that for all , , and ,
[TABLE]
Remark 2.5*.*
This kind of lemma is useful in the proof of peeling lemmas: see Lemma B.3 of [1], Lemma 6.1 of [34] and Lemma A.6 of [40]. A slight difference here is that the terminal interval is instead of in the previous results. Furthermore, in these papers, they take with . This would have been sufficient for the proof of our peeling lemma, but an anonymous referee asked us whether the result holds for any and this lemma answers in the affirmative.
2.3 Convergence towards the Brownian meander
We define the rescaled trajectory of the random walk until time : for each ,
[TABLE]
We state the following convergence result for the trajectory conditioned to stay nonnegative, with uniformity in the starting point of the random walk.
Proposition 2.6**.**
Let be a sequence of positive numbers such that as . Then, for all , we have
[TABLE]
as , uniformly in , where denotes the Brownian meander of length .
This invariance principle has been proved in the case by Iglehart [28], Bolthausen [14] and Doney [25]. The case where is a function of the terminal value of the trajectory is already showed in (2.6). We give a short proof of this generalization in Subsection B.1, that relies on the invariance principle by Caravenna and Chaumont [15], for random walk conditioned to stay nonnegative for all time.
We present also a corollary of Proposition 2.6, which holds under an additional assumption on the random walk that is equivalent to assumption (1.5) by the many-to-one lemma. It will be used in Section 4 and its proof is postponed to Subsection B.1.
Corollary 2.7**.**
Assume that it exists such that . Then, for all and , we have
[TABLE]
as , where denotes the Brownian meander of length .
2.4 Convergence towards the Brownian excursion
In this subsection, our interest is the convergence of , conditioned to stay above two successive barriers and to end up in a small interval, towards the normalized Brownian excursion, with uniformity with respect to the barriers’ positions and to the endpoint as long as they are much smaller than . This result will be used repetitively in Section 5 for the proof of part (ii) of Theorems 1.1 and 1.2.
At this scale for the endpoint , the random walk behaves differently in the lattice and nonlattice cases, so they have to be distinguished. Moreover, we need some new notation: let be a random walk such that, under , a.s. and has the same law as . All objects referring to (, , , …) have their analogue for denoted with a superscript (, , , …).
We can now state our result, which generalizes Lemma 2.6 of Chen, Madaule and Mallein [22] and is proved in Subsection B.5.
Proposition 2.8**.**
Let be a sequence of positive numbers such that as and be the normalized Brownian excursion.
- (i)
If the law of is nonlattice, then for all , and ,
[TABLE]
as , uniformly in and . 2. (ii)
If the law of is -lattice, then for all and ,
[TABLE]
as , uniformly in , and .
3 The critical window
We prove here cases (ii) and (iii) of Theorems 1.1 and 1.2, where with . This proof is based on Theorem 1.2 of Madaule [35], recalled in (1.14).
Proof of part (ii) of Theorems 1.1 and 1.2.
Our aim is to show that, for all , we have
[TABLE]
Then, using (1.12) and the case , part (ii) of Theorems 1.1 and 1.2 follows. Note also that it is sufficient to show (3.1) for nonnegative.
We first deal with the case . We consider some nonnegative function and . By dominated convergence, we have that tends towards as , so we can choose such that
[TABLE]
Since and, under , \min_{\mathopen{}\mathclose{{}\left\lvert z}\right\rvert=n}V(z)\to\infty a.s., it exists such that, for all , we have and also \mathbb{P}^{*}(\min_{\mathopen{}\mathclose{{}\left\lvert z}\right\rvert=n}V(z)<0)\leq\varepsilon. Moreover, we set, for , and : if , then we have . Thus, we get, for all ,
[TABLE]
Therefore, we have
[TABLE]
using (3.2). Then, applying (1.14), we get that both probabilities in (3.3) tends to 0, because and it concludes the proof of (3.1) in the case .
Finally, for the case , we proceed in the same way as for , taking here such that , defined as before and . Then, the same inequalities hold. ∎
Remark 3.1*.*
In case (iii), we work under assumption (1.5) and we will use Proposition 3.8 of Madaule [35], whereas Madaule works in [35] under the stronger assumption (1.6). But, for the proof of his Proposition 3.8, he only uses Assumption (1.6) in the proof of his Lemma A.2, in order to have that is finite on a left-neighbourhood of 1 and as , and this holds also under our assumption (1.5).
Proof of part (iii) of Theorems 1.1 and 1.2.
Applying Proposition 3.8 of Madaule [35] (see Remark 3.1 above), we get that in -probability with for any , although is not bounded. Combining this with (1.14), it is straightforward to extend this convergence to functions of the type with and .
Then, we prove that, for all nonnegative,
[TABLE]
using the same method as for the proof of case (ii): we approach function from above and from below, by considering here and when and the same function but with when . Finally, part (iii) of Theorems 1.1 and 1.2 follows from (3.4). ∎
4 The near-critical window in the weak disorder regime
In this section, we deal with the case where and and prove successively convergence of the rescaled partition function and then convergence of the trajectories. We work here under assumption (1.6) so is analytic on an open interval containing 1. Moreover, by Lemma 4.3 of Madaule [35], assumption (1.6) implies that there exist and such that for any and . Thus, for any and , we have, with ,
[TABLE]
by using Jensen’s inequality.
We need to introduce a more general statement of the many-to-one lemma. For , we define another random walk starting at [math] under and such that
[TABLE]
Then, is centred and has variance . Moreover, we have the following analogue of the many-to-one lemma (see Shi [40]): for all and all measurable function , we have
[TABLE]
One can see as a discrete Girsanov transform of .
We first establish a preliminary lemma.
Lemma 4.1**.**
Let and be sequences such that and for any . Then, for any ,
[TABLE]
Proof.
We first apply the many-to-one lemma to get that
[TABLE]
Then, we set and, applying the Markov property at time , we have
[TABLE]
using Corollary 2.7 to bound the first expectation in the middle part of (4) and the many-to-one lemma for the second expectation. Using that , it proves the lemma. ∎
Proof of part (iv) of Theorem 1.1.
We set and want to show that in -probability. It will prove part (iv) of Theorem 1.1, since in -probability by (1.13).
We first follow the proof of Lemma 4.2 of Madaule [35]. We set and . For , we have
[TABLE]
using that \mathbb{P}\mathopen{}\mathclose{{}\left(\xi_{n}^{p_{n}}\geq\varepsilon^{p_{n}}\mathrel{}\middle|\mathrel{}\mathscr{F}_{n}}\right)\leq\varepsilon^{-p_{n}}\mathbb{E}[\xi_{n}^{p_{n}}|\mathscr{F}_{n}]=\varepsilon^{-p_{n}}\xi^{\prime}_{n}. By the branching property at time , we have
[TABLE]
where, conditionally on , the for \mathopen{}\mathclose{{}\left\lvert x}\right\rvert=n are independent variables with the same law as . Then, using that for any sequence of independent centred variables and any we have (see [42]), we get
[TABLE]
by using (4.1) for large enough such that . Now, we choose such that and, by (4.5) and Markov’s inequality, we get
[TABLE]
As , we have , and, by a Taylor expansion, . Thus, applying Lemma 4.1 with and , we showed that . Coming back to (4.4), it concludes the proof. ∎
Proof of part (iv) of Theorem 1.2.
By Lemma A.1, we can reduce the proof to the case . Moreover, by considering F-\mathbb{E}\mathopen{}\mathclose{{}\left[F(B)}\right] instead of , we can assume that \mathbb{E}\mathopen{}\mathclose{{}\left[F(B)}\right]=0. By Theorem 1.1, we have in -probability with -a.s., so it is sufficient to prove that
[TABLE]
in -probability, where for . For some , we set and, for each , . Let . In order to prove (4.6), it is sufficient to prove that we can choose such that
[TABLE]
The assumption that \mathbb{E}\mathopen{}\mathclose{{}\left[F(B)}\right]=0 is only needed for (4.9). For the sequel, we fix some such that .
We first prove (4.8). We set \zeta_{n}\coloneqq U_{n}(F_{n})-\mathbb{E}\mathopen{}\mathclose{{}\left[U_{n}(F_{n})\mathrel{}\middle|\mathrel{}\mathscr{F}_{k_{n}}}\right] and proceed in a similar way as for the proof of part (iv) of Theorem 1.1, by setting and \zeta^{\prime}_{n}\coloneqq\mathbb{E}[\mathopen{}\mathclose{{}\left\lvert\zeta_{n}}\right\rvert^{p_{n}}|\mathscr{F}_{n}]. By (4.4), we have \mathbb{P}^{*}(\mathopen{}\mathclose{{}\left\lvert\zeta_{n}}\right\rvert\geq\varepsilon)\leq\varepsilon\mathbb{P}(S)^{-1}+\mathbb{P}^{*}(\zeta^{\prime}_{n}\geq\varepsilon^{1+p_{n}}). By the branching property at time , we have
[TABLE]
where, conditionally on , the for \mathopen{}\mathclose{{}\left\lvert x}\right\rvert=k_{n} are independent variables with the same law as defined by
[TABLE]
Since the \Upsilon_{n}^{(x)}-\mathbb{E}\mathopen{}\mathclose{{}\left[\Upsilon_{n}}\right] are also centred, we get, in the same way as for (4.5),
[TABLE]
by using the following bound
[TABLE]
where we used (4.1) for large enough such that . Now, using that and Markov’s inequality, we get
[TABLE]
so, using that, as , we have (2\mathopen{}\mathclose{{}\left\lVert F}\right\rVert\alpha_{n}/\varepsilon)^{p_{n}}\sim 2\mathopen{}\mathclose{{}\left\lVert F}\right\rVert\alpha_{n}/\varepsilon, and , and applying Lemma 4.1 with , we get
[TABLE]
by choosing large enough. This proves (4.8). The constant is now fixed.
We now prove (4.7). Using that and the Markov inequality, we get, with ,
[TABLE]
For , we define . Then, using the many-to-one lemma and the triangle inequality, we get that, for any , the expectation in (4.13) is smaller than
[TABLE]
Note that, using (A.7) with here , we have, for any ,
[TABLE]
and, thus, the first term in (4.14) is smaller than
[TABLE]
using (4) to bound the expectation and recalling that . On the other hand, using Markov property at time , the second term in (4.14) is smaller than
[TABLE]
applying Corollary 2.7 and recalling that . Coming back to (4.13), we finally get
[TABLE]
by choosing large enough ( and being fixed). It proves (4.7).
Finally, we prove (4.9). Using the branching property in the same way as for (4.10), we have \mathbb{E}\mathopen{}\mathclose{{}\left[U_{n}(F_{n})\mathrel{}\middle|\mathrel{}\mathscr{F}_{k_{n}}}\right]=\alpha_{n}\widetilde{W}_{k_{n},\beta_{n}}\mathbb{E}\mathopen{}\mathclose{{}\left[\Upsilon_{n}}\right], where is defined in (4.11). By (4.2), we get
[TABLE]
where we introduced and where and satisfy and , using that , and . Now, note that in law. This is not a direct consequence of Donsker’s theorem because here the law of the random walk changes for each . However, we can apply a strong invariance principle like Equation (11) of Sakhanenko [39]: it exists such that, for any and , it exists a Brownian motion such that
[TABLE]
Since and \mathbb{E}[\mathopen{}\mathclose{{}\left\lvert S_{1,\beta_{n}}}\right\rvert^{3}]\to\mathbb{E}[\mathopen{}\mathclose{{}\left\lvert S_{1}}\right\rvert^{3}]<\infty as , this proves that in law. Then, applying Lemma A.3, we get that \mathbb{E}\mathopen{}\mathclose{{}\left[\Upsilon_{n}}\right]\to\mathbb{E}\mathopen{}\mathclose{{}\left[F(B)}\right]=0 as . On the other hand, using that and the Markov inequality, we get
[TABLE]
and it proves (4.9) by using Lemma 4.1 and that \mathbb{E}\mathopen{}\mathclose{{}\left[\Upsilon_{n}}\right]\to 0 as . ∎
5 The near-critical window in the strong disorder regime
In this section, we prove case (i) of Theorems 1.1 and 1.2 and of Corollary 1.3, where with . This case constitutes the main part of this paper.
5.1 Change of probabilities
We introduce a first barrier by setting, for and ,
[TABLE]
so that for large this two variables are equal with high probability. Moreover, recall that, for , and
[TABLE]
and that, using the martingale , we defined the modified probability measure . We will work under this measure to study the asymptotic behaviour of .
Proposition 5.1**.**
For all and , we have the convergence
[TABLE]
This proposition will be proved in the following subsections, using a second moment argument similar to the one used by Aïdekon and Shi [3].
Proof of part (i) of Theorems 1.1 and 1.2.
We consider and we are going to show here that
[TABLE]
Using (2.5), it proves part (i) of Theorem 1.1 by taking . Moreover, noting that and -a.s., it proves part (i) of Theorem 1.2 in the case . The general case follows by Lemma A.1. Thus, it is now sufficient to prove (5.1) and we can for this purpose assume that is nonnegative.
Let . We first fix such that . Combining that \min_{\mathopen{}\mathclose{{}\left\lvert x}\right\rvert=n}V(x)\to\infty -a.s. as and that as , we get that, on the event , -a.s. Thus, considering the event
[TABLE]
we can fix such that . We now introduce the event
[TABLE]
Then, using that on we have , we get, for ,
[TABLE]
applying Proposition 5.1. Since , it follows that in -probability. Using again that, on , -a.s., we get that . Recalling that , we showed that \limsup_{n\to\infty}\mathbb{P}^{*}\mathopen{}\mathclose{{}\left(E_{n}}\right)\leq 2\eta and, therefore, it proves (5.1). ∎
5.2 Proof of Proposition 5.1 and of part (i) of Corollary 1.3
The aim of this section is to break down the proof of Proposition 5.1 into the proof of several lemmas. Our goal is to use a second moment argument, but the first moment of under does not have the right order and the second moment is not even necessarily finite. Thus, we first need to come down to another random variable that is close to with high probability and that has first and second moments of the right order, by eliminating some rare particles with a too strong weight in the expectations. The lemmas stated in this subsection will also allow us to prove part (i) of Corollary 1.3.
Until the end of the paper, is a fixed positive constant. We first add a second barrier between times and at position , by setting, for ,
[TABLE]
This first lemma shows that and are close with high probability and will be proved in Subsection 5.4. Note that this step is superfluous when : since the particles contributing to are of order , it is approximately the same difficulty for them to stay above the two barriers than only above the first (see Proposition 2.8) and, thus, and have asymptotically the same first moment in that case.
Lemma 5.2**.**
*For all and , it exists such that, for each , *
[TABLE]
We now consider a fixed . We will see in Lemma 5.4 that has the right first moment, but we still need to remove other particles for the second moment. Let and be sequences of positive real numbers and be a sequence of integers such that
[TABLE]
[TABLE]
when . We add some controls on the trajectory of the particle’s lineage, by considering
[TABLE]
where we set A_{n}\coloneqq\{\mathopen{}\mathclose{{}\left\lvert z}\right\rvert=n:\forall\,j\in\llbracket 0,n\rrbracket,V(z_{j})\in I_{n,j}\} (see Figure 1), with
[TABLE]
Note that the second barrier is here simply at : indeed, Lemma 5.4 shows that it does not change the first moment (and we could even have taken with any ).
But, in order to compute the first moment of , we will first need to take the conditional expectation given and, thus, we want to show that does not asymptotically depend on what happen before time . For this, we consider for each a slight modification of function such that
[TABLE]
More formally, we set, for each , , . The following lemma, proved in Subsection 5.5, shows that we can replace with . It does not play a crucial role, but makes the calculations easier for the next lemmas.
Lemma 5.3**.**
For all and , we have
[TABLE]
as .
Noting that, for nonnegative, , Lemma 5.3 combined with the following lemma shows that the first moments under of , , and (divided by ) have the same equivalent as . It will be proved in Subsection 5.3.
Lemma 5.4**.**
*For all and nonnegative, we have *
[TABLE]
However, this is still not sufficient for controlling the second moment of and we need to introduce a new random variable . We consider the sequences
[TABLE]
Then, for some fixed sequence that tends to infinity such that , we define the following set, that will allow us to control the offspring of the spine in the second moment calculation (see Aïdékon [1] for the first use of this method),
[TABLE]
where is the set of brothers of , and we set
[TABLE]
The following lemma shows that this new random variable is close to . Its proof relies on the peeling lemma stated in Subsection 5.6 in a more general feature.
Lemma 5.5**.**
For all and , we have
[TABLE]
as .
By considering , we can now control the second moment properly, as stated in this last lemma, proved in Subsection 5.7. This second moment is exactly the square of the first moment in Lemma 5.4.
Lemma 5.6**.**
For all and nonnegative, we have
[TABLE]
Using these lemmas, we can now prove Proposition 5.1.
Proof of Proposition 5.1.
We consider here the case and the general case follows by taking the positive and negative parts of . We first fix . Using Lemmas 5.4 et 5.5, we have
[TABLE]
and so, by applying Bienaymé-Chebyshev inequality and Lemma 5.6, we get that
[TABLE]
Combining (5.3) with Lemmas 5.4, 5.3 and 5.5, we deduce that for all ,
[TABLE]
Now, for , by Lemma 5.2, we can choose such that
[TABLE]
and, thus, we get, using the triangle inequality, (5.4) and (5.5),
[TABLE]
This concludes the proof of Proposition 5.1. ∎
We conclude this subsection by proving part (i) of Corollary 1.3, which is a consequence of the fact that considering only particles in does not change the first moment asymptotic.
Proof of part (i) of Corollary 1.3.
First note that it is sufficient to prove that, for all sequences and such that , we have
[TABLE]
Indeed, if we assume that part (i) of Corollary 1.3 is false, then it exists such that for any , it exists (with ) such that
[TABLE]
Setting , we have and . Thus, with and , (5.7) implies the negation of (5.6). Therefore, we now want to prove (5.6).
The left-hand side of (5.6) is larger than , therefore, it is sufficient to show that in -probability. Combining (5.3) with Lemma 5.5, we first have
[TABLE]
and, in the same way as in the proof of part (i) of Theorems 1.1 and 1.2, where we showed (5.1) from Proposition 5.1, it follows from (5.8) that
[TABLE]
Using (5.9) and (5.1) with , we get that in -probability and so (5.6) is proved. ∎
5.3 First moments of and
We start with the proof of Lemma 5.4 in this subsection, because this first moment calculation will be a kind of routine at which we will refer for the proof of other lemmas. In this calculations, sums of general term appear regularly, with a constant and as . Therefore, we state the following result, proved by explicit computation: if and are sequences with values in such that as , then we have, for any constant ,
[TABLE]
Proof of Lemma 5.4.
Recall that we consider nonnegative. We begin with the control of . Note that . Then, using the many-to-one lemma, we get
[TABLE]
We cut this expectation in two pieces depending on whether or . Let start with the case : we have, by cutting the interval in pieces of length ,
[TABLE]
by using (2.8). Since , it follows from (5.10) that the right-hand side of (5.11) is a . We now control the term corresponding to the case , which is the dominant term in . This time, we cut the segment in pieces of length , where is any real number if is nonlattice and is the span of the lattice if is lattice, and thus we get
[TABLE]
by using Proposition 2.8 in both lattice and nonlattice cases, with uniformity in because . Then, by applying (2.1) to , for , it exists such that, for all , and thus we get
[TABLE]
by applying (5.10). Coming back to (5.12), we get
[TABLE]
by applying (2.5) to constants and . Combining (5.11) and (5.14), we conclude that
[TABLE]
and it shows the first part of Lemma 5.4.
We now want to prove the lower bound for . We use the branching property at time to get
[TABLE]
where we set, for all ,
[TABLE]
We note and fix , then we have for large enough because . Using the many-to-one lemma, we get, for ,
[TABLE]
Then, we cut the segment in pieces of length , where is any real number if is nonlattice and is the span of the lattice if is lattice, and we get that is larger than
[TABLE]
where the o\mathopen{}\mathopen{}\mathclose{{}\left(1}\right) is uniform in and , by using Proposition 2.8, because we have and . Thus, we get, using that and uniformly in and ,
[TABLE]
uniformly in , where we used (2.5) twice and also (5.10). Coming back to (5.15), we get that \mathbb{E}\mathopen{}\mathclose{{}\left[Y_{n}(F_{n})}\right] is larger than
[TABLE]
Using the many-to-one lemma, the expectation in (5.16) is equal to
[TABLE]
for all . We then choose a function continuous and bounded such that, for all ,
[TABLE]
and (5.17) is larger than
[TABLE]
by applying (2.6). Coming back to (5.16), we get
[TABLE]
using that . Since , it concludes the proof of Lemma 5.4. ∎
5.4 Addition of the second barrier
In this section, we prove Lemma 5.2 with a method similar to the one used by Madaule [34] for his Lemma 4.9 (or Lemma 3.3 of Aïdékon [1]). The main difference is that we do not only consider particles that are at a distance of order from , but we can nevertheless apply some of Madaule’s results.
Proof of Lemma 5.2.
We fix and . For all and , we have \lvert W^{(L)}_{n,\beta_{n}}(F)-W^{(L,K)}_{n,\beta_{n}}(F)\rvert\leq 2\mathopen{}\mathclose{{}\left\lVert F}\right\rVert\lvert W^{(L)}_{n,\beta_{n}}-W^{(L,K)}_{n,\beta_{n}}\rvert, where and . Therefore, it is sufficient to show that we have
[TABLE]
for large enough.
Using Proposition A.3 of Aïdékon [1], we know that converges in to under , so we can choose large enough such that, for all , . Therefore, the probability in (5.18) is smaller than
[TABLE]
because . Thus, we now want to prove that, for some large enough, we have
[TABLE]
with and . Moreover, using (1.8), we can fix such that \mathbb{P}(\min_{\mathopen{}\mathclose{{}\left\lvert z}\right\rvert=n}V(z)<\frac{3}{2}\log n-K^{\prime})\leq\eta^{\prime}. Thus, our aim is now to show that we have
[TABLE]
for some large enough.
Now, following Madaule’s [34] proof of his Lemma 4.9, we introduce the intervals , for , and the events, for and ,
[TABLE]
and, denoting for some fixed ,
[TABLE]
Then, we have
[TABLE]
On the one hand, by Madaule’s [34] proof of his Lemma 4.9, we have the inequality
[TABLE]
and, therefore,
[TABLE]
On the other hand, by using the many-to-one lemma, we get, for , and ,
[TABLE]
where we set
[TABLE]
We recall Equation (4.27) of Madaule [34]:
[TABLE]
Applying (5.25), we get
[TABLE]
Using (5.22), (5.26) and that \mathbb{P}\mathopen{}\mathclose{{}\left(E_{i,k,\ell}}\right)=0 for , we have
[TABLE]
We can bound the sum on in (5.27) by (1+\ell)\alpha_{n}^{2}(1+o\mathopen{}\mathopen{}\mathclose{{}\left(1}\right)) uniformly in . Moreover, taking large enough such that , we have for all . Thus, we get that (5.27) is smaller than
[TABLE]
and, thus, with the Markov inequality, we have
[TABLE]
Finally, (5.19) follows from (5.20), (5.21) and (5.28) by taking large enough and it concludes the proof of Lemma 5.2. ∎
5.5 From to
We prove here that considering instead of does not change significantly the first moment.
Proof of Lemma 5.3.
To control the first moment of W^{(L,K)}_{n,\beta_{n}}(\mathopen{}\mathclose{{}\left\lvert F-F_{n}}\right\rvert)/D_{n}^{(L)} under , we follow the proof of Lemma 5.4 for the upper bound of the first moment of , but, instead of applying directly Proposition 2.8 as in (5.12), we use that
[TABLE]
uniformly in , by using Proposition 2.8 combined with Lemmas A.3 and A.5, because . The result follows with the same calculations as in the proof of Lemma 5.4. ∎
5.6 The peeling lemma
The aim of this subsection is to prove Lemma 5.5, which shows that introducing the event does not change the first moment. This proof is based on the so-called peeling lemma (see Shi [40]), which controls that the spine, conditioned to have a specific trajectory, does not have too many and too low children. Such lemmas have been proved in the case where the spine ends up at a distance of constant order from (see [1, Lemma C.1], [34, Lemma 7.1] and [40, Theorem 5.14]) and also when it ends up at a position of order (see [3, Lemma 4.7]). Here we have to deal with the intermediate case where the spine end up far above and far below .
In order to state the peeling lemma in a general setting, we introduce some notation. For and , we set
[TABLE]
We consider
[TABLE]
and the following set
[TABLE]
We can now state our version of the peeling lemma, which covers the case where the spine ends up far below and is therefore more general than the peeling lemmas in [1, 34, 40].
Lemma 5.7** (Peeling lemma).**
For all , there exist and such that, for all , and ,
[TABLE]
Remark 5.8*.*
We present here the peeling lemma in terms of probability measure , because it simplifies somehow the proof (for example, under , the reproduction law along the spine does not depend on the position), but it is a direct consequence that, for all , there exist and such that, for all , and ,
[TABLE]
Before proving the peeling lemma, we first use it to show Lemma 5.5.
Proof of Lemma 5.5.
We set and and, since is bounded, it is sufficient to show that
[TABLE]
We first change probabilities from to : we have
[TABLE]
Then, setting and and cutting the segment in pieces of length 1, we get that (5.29) is smaller than
[TABLE]
using Lemma 5.7 uniformly in , noting that and . Using then (2.2) and (5.10), we get that (5.30) is a and it concludes the proof of Lemma 5.5. ∎
Proof of Lemma 5.7.
By Lemma 2.4 and (2.2), it exists such that, for all , and , we have
[TABLE]
Thus, it is sufficient to show that
[TABLE]
for large enough, and . Therefore, we now prove (5.31). We first set, for ,
[TABLE]
Since for all , , we have
[TABLE]
where we set, for , by noting the parent of and ,
[TABLE]
Thus, we have, on event ,
[TABLE]
and, if we are moreover on event so that , it implies that
[TABLE]
where . Therefore, we get
[TABLE]
where we set, for ,
[TABLE]
and . From now, we choose such that .
We first consider the case . Since , we have
[TABLE]
where we set, for ,
[TABLE]
using successively Proposition 2.2 (ii), (2.8) and (2.2). We thus get
[TABLE]
But, under , is independent of and has moreover the same law as , where and are defined in (1.4). Therefore, we get, by integrating first on ,
[TABLE]
where we set, for ,
[TABLE]
by using Proposition 2.2 (ii) and that . Then, applying (2.9), we have
[TABLE]
using also (2.2). Coming back to (5.33) and noting that, since , we have , this gives
[TABLE]
and concludes the case .
We now consider the case , so . On event , we have , and, for all , . Moreover, we set
[TABLE]
and note that, on event , for large enough, we have , because and . Thus, we get
[TABLE]
Under , is independent of , so the right-hand side of (5.35) is equal to
[TABLE]
where we set, for ,
[TABLE]
using successively Proposition 2.2 (ii), (2.8) and (2.2) as before. On the event , using that , we have and also , therefore, (5.36) is smaller than
[TABLE]
Note then that, under , is independent of and has the same law as , so (5.37) is equal to
[TABLE]
where we set, for ,
[TABLE]
by applying Proposition 2.2 (ii) and then reversing time. Thus, using (2.9) and (2.2), we get
[TABLE]
On event , we have so, coming back to (5.38), we get
[TABLE]
Finally, using (5.34), (5.39) and that \mathbb{E}_{\mathbb{Q}}\mathopen{}\mathclose{{}\left[V(w_{1})_{+}\mathrel{}\middle|\mathrel{}\mathscr{F}_{1}}\right]=\widetilde{X}/X by Proposition 2.2 (i), we get
[TABLE]
Using (1.3), we can choose large enough such that the expectation in (5.40) is smaller than and, recalling (5.32), it proves (5.31) and concludes the proof of Lemma 5.7. ∎
5.7 Second moment of
For , by decomposing along the spine, is equal to
[TABLE]
We cut this sum in two pieces, depending on whether the lineage of the considered particle splits off from the spine’s lineage before or after time :
[TABLE]
We define in the same way and . Then, by Aïdékon and Shi [3, Lemma 4.7], since , we have
[TABLE]
Proof of Lemma 5.6.
First note that, using Proposition 2.3 (i),
[TABLE]
and, thus, we have
[TABLE]
by splitting . The first part will give the right order and constant and the second part will be negligible. Recall that is assumed to be nonnegative.
We begin by bounding . Using and , we get
[TABLE]
where we set, for and with ,
[TABLE]
for some fixed and large enough, by applying Proposition 2.3 (ii). Then, proceeding in the same way as for the lower bound of in the proof of Lemma 5.4 (but with a sum on from to ), we get the upper bound
[TABLE]
uniformly in , using (2.1) and (2.5) for the last equality. Coming back to (5.42), we showed that
[TABLE]
and thus we now want to bound the expectation in (5.44). We proceed in a way similar to the proof of Lemma 4.5 of Aïdekon et Shi [3], by introducing the event :
[TABLE]
Thus, applying (5.41), we get that the expectation in (5.44) is smaller than
[TABLE]
using that Y_{n}^{\prime[0,k_{n})}(F_{n})/D_{n}^{(L),[0,k_{n})}\leq\mathopen{}\mathclose{{}\left\lVert F}\right\rVert and applying the Markov inequality for the second term. Using that , and then Lemmas 5.4 and 5.3, we get that (5.45) is smaller than
[TABLE]
because for large enough. Thus, coming back to (5.44), we proved that
[TABLE]
We now want to show that and, by bounding F\leq\mathopen{}\mathclose{{}\left\lVert F}\right\rVert, it is sufficient to deal with the case . Using that , and breaking down along the spine, we first have that is smaller than
[TABLE]
Noting that, on the event , we have , the first term in (5.47) is smaller than
[TABLE]
by conditioning with respect to and noting that, given , is independent of (V(z),\mathopen{}\mathclose{{}\left\lvert z}\right\rvert=n,z\geq y,y\in\Omega(w_{i+1}^{(L)})). Noting that we are on the event , (5.48) is smaller than
[TABLE]
because and therefore the sum for is a . For the second term in (5.47), we use that, on the event , , and, thus, combining with (5.49), we get
[TABLE]
where the function has been defined previously in the proof. Using (5.43) again, we get
[TABLE]
Proceeding in the same way as before by using (5.41) to introduce the event , the expectation in (5.50) is smaller than
[TABLE]
As for (5.45), (5.51) is equal to (1+o\mathopen{}\mathopen{}\mathclose{{}\left(1}\right))/R(L) and, coming back to (5.50), we get
[TABLE]
because . This concludes the proof of Lemma 5.6. ∎
Appendix A Convergence of random measures
In this section, we present some results concerning convergence of random or deterministic probability measures on a polish space or more specifically on . Some of these results are classical, but we state them here with uniformity in some parameter .
A.1 General space
Let , and be Polish spaces. We consider some set . In the sequel, for each , will denote a sequence of random probability measures on and a sequence of deterministic probability measures on . Moreover, and will be probability measures on , that are deterministic and do not depend on .
Lemma A.1**.**
Assume that for all Lipschitz and , as uniformly in . Then, the same convergence holds for all .
Note that in the case where we consider a deterministic sequence , it simply means that uniformly in . It is necessary that the limit does not depend on .
Proof.
We follow the proof of Portmanteau Theorem in Billingsley [13, Theorem 2.1]. Thus, we first consider a closed set and and we want to show that
[TABLE]
We consider, for each , the function that is Lipschitz and such that . Thus, by dominated convergence, we have as . We fix such that . Since , we get
[TABLE]
by using the assumption of the lemma. From (A.1), we get that, for all set such that -a.s. and all ,
[TABLE]
We now consider and we can assume that is nonnegative. We fix and set M\coloneqq\mathopen{}\mathclose{{}\left\lVert F}\right\rVert. Firstly, we have
[TABLE]
and, therefore, for almost every (in the sense of the Lebesgue measure), -a.s., . Thus, for all , we can fix a subdivision such that for all , and for all , -a.s. Since and is nonincreasing, we have
[TABLE]
and the same holds for instead of , for all and . Since, in (A.3), the left-hand side and right-hand side of (A.3) tend to as , we can choose large enough such that they are at most at distance from . Then, using (A.3) for , it follows that
[TABLE]
by noting that, in the sums, the term for is zero, since t_{N}=M=\mathopen{}\mathclose{{}\left\lVert F}\right\rVert, and the term for is smaller than , if we choose large enough. Then, we have
[TABLE]
by using (A.2), because for all , -a.s. ∎
Lemma A.2**.**
We consider the product space . Assume that, for all and , as uniformly in , where . Then, for all , uniformly in .
Proof.
Using Lemma A.1, it is sufficient to consider . Let . Since is uniformly continuous, it exists such that for any that verify , we have \mathopen{}\mathclose{{}\left\lvert F(x)-F(y)}\right\rvert\leq\varepsilon. Since is separable for the metric , it exists such that , where denotes the open ball of radius centred at in . Now we consider a compact set such that and the image of under the canonical projection . Since is a compact set of , we can extract a finite cover . Using again the compacity of , there exist nonnegative Lipschitz functions such that, for all , , and on . Finally, we set , so that is a Lipschitz continuous function from and on .
We can now construct some functions of the form to approach . For , we set and . By the triangle inequality, we have that is smaller than
[TABLE]
The second term in (A.4) tends to 0 as uniformly in by the assumption of the lemma. On the other hand, we have
[TABLE]
because of the choice of . Thus, we get that the first and third terms of (A.4) are smaller than 2\varepsilon+\mathopen{}\mathclose{{}\left\lVert F}\right\rVert(\xi^{\theta}_{n}(1-\chi)+\xi(1-\chi)). Since , we can use again the assumption of the lemma to get that as uniformly in . Noting that and , it proves Lemma A.2. ∎
A.2 Weak convergence in
We keep here the notation of the previous subsection, but we take . Recall the definition of the Skorokhod distance on : for ,
[TABLE]
where we set \Lambda\coloneqq\mathopen{}\mathclose{{}\left\{\lambda\colon[0,1]\to[0,1]\mathrel{}\middle|\mathrel{}\lambda(0)=0,\lambda(1)=1,\lambda\text{ continuous and increasing}}\right\}, and that, equipped with this distance, is a polish space (see Billingsley [13]).
Lemma A.3**.**
Assume that, for all , we have as uniformly in . We consider , , and such that , and . We set . Then, we have uniformly in .
Proof.
For , we first have
[TABLE]
Now, we consider and we fix such that and some Lipschitz function such that for all (this is possible since is Lipschitz on ). Thus, we have
[TABLE]
On the one hand, for , we have, using and (A.5),
[TABLE]
so we get \lvert\xi^{\theta}_{n}(\chi(F_{n}-F))\rvert\leq\mathopen{}\mathclose{{}\left\lVert\chi(F_{n}-F)}\right\rVert\to 0 as uniformly in . On the other hand, by using the assumption of the lemma with the function , the first and third terms in the right-hand side of (A.6) tend towards 2\mathopen{}\mathclose{{}\left\lVert F}\right\rVert\xi(1-\chi) uniformly in . Since , it concludes the proof of Lemma A.3. ∎
Remark A.4*.*
In Lemma A.3, if the limit measure depends on , then the result is still true under the additional assumption that as , so that in the proof and could be chosen independently of .
Lemma A.5**.**
Assume that and that, for all , we have as uniformly in . We consider and such that . We set . Then, we have \xi^{\theta}_{n}(\mathopen{}\mathclose{{}\left\lvert F_{n}-F}\right\rvert)\to 0 uniformly in .
Proof.
The function is not bijective from to , so we consider a function such that for and that is linear on . Then, for , we have
[TABLE]
Let . Since , it exists such that \xi(\{\max_{[0,\delta]}\mathopen{}\mathclose{{}\left\lvert x}\right\rvert>\varepsilon\})\leq\varepsilon. Let be a Lipschitz function such that \mathbbm{1}_{\max_{[0,\delta]}\mathopen{}\mathclose{{}\left\lvert x}\right\rvert<\varepsilon}\leq\chi(x)\leq\mathbbm{1}_{\max_{[0,\delta]}\mathopen{}\mathclose{{}\left\lvert x}\right\rvert<2\varepsilon} for all . Thus, we have, using the triangle inequality and then (A.7),
[TABLE]
for large enough such that (so independent of ). Finally, using the assumption of the lemma with the function and recalling that , it concludes the proof. ∎
Appendix B Proofs of the preliminary results concerning random walk
In this section, we prove the results stated in Section 2. Subsection B.1 is devoted to the proof of Proposition 2.6 and Corollary 2.7, and Subsection B.2 to the proof of Lemma 2.4. Subsections B.3 and B.4 contain preliminary results for the proof of Proposition 2.8 in Subsection B.5.
B.1 Convergence towards the 3-dimensional Bessel process and the Brownian meander
We first recall a known invariance principle for the random walk conditioned to stay nonnegative for all time. For all , and , we set
[TABLE]
It defines a probability measure , that is called the law of the random walk started at and conditioned to stay nonnegative for all time. Then we have the following invariance principle, by Theorem 1.1 of Caravenna and Chaumont [15]: for any and such that as and for any ,
[TABLE]
where denotes the 3-dimensional Bessel process on .
Proposition 2.6 follows from (B.2) and from the following link between the 3-dimensional Bessel process and the Brownian meander (see Imhof [29]): for all , we have
[TABLE]
Proof of Proposition 2.6.
We can assume that is nonnegative. For , we consider continuous such that . On the one hand, we have
[TABLE]
uniformly in , using (2.6). On the other hand, we have
[TABLE]
where we set, for , . Using (2.1) and , we have \limsup_{n\to\infty}\sup_{z\in\mathbb{R}_{+}}\mathopen{}\mathclose{{}\left\lvert h_{n}(z)-h(z)}\right\rvert=0, with . Thus, we get, uniformly in because and do not depend on ,
[TABLE]
Moreover, using (B.2), (B.3) and (2.5), we have
[TABLE]
uniformly in . Coming back to (B.4) and (B.5) and using that the density of is , we showed that
[TABLE]
which tends to 0 as , so it concludes the proof. ∎
Proof of Corollary 2.7.
By Lemma A.2 of Madaule [35] (that holds under the assumption of this corollary, see Remark 3.1), it exists such that for all large enough and ,
[TABLE]
We consider some and fix large enough such that c_{27}(L,C)\mathrm{e}^{-CK/2}\leq\varepsilon\theta R(L)/\mathopen{}\mathclose{{}\left\lVert F}\right\rVert and also \mathbb{E}[\mathrm{e}^{C\sigma\mathcal{M}(1)}\mathbbm{1}_{\mathcal{M}(1)\geq K}]\leq\varepsilon/\mathopen{}\mathclose{{}\left\lVert F}\right\rVert. Considering a continuous function such that and using the triangle inequality, we get
[TABLE]
Then, note that in Proposition 2.6, we can replace by : indeed, it works when and we extend to by Lemma A.1. Thus, applying Proposition 2.6 with the function which belongs to we get that the right-hand of (B.7) tends to as and it concludes the proof of Corollary 2.7. ∎
B.2 Lower envelope for the random walk above two barriers
We prove here Lemma 2.4. Firstly, by Equation (A.9) of Shi [40], we have: it exists such that, for all , and ,
[TABLE]
Secondly, it exists such that for all , , and ,
[TABLE]
Indeed, the left-hand side of (B.9) is equal to
[TABLE]
by using (B.8), and then we get (B.9) by applying (2.4).
Proof of Lemma 2.4.
Recall that, for , and , we have
[TABLE]
Moreover, for , we set
[TABLE]
and write simply when the parameters are obvious. For any and , we want to prove that for large enough, for any , , and , . For this, it is sufficient to prove that
[TABLE]
Indeed, we have and, setting ,
[TABLE]
where is the analogue of for the random walk . Then, using (LABEL:za) for and for , we get the wanted bound for .
We now prove (LABEL:za). We first have , for some that will be chosen afterwards. We take , so that for , and thus . We set and get
[TABLE]
where, for , we set
[TABLE]
We first consider the case and the case will be treated after. Applying the Markov property at time and noting that for , , we have
[TABLE]
If , we can use (2.8) to bound the probability in (B.11) (because ), but, if , we use instead (B.9). Thus, the probability in (B.11) is smaller than
[TABLE]
Setting , we get
[TABLE]
Cutting the interval in pieces of length 1, the expectation in (B.12) is smaller than
[TABLE]
by noting that is independent of and applying (2.7). Recalling that and coming back to (B.12), we get
[TABLE]
Moreover, we have
[TABLE]
using that has the same law than . We now deal with the case : we have
[TABLE]
by using first (2.8) and then (2.4). Combining this with (B.13) and (B.14), we get
[TABLE]
where we used that is an increasing sequence and the constants depend only on . Since furthermore , we have . Thus, we can choose such that for all , . Moreover, we can choose such that . This concludes the proof of (LABEL:za) and, therefore, of the lemma. ∎
B.3 Local limit theorems
We first recall the classical Stone’s [41] local limit theorem: letting be any real number if is nonlattice and be the span of the lattice if is lattice, we have
[TABLE]
as , uniformly in . Thus we have the following uniform bound: it exists such that, for all and ,
[TABLE]
Now, we state a local limit theorem for the random walk staying above a barrier, in the case where the starting point is at distance of order from the barrier and the endpoint at distance .
Lemma B.1**.**
Let be a sequence of positive numbers such that as . We set .
- (i)
If the law of is nonlattice, then, for all ,
[TABLE]
*as , uniformly in and in . * 2. (ii)
If the law of is -lattice, then,
[TABLE]
as , uniformly in and .
Proof.
First note that, for each , by Propositions 11, 18 and 24 of Doney [26], both estimates of Lemma B.1 holds uniformly in and 555Doney states his results in terms of the renewal function for the first weak increasing ladder height process of and of . Our formulation follows from Remark 4.6 of Caravenna and Chaumont [16] (although they work only in the lattice and absolutely continuous cases, this remark does not rely on these assumptions).. Noting also that tends to [math] at [math] and at infinity, it is sufficient to prove that, for each and , there exist and large enough such that \mathbb{P}\mathopen{}\mathclose{{}\left(S_{n}\in[u-b,u-b+h),\underline{S}_{n}\geq-b}\right)\leq\varepsilon R^{-}(u)/n for all and . Reversing time, we can equivalently prove that \mathbb{P}\mathopen{}\mathclose{{}\left(S_{n}\in[b-u,b-u+h),\underline{S}_{n}\geq-u}\right)\leq\varepsilon R(u)/n.
For , using (2.7) and (2.2), we get \mathbb{P}\mathopen{}\mathclose{{}\left(S_{n}\in[b-u,b-u+h),\underline{S}_{n}\geq-u}\right)\leq c_{37}R(u)(1+D^{-1}\sqrt{n})/n^{3/2}. For and large enough and independent of , this is smaller than .
For , we cancel the lower barrier between times and so that we get
[TABLE]
where we set for , . Using (B.15), we have, uniformly in and ,
[TABLE]
where , obtained by taking the supremum on , is continuous and converges simply to 0 as . Thus, using (2.6) and (2.4), we get that (B.17) is smaller than
[TABLE]
uniformly in and . By the dominated convergence theorem, the integral in (B.18) tends to 0 as and, thus, (B.17) is smaller than for and large enough and independent of . ∎
B.4 Convergence towards the Bessel bridge
In this subsection, we are going to prove the following generalization of Lemma 2.4 of Chen, Madaule and Mallein [22]. It proves that conditioned by the event of Lemma B.1 the trajectory converges to the Bessel bridge. This is a first step in the proof of Proposition 2.8.
Lemma B.2**.**
Let be a sequence of positive numbers such that as . We set and we denote by the 3-dimensional Bessel bridge of length from to [math].
- (i)
If the law of is nonlattice, then, for all and ,
[TABLE]
as , uniformly in and in . 2. (ii)
If the law of is -lattice, then, for all ,
[TABLE]
as , uniformly in and .
Proof.
We will treat only the nonlattice case, because the proof in the lattice case is exactly the same, with instead of . Moreover, since is bounded, as and Lemma B.1 deals with the case , it is sufficient to show that, for each , the estimate holds uniformly in instead of .
We first assume that , for some . Thus, is -measurable with and we have
[TABLE]
where we set, for ,
[TABLE]
uniformly in and , using Lemma B.1. Therefore, (B.19) is equal to
[TABLE]
uniformly in and . Thus, it is now sufficient to prove that
[TABLE]
uniformly in . On the one hand, by Equation (2.30) of Chen, Madaule and Mallein [22], we have
[TABLE]
On the other hand, recalling (B.1), we get
[TABLE]
where we set, for ,
[TABLE]
Since as and using (2.1), it is clear that
[TABLE]
so we get, combining (B.21), (B.22) and (B.23),
[TABLE]
which is equal to 0 by applying (B.2). It proves (B.20) and so it concludes the case where for some .
We now want to extend the result to the case . For and , we define by and , so that satisfies the assumption of the particular case that is already proved. Thus, it is now sufficient to show that, for each , it exists such that
[TABLE]
We first prove (B.25): we have
[TABLE]
Since the function is nondecreasing and , the left-hand side of (B.25) is smaller than and, thus, it tends to 0 as by dominated convergence, because is bounded. Now, we prove (B.24): using that and reversing time, we get that the expectation in (B.24) is smaller than
[TABLE]
Conditioning with respect to and applying (B.16), we get that (B.26) is smaller than
[TABLE]
uniformly in et , by using Lemma 2.6 to get the last equality. The expectation in the right-hand side of (B.27) does not depend on , and and tends to 0 as by dominated convergence, so it shows (B.24) and concludes the proof. ∎
B.5 Convergence towards the Brownian excursion
We prove here Proposition 2.8, in a similar way as Lemma 2.5 of Chen, Madaule and Mallein [22], but directly with a first barrier that can be different of 0. Following [22], we fix some and, for , and , we set
[TABLE]
Then, by [22, Lemma 2.3], we have, for each and ,
[TABLE]
where is the normalized Brownian excursion, is the Brownian meander of length and is a family of Bessel bridges of length from to [math], independent of .
Proof of Proposition 2.8.
Using Lemma A.2, it is sufficient to consider the case where for some and . The proof will be only treated in the lattice case: in the nonlattice case, the reasoning is exactly the same with instead of . We set and take the conditional expectation according to to get that
[TABLE]
where we set , for each such that . Then, since is uniformly continuous, using Lemma B.2 combined with Lemma A.3666Here the limit measure is the law of that depends on the parameter . Thus, using Remark A.4, we should prove that as uniformly in , but it is obviously false. However, since as , it is sufficient to have, for each , as uniformly in and this is clearly true., we get that is equal to
[TABLE]
uniformly in , and such that (note that if , the fact that the Bessel bridge is not well-defined is not a problem, because so the first term is zero). Since the last expectation is equal to by scaling properties of the Bessel bridge, (B.29) is equal to
[TABLE]
where is independent of . Then, since the function is continuous, using Lemma 2.6 combined with Lemma A.3, we get that (LABEL:bj) is equal to
[TABLE]
where is the Brownian meander of length , independent of . Finally, recalling the definition of and using (B.28), it concludes the proof of Proposition 2.8. ∎
Appendix C Trajectories in the weak disorder regime
We prove here (1.16). We consider some and such that . Using also (1.2), (1.3) and the convexity of , we get that is finite and nonincreasing on and positive on . Moreover, we can assume that and that . Then, we have and, by Theorem 1 of Biggins [9], we get that in . The proof follows the lines of the proof of part (iv) of Theorem 1.2, without having to introduce the barrier at .
By Lemma A.1, we can reduce the proof to the case . Since is positive -a.s., it is sufficient to prove that
[TABLE]
in -probability, where for . We consider a sequence of integers such that and, for each , . Then, using (4.2) and setting , we get
[TABLE]
by Lemma A.5. Thus, in order to prove (C.1), it is now sufficient to prove that U_{n}(F_{n})\to\widetilde{W}_{\infty,\beta}\mathbb{E}\mathopen{}\mathclose{{}\left[F(B)}\right] in -probability.
First, we prove that \zeta_{n}\coloneqq U_{n}(F_{n})-\mathbb{E}\mathopen{}\mathclose{{}\left[U_{n}(F_{n})\mathrel{}\middle|\mathrel{}\mathscr{F}_{k_{n}}}\right]\to 0 in -probability. We set \zeta^{\prime}_{n}\coloneqq\mathbb{E}[\mathopen{}\mathclose{{}\left\lvert\zeta_{n}}\right\rvert^{p}|\mathscr{F}_{n}] and, by (4.4), we have \mathbb{P}^{*}(\mathopen{}\mathclose{{}\left\lvert\zeta_{n}}\right\rvert\geq\varepsilon)\leq\varepsilon\mathbb{P}(S)^{-1}+\mathbb{P}^{*}(\zeta^{\prime}_{n}\geq\varepsilon^{1+p_{n}}). By the branching property at time , we have
[TABLE]
where, conditionally on , the for \mathopen{}\mathclose{{}\left\lvert x}\right\rvert=k_{n} are independent variables with the same law as defined by
[TABLE]
Since and , we get, in the same way as for (4.12),
[TABLE]
Using that and -a.s., we get -a.s. and, therefore, in -probability.
Finally, we prove that \mathbb{E}\mathopen{}\mathclose{{}\left[U_{n}(F_{n})\mathrel{}\middle|\mathrel{}\mathscr{F}_{k_{n}}}\right]\to\widetilde{W}_{\infty,\beta}\mathbb{E}\mathopen{}\mathclose{{}\left[F(B)}\right] -a.s. Using the branching property in the same way as for (C.2), we have \mathbb{E}\mathopen{}\mathclose{{}\left[U_{n}(F_{n})\mathrel{}\middle|\mathrel{}\mathscr{F}_{k_{n}}}\right]=\widetilde{W}_{k_{n},\beta}\mathbb{E}\mathopen{}\mathclose{{}\left[\Upsilon_{n}}\right], where is defined in (C.3). By (4.2) and recalling that , we have
[TABLE]
where and satisfy and . Using that converges in law towards the Brownian motion and applying Lemma A.3, we get that \mathbb{E}\mathopen{}\mathclose{{}\left[\Upsilon_{n}}\right]\to\mathbb{E}\mathopen{}\mathclose{{}\left[F(B)}\right] as . On the other hand, we have -a.s. Therefore, \mathbb{E}\mathopen{}\mathclose{{}\left[U_{n}(F_{n})\mathrel{}\middle|\mathrel{}\mathscr{F}_{k_{n}}}\right]\to\widetilde{W}_{\infty,\beta}\mathbb{E}\mathopen{}\mathclose{{}\left[F(B)}\right] -a.s. and it concludes the proof.
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