Occupation densities of Ensembles of Branching Random Walks

We study the limiting occupation density process for a large number of critical and driftless branching random walks. We show that the rescaled occupation densities of $\lfloor sN\rfloor$ branching random walks, viewed as a function-valued, increasing process $\{g_{s}^{N}\}_{s\ge 0}$, converges weakly to a pure jump process in the Skorohod space $\mathbb D([0, +\infty), \mathcal C_{0}(\mathbb R))$, as $N\to\infty$. Moreover, the jumps of the limiting process consist of i.i.d. copies of an Integrated super-Brownian Excursion (ISE) density, rescaled and weighted by the jump sizes in a real-valued stable-1/2 subordinator.


Introduction
In a branching random walk on the integers, individuals live for one generation, reproduce as in a Galton-Watson process, giving rise to offspring which then independently jump according to the law of a random walk. A branching random walk is said to be critical if the offspring distribution ν has mean 1, and driftless if the jump distribution F has mean 0 and finite variance. We will assume throughout that (i) the offspring distribution ν has mean one (so that the Galton-Watson process is critical) and finite, positive variance σ 2 ν ; and (ii) the step distribution F for the random walk has span one, mean zero and finite, positive variance σ 2 F . (Thus, the spatial locations of individuals will always be points of the integers Z.) To any branching random walk can be associated a randomly labeled Galton-Watson tree T , where the Galton-Watson tree describes the lineage of the individuals and the label of each vertex marks the spatial location of the corresponding individual. This labeled tree is generated as follows.
(i) Let T be the genealogical tree of a Galton-Watson process with a single ancestral individual and offspring distribution ν, with the root node ρ representing this ancestral individual. Since ν has mean 1, the tree T is finite with probability one. Given the labeled tree T associated with the branching random walk, the occupation measures can be recovered as follows. For any time n ∈ Z + and any site x ∈ Z, the number Z n (x) of individuals at location x at time n is the number of vertices v ∈ T at height n (i.e., at distance n X(j; T i ) (1.1) to be the total number of vertices in the first m trees with label j ∈ Z, and defineX m (x) to be the linear interpolation to x ∈ R. Observe that X m can be viewed as the occupation measure of the branching random walk initiated by the m ancestral particles that engender the branching random walks Z 1 , Z 2 , · · · Z m . Clearly, the functionX m (x) is an element of C 0 (R). Finally, define (1.2) Theorem 1.1. As N → ∞, the rescaled density processes {g N s } s≥0 converge weakly in the Skorohod space D := D([0, +∞), C 0 (R)) to a process {g s } s≥0 . Moreover, the limiting process satisfies where {Y s (t, x), x ∈ R} t≥0 is the density process for a super-Brownian motion {Y s t } t≥0 with variance parameters (σ 2 ν , σ 2 F ), started from the initial measure Y s 0 = sδ 0 . Remark 1.2. Super-Brownian motion {Y t } t≥0 is, by definition (see for instance [7], ch. 1) a measure-valued stochastic process that can be constructed as a weak limit of rescaled counting measures associated with branching random walks. In one dimension, for each t > 0, the random measure Y t is absolutely continuous relative to the Lebesgue measure, and the Radon-Nikodym derivative Y (t, x) is jointly continuous in (t, x) [9]. Super-Brownian motion is singular in higher dimensions and thus the representation (1.3) does not exist in higher dimensions. When the dependence on the variance parameters σ 2 ν and σ 2 F must be emphasized, we do so by adding them as extra superscripts, i.e., When σ 2 ν = σ 2 F = 1, the measure-valued process associated with Y s,1,1 (t, x) is a standard super-Brownian motion. The density processes for different variance parameters obey a simple scaling relation: Thus, we can rewrite (1.3) in terms of the density function of standard super-Brownian motion as follows: For any fixed s > 0 and each integer N ≥ 1, the random function g N s (·) is the (rescaled) occupation density of the branching random walk gotten by amalgamating the branching random walks generated by the first sN initial particles. Because this sequence of branching random walks is governed by the fundamental convergence theorem of Watanabe [14] and its extension to densities by Lalley [10], the limiting random function g s (·) must (after the appropriate scaling) be the integrated occupation density of the super-Brownian motion with initial measure sδ 0 . This explains relation (1.3). But even for fixed s > 0 the weak convergence x) dt does not follow directly from the local convergence of the density process proved in [10, Theorem 2], for two reasons. First, the local convergence result in [10] requires that the initial densities must, after Feller-Watanabe rescaling, converge to a density function Y s (0, ·) ∈ C 0 (R). In Theorem 1.1, however, the limiting initial density Y s 0 = sδ 0 is not absolutely continuous with respect to the Lebesgue measure. Second, even if the local convergence could be shown to remain valid under the initial condition Y s 0 = sδ 0 , the indefinite integral operator on C([0, +∞), C 0 (R)) is not bounded, and so it would not follow, at least without further argument, that the integral of the discrete densities would converge to that of the super-Brownian motion density over the time interval [0, +∞). For each N ≥ 1, the process {g N s } s≥0 is nondecreasing 1 in s (relative to the natural partial ordering on C 0 (R)) and has stationary, independent increments. Therefore, the limiting process {g s } s≥0 must also be nondecreasing, with stationary, independent increments. We prove the following properties of the limiting process.
Theorem 1.4. The limiting function-valued process {g s } s≥0 has the following properties.
(i ) It obeys the scaling relation g s (x) The real-valued process {I s } s≥0 , where I s := g s (0) is the occupation density at zero, is a stable subordinator with exponent α = 2/3. (iii ) The real-valued process {θ s } s≥0 , where θ s := ∞ −∞ g s (x)dx is the total rescaled occupation density, is a stable subordinator with exponent α = 1/2. (iv ) {g s } s≥0 is a pure-jump subordinator in the Banach lattice C 0 (R) (see Definition 3.1).
As we will show, the limiting process g s in Theorem 1.1 has jump discontinuities, that is, there are times t > 0 such that the function g t − g t− is non-zero (and hence positive) over some interval. The jumps that occur before time t = 1 can be ordered by total area (g t − g t− )(x) dx, i.e., the jump size in the stable-1/2 process {θ s } s≥0 . Denote these jumps (viewed as elements of C 0 (R)) by (In Section 3, we will see that no two jump sizes can be the same.) For each N , the Galton-Watson trees T i with i ≤ N can also be ordered by their size (i.e., the number of vertices). The corresponding jumps in the (rescaled) occupation density g N s will be denoted by where the weak convergence is relative to the m-fold product topology on C 0 (R).
Proof. This is an immediate consequence of Theorem 1.1, because weak convergence in the Skorohod topology on D implies weak convergence of the ordered jump discontinuities.
Consequently, the process g s can be written as The remainder of the paper is organized as follows. Section 2 is devoted to the proof of Theorem 1.1, where we make use of Aldous' stopping time criterion [1] to show the tightness of the sequence of processes {g N s } s≥0 . In Section 3, we prove the properties of the limiting process g s enumerated in Theorem 1.4 and the Lévy-Khintchine representation (1.6).
2. Proof of Theorem 1.1 2.1. Preliminaries on the Skorohod space D. Let (S, d) be a separable and complete metric space, and let D(S) · · = D([0, +∞), (S, d)) be the spaces of all S-valued càdlàg functions f with domain [0, +∞), i.e., f ∈ D(S) is right-continuous and has left limits. The space D(S) is metrizable, and under the usual Skorohod metric, the space D(S) is complete and separable. We refer to [4,Section 13] for details on the Skorohod topology. Here, we quote the following theorem, which gives a sufficient condition for the weak convergence in D(S).
) be a sequence of S-valued stochastic processes. A sufficient condition for tightness of the sequence X N s s≥0 is that the following two conditions hold: , and Condition 2 • . For any L > 0, any sequence of constant δ N ↓ 0, and any sequence of stopping times Proof of Tightness. In this section, we prove that the sequence {g N s } N ∈N is tight in D(C 0 (R)) by verifying Condition 1 • and Condition 2 • .
To verify Condition 1 • , we will show that for any s ≥ 0 fixed and any > 0, we can find a compact subset K ⊂ C 0 (R) such that P(g N s ∈ K) > 1 − for all N large. Let ζ sN be the extinction time of the branching random walk gotten by amalgamating the branching random walks (Z 1,n ) n≥0 , (Z 2,n ) n≥0 , · · · , (Z sN ,n ) n≥0 , that is, ζ sN is the maximum of the extinction times of the branching random walks initiated by the first sN ancestral particles. By a fundamental theorem of Kolmogorov, consequently, for every > 0, there exists H = H s, > 0 such that Therefore, it suffices to prove that there is a compact set K ⊂ C 0 (R) such that for all N large, To establish inequality (2.4) we will use Kolmogorov-Čentsov criterion (see, e.g., [8, Chapter 2, Problem 4.11]). It suffices to prove that

5a)
and that for some m ≥ 3, there exists C = C(s, m, H) > 0 such that for all x, y ∈ R and for all N sufficiently large, Note that the requirement m ≥ 3 in (2.5b) ensures that the exponent 2m/5 is larger than 1, as is needed for the Kolmogorov-Čentsov criterion. We will rely on the following estimates of [10] to compute these bounds.
Proposition 2.4. [10, Proposition 5] Let Z n (x) be the number of particles at location x ∈ Z and time n ∈ Z + in a branching random walk, started from a single particle at 0 ∈ Z, with offspring distribution ν and step distribution F . For each m ∈ N, there is constant C m such that for all x, y ∈ Z and all n ≥ 1, Proof. Suppose first m is even. Then by a trivial extension of Hölder's inequality (see, e.g., [13, pp. 4]) and Proposition 2.4, The case when m is odd is similar.
Proof of Condition 1 • . The bound in (2.5a) is easy to check using (2.6) with m = 1 and x = 0 and the linearity of expectation. In particular, For (2.5b), first of all, by triangle inequality and the assumption thatX N (x) is defined by linear interpolation, we need only consider x, y ∈ Z/ √ N in (2.5b).
Let Z i,n (x) be the number of particles at site x ∈ Z in generation n of the i-th ancestral particle. The left side of (2.5b) is clearly bounded by We expand the product under the expectation sign and write it as a sum of expectations: When the product inside the expectations is expanded, each term is a product of 2m differences of occupation counts in one of the branching random walks Z i j in some generation n j . Observe that repetitions of the indices i j and n j are allowed. Note that Proposition 2.4 applies only for generation n ≥ 1, whereas n 1 , . . . , n 2m in (2.9) run from generation 0. However, since originally all particles are placed at the origin, we lose nothing by summing from 1 to N H as long as xy = 0. The case when xy = 0 will be treated separately at the end. Suppose x = 0, y = 0. If i j 1 = i j 2 are indices of two distinct ancestral individuals, then the dif- Let r be the number of distinct i j 's inside the expectation in (2.9); then (2.9) can be written as For a particular term with r distinct ancestors i 1 , . . . , i r in which i j occurs m j times (j = 1, 2, . . . , r), the expectation can be factored as a product of r expectations, where each expectation is an expectation of the differences involving the offspring of only one ancestor at time 0. Thus, we always have r j=1 m j = 2m. For each bracketed factor in (2.10), for each ancestor i j , the summation is over all possible choices of the generations n j 1 , . . . , n j m j ; this can be bounded using Corollary 2.5 above. It follows that  Finally, we must deal with the case when xy = 0. If x = y = 0, then both sides of (2.5b) are zero. If x = 0 and y = 0, then, because all initial sN particles are placed at zero, we can write the left side of (2.5b) as .
It is not difficult to see that for large N , the first term dominates, because this term can be handled exactly as in the case xy = 0. This proves that Condition 1 • holds.
Proof of Condition 2 • . For each N the process g N s is piecewise constant in s, with jumps only at times s that are integer multiples of 1/N . Consequently, in verifying Condition 2 we may restrict attention to stopping times τ N such that N τ N is an integer between 0 and N L. It is obvious from its definition that the discrete-time process g N s , with s = 0, 1/N, 2/N, · · · , is non-decreasing and has stationary, independent increments; therefore, for any stopping time τ N and any constant δ > 0, the increment g N τ N +δ − g N τ N has the same distribution as g N δ . Therefore, to prove Condition 2 • it is enough to show that for any > 0 there exists δ > 0 such that for all N sufficiently large, But we have already proved, in Condition 1 • , that the sequence of C 0 (R)−valued processes g N 1 is tight, so there exists K = K < ∞ so large that for all m ∈ N, By choosing δ > 0 so small that /δ 3/2 > K, we obtain (2.11) for N sufficiently large.

Uniqueness of the Limit Process.
Since {g N s } s≥0 has stationary and independent increments, any weak limit will also have these properties. Therefore, to prove the uniqueness of the limit process it suffices to show that for any fixed time s > 0 there is only one possible limit for the sequence {g N s } N ∈N . For any N ∈ N, the random function g N s (·) is defined by rescaling the occupation measure X sN of the branching random walk initiated by the first sN ancestral individuals (cf. equation (1.2)). The occupation measure X sN is defined by (1.1), which can be rewritten as where Z i,n (j) is the occupation counts at location j of individuals in the n-th generation of the i-th labeled tree. As discussed in Remark 1.3, to avoid invoking an indefinite integral operator in the weak limit, we consider the truncated occupation counts and the associated occupation density up to the N H -th generation for some H > 0 fixed. Define where, as earlier, the bar denotes the function obtained by linear interpolation. The same calculations as in the proof of Condition 1 • show that for any fixed H > 0 and s > 0 the sequence g N s,H is tight.
Watanabe's convergence theorem states that for any s > 0, the rescaled measure-valued process converges weakly to the super-Brownian motion, i.e., Viewing the measure-valued process Y s,N t as nonnegative continuous functions over R, we definē Y s,N t ∈ C 0 (R) by settingȲ and then doing a linear interpolation. The above convergence implies that the weak convergence of the rescaled total occupation measure of the first N H generations: Notice that the left side is indeed X sN ,H ( , which has densities g N s,H . Consequently, any possible weak subsequential limit of {g N s,H } N ≥1 in the function space C 0 (R) must be a density for the occupation measure of the super-Brownian motion, that is, as N → ∞ But by inequality (2.3), for any > 0 there exists H = H < ∞ so large that for any N ,

Properties of the Limiting Process
In this section, we prove properties of the limiting process {g s } s≥0 (Theorem 1.4) and characterize it using a Poisson point process (Theorem 1.6). In order to make sense of the notion of a "subordinator" on the function space C 0 (R), we first briefly review the definition of a Banach lattice.
Definition 3.1. A Banach lattice is a triple (E, · , ≤) such that (a). (E, · ) is a Banach space with norm · ; (b). (E, ≤) is an ordered vector space with the partial ordering ≤; (c). under ≤, any pair x, y ∈ E has a least upper bound denoted by x ∨ y and a greatest lower bound denoted by x ∧ y (this is the "lattice" property); and (d). Set |x| · · = x ∨ (−x). Then |x| ≤ |y| implies x ≤ y , ∀x, y ∈ E (i.e., · is "a lattice norm"). Example 1. The Banach space (C 0 (R), · ∞ ) has a natural partial ordering, defined by The triple (C 0 (R), · ∞ , ≤) clearly satisfies (a) and (b) in Definition 3.1. The least upper bound and the greatest lower bound are defined pointwise: .

Condition (d) can be verified easily.
Definition 3.2. Let (E, · , ≤) be a Banach lattice. An E-valued stochastic process {X t } t≥0 is a subordinator if {X t } t≥0 is a Lévy process (that is, {X t } t≥0 has stationary, independent increments) and with probability one, for all t ≥ s ≥ 0, A subordinator {X t } t≥0 is a pure jump process if for every t, Proof of Theorem 1.4. For (i), we have for each N ≥ 1, Taking N → ∞ gives (i). The claim that {I s } s≥0 is a stable-2/3 subordinator follows from monotonicity of g s and the scaling relation above at x = 0, which yields For (iii), recall that a version of the stable-1 2 subordinator on R is the inverse local-time process of a standard Brownian motion {B t } t≥0 Now consider a sequence of independent critical Galton-Watson trees T i with offspring distribution ν, initiated by particles i = 1, 2, 3, . . . . Let |T i | be the size (number of vertices) of the i-th tree, and set A N · · = N i=1 |T i |, the total number of vertices in the first k trees. Then by a theorem of Le Gall [11], as N → ∞, where the last equality follows from the scaling rule of a stable-1 2 process. Next, suppose that branching random walks are built on the Galton-Watson trees T i by labelling the vertices, as described earlier. Then clearly By Theorem 1.1, {g N s } s≥0 ⇒ {g s } s≥0 in D([0, +∞), C 0 (R)). Considering the space-truncated occupation density g N s (x)1 [−B,B] (x) (i.e., truncated in space) for sufficiently large B > 0 and following the same strategy as when proving the uniqueness of the limiting process {g s } s≥0 , one would obtain where the left side is indeed Consequently, the processes {θ s } s≥0 and {σ −2 ντ s } s≥0 have the same law, and so {θ s } s≥0 is a stable-1/2 subordinator.
For (iv), we have already observed that g s has stationary, independent increments and increasing sample paths relative to the natural partial order ≤ on C 0 (R). To show that {g s } s≥0 has pure jumps, we make use of the fact that the total area process {θ s } s≥0 is a stable-1 2 subordinator and thus has pure jumps. Let J be the set of jump times of the process {θ s } s≥0 , that is, the set of all t ≥ 0 for which θ(t) − θ(t−) > 0. Defineg a process that collects the changes in {g s } s≥0 at those times when the limiting total area process {θ s } s≥0 makes jumps. Clearly, the processg s is an increasing process in C 0 (R), and sinceg s only gathers the jumps of g s , we haveg s ≤ g s for every s ≥ 0. But since the area process θ s is pure jump, g s andg s bound the same total area for every s, that is, By continuity of both g s andg s , we have g s =g s for every s, and thus the process g s is a pure jump process in C 0 (R).
Proof of Theorem 1.6. We have already proved in Theorem 1.4 that the process g s consists of pure jumps. It remains to show that the point process of jumps is a Poisson point process with intensity given by (1.5) and then the representation (1.6) would follow automatically.
Consider the point process of jumps of g s for s ≤ 1 (the case s ≤ s * , for arbitrary s * > 0, can be handled in analogous fashion). Let J 1 , J 2 , . . . be the jumps ordered by size from largest to smallest, as in Corollary 1.5. Since by Theorem 1.4, the limiting process {g s } s≥0 is a pure jump subordinator, we have where J N 1 , J N 2 , . . . are the ordered jumps in the (rescaled) occupation density processes g N s for s ≤ 1 for the branching random walk obtained by amalgamating the first N trees. Consequently, the joint distribution of the random variables |J N i | = |T N (i) |/N 2 (where T N (i) is the i-th largest tree among the first N trees) converges to the joint distribution of the sizes |J i |. In particular, the largest, second largest, etc., trees among the first N trees have sizes of order N 2 -and so as N → ∞, these will be large.
To identify the limiting distribution of the rescaled jumps, we now make use of Theorem 1.1 in [6], which states that the occupation density of a conditioned branching random walk scaled by the size of the tree converges to that of the ISE density f ISE , as the size of the tree becomes large. This implies, for each i = 1, 2, . . . , as N → ∞, and the limiting ISE densities, f ISE , . . . are i.i.d copies of f ISE . By (3.2) and (3.3), we can describe the joint distribution of J 1 , J 2 , . . . as follows: (a) let ε 1 > ε 2 > ε 3 > . . . be the ordered excursion lengths of a standard Brownian motion run until the first time t such that L 0 t = 1; (b) let f 1 , f 2 , f 3 , . . . be i.i.d. copies of the ISE density f ISE which are independent of the ε i 's; and (c) set J i (·) · · = (σ −2 ν ε i ) 3/4 γf i (γ(σ −2 ν ε i ) −1/4 · ) Since the ordered excursion lengths ε 1 > ε 2 > ε 3 > . . . have the distribution of the ordered points in a Poisson point process on R + with intensity measure dy √ 2πy 3 , the representation (1.6) follows from (3.1).