GEOREFERENCIACIÓN DE LA CANTERA 2 (TOCOGUA)

6.1 Compactness

In order to use Theorem 3.4, we must verify that our various set of paths really are subsets of_K(M). That is, we need to show that they are compact subsets ofMf, which is the content of this subsection. We concentrate on forwards paths; analogous arguments apply to backwards paths.

We require three preparatory lemmas. The first two of these embody the key features of the argument; at any given time, within bounded intervals of space, the number of ancestral lines at distinct spatial locations is finite, and there do not exist ancestral lines that move arbitrarily fast across space.

Lemma 6.1.Letn_∈N. Then, almost surely, for all (random)a, b, t_∈Rthe setEa,b,t=

[a, b]∩ {f(t) ; f ∈Pe↑

Proof. LetK_∈(0,_∞). Then, the probability that Πn∩ [4kRn,4(k+ 1)Rn]×[−K, K]×[0,Rn] =∅

is positive and does not depend onk. Consequently, the probability that there exists a sequence(xm, tm, rm)∞m=1⊆Πnsuch thatlimm→∞xm=∞andsupm|xm+1−xm| ≤4Rn, withsup_m|tm| ≤K, is zero. SinceKwas arbitrary, the result follows.

Recall that our ultimate goal is to prove Theorem 3.5, which claims convergence in distribution. In view of this, from now on we will (abuse notation slightly and) assume that the conclusions of Lemma 6.2 and 6.1 holdsurely.

The next lemma asserts that any convergent sequence of paths that becomes close, in space, to touching_∞within some bounded interval of time, must converge to a constant path at_∞.

Lemma 6.3.Letn ∈ N. Let(fm)∞m=1 ⊆ Pn↑(Dn)be a sequence of paths and suppose σ(fm)converges tov ∈[−∞,∞]. Suppose also that there exists a bounded sequence

(tm)withtm≥σ(fm)for whichfm(tm)→ ∞asm→ ∞.

Letf∞∈M be the path defined byσ(f∞) =vandf∞(s) =∞for alls∈[v,∞]. Then

fm→f∞inM.

Moreover, if instead(fm)⊆Pen↑(Dn), then under the same hypothesisfm→f∞inMf.

Proof. Fix K ∈ (0,∞), large enough that supm|tm| ≤ K, and define x(m∗, K) =

inf{fm(s) ;m≥m∗, σ(fm)≤s,|s| ≤K}.

Suppose, for a contradiction, thatx(m∗_{, K)does not tend to∞asm}∗ _{→ ∞. Then,}

there existsX∈(−∞,∞)and infinitely manym∗for whichx(m∗, K)≤X. For all such

m∗ _{we have somem}

≥m∗_and

|s_{| ≤} K such thatfm(s)≤X, and (by our hypothesis) asm∗

→ ∞ we have also fm(tm) → ∞; since fm ∈ P↑(Dn)this is a contradiction to Lemma 6.2.

So,x(m∗_{, K)}

→ ∞asm∗

→ ∞. Thus, for anyK, X _∈(_−∞,_∞)we can findm∗

∈N

such that, for allm_≥m∗_ands

≥σ(fm)such that|s| ≤K, we havefm(s)≥X. With this in hand, the stated results follow easily from (5.18) and (5.21).

Recall that, by (4.1), the pathf∞ in the statement of Lemma 6.3 is an element of

both_P↑_(D

n)andPe↑(Dn). Recall also that, in the notation of (4.1), both these sets also contain paths that are infinite extenders.

Lemma 6.4.Let⋆_{∈ {↑},_↓},_{† ∈ {}l, r_}andn_∈N. Then,_Pn⋆(Dn)andPn⋆,†(Dn)are compact

subsets ofM, also_Pe⋆

n(Dn)andPen⋆,†(Dn)are compact subsets ofMf.

Proof. As usual, it suffices to consider the case of forward paths. Since Lemma 6.3, on which the following argument relies, holds in bothM andMf, it will suffice to consider only cases inM. Moreover, the arguments required the case of_P↑,†

identical to those required for the case_Pn↑(Dn); thus, we aim to show thatPn↑(Dn)is a sequentially compact subset of (the metric space)M.

Let (fm)∞m=1 ⊆ Pn↑(Dn) be a sequence of paths. We must show that (fm) has a convergent subsequence, with limit in_P↑

n(Dn). We now split into several cases.

1. If(σ(fm))m≥1has a subsequence that converges to∞then, along this subsequence,

fmconverges to the degenerate pathf withσf =∞andf(∞) = 0.

2. If (σ(fm))m≥1 has a bounded subsequence, then consider the sequence xm = fm(σfm).

(a) If(xm)m≥1 is bounded, then sinceDn is locally finite there must be a subse- quence along which(σ(fm), xm)is eventually constant. Any given ancestral line moves to one of at most two locations in a reproduction event, thus

(fm)m≥1has a convergent subsequence; to construct the limit path we suc-

cessively follow parent points that were followed by infinitely many of our

fm.

(b) If(xm)m≥1is not bounded, then without loss of generality we pass to a subse-

quence and assume that bothxm→ ∞andσ(fm)converges. It then follows immediately from Lemma 6.3 thatfmconverges along this subsequence. 3. If(σ(fm))m≥1has a subsequence that converges to−∞, then pass to that subse-

quence and sett= supmσ(fm)<∞.

(a) If_{m;|fm(t)| ≤K}is finite for allK <∞, then essentially the same argument as in 2(b), reliant on Lemma 6.3, shows thatfmhas a convergent subsequence. (b) If, for some K < _∞the set _{m; _|fm(t)| ≤ K} is infinite, then pass to the

subsequence offmsuch that|fm(t)| ≤K. By Lemma 6.1, the set[−K, K]∩

{fm(t) ; m∈N}is finite. Hence, there is some|z| ≤Kthrough which infinitely manyfmpass. Using the same method as in 2(a), we can construct a path f : [t,∞]with f(t) =zthat is followed by infinitely manyfm. So, pass to a further subsequence and assumefm(s) =f(s)for alls≥t.

We extendf backwards in time as follows. From location(z, t), look backwards in time until the most recent reproduction event (strictly) beforetthat affected

z, sayp= (x, t′_{, r). By Lemma 6.1, the set}

{fm(t′−) ; paffectsfm, m∈N} is finite. Pick some elementz′_{of this set, and restrict tof}

mfor whichfm(t′−) = z′_{. Set f(s) = z for s}

∈ [t′_{, t) set f(s) = z. Then, look back from (z}′_{, t}′_),

and repeat (in the language of (4.1), f is an ‘infinite extender’). Thus, a subsequence of(fm)m≥1converges tof.

Since(σ(fm))m≥1must have a subsequence that converges in[−∞,∞], at least one

of the above cases occurs. This completes the proof.

6.2 Convergence of multiple left/right paths

We now extend Proposition 4.9 to larger collections of left and right-most paths. Let

N ∈N. Given a finite setD={(yi, si)∈R2;i= 1, . . . , N}of distinct points inR2and a function O : {1, . . . , N} → {l, r}, [24], Section 2.2, construct a system of left-right coalescing Brownian motions started from the points ofD. (Recall that two left-most paths coalesce on meeting, as do two right-most paths, and that (3.4) describes the interaction between left-most and right-most paths.) We write

P↑(D, O) =_{B↑,(yi,si)_{;i= 1, . . . , N}

converges weakly in M to _P (D, O) and _Pen(D , O) converges weakly in Mf to

P↑_{(D, O).}

Proof. The argument is essentially identical to that of the proof of Proposition 5.2 of [24]. The construction of_P↑(D, O)in Section 2.2 of [24] is an inductive construction that views_P↑_{(D, O)as made up of several independent pieces consisting of segments}

of either single left-most paths, single right-most paths or a pair of left/right paths. The same inductive construction breaks down_P↑

n(D(n), O)into corresponding pieces. The stopping times used in this construction are continuous functionals onMN _with respect to the law of independent evolutions of paths within each such piece, so the first part of the lemma follows from Proposition 4.9 and Lemma 5.10. Similarly,_Pen↑(D(n), O) converges weakly to_Pe↑_{(D, O).}

6.3 Proof of Theorem 3.5

We complete the proof of Theorem 3.5 in three steps. Recall that the Brownian net is denoted by_N, and recall the function_Hcrossdefined in Section 3.2.

Lemma 6.6.Asn→ ∞, we have that

Hcross e P↑,l n (Dn)∪Pen↑,r(Dn) → N, in distribution in_K(Mf).

Proof. We verify the conditions(A_)-(D_{)of Theorem 3.4. This theorem is applied with}

X†

n =Pen↑,†(Dn)andXˆn† = Pen↓,†(Dn), where † ∈ {l, r}. By Lemma 6.4, all these sets of paths are (almost surely) elements of_K(Mf)(after rotation by180degrees about(0,0)for the backwards paths). We defineXn=Hcross(Pen↑,l(Dn)∪Pen↑,r(Dn))and similarly forXˆn.

We now check the conditions in turn. For (A_{), the required statements about}

non-crossing paths are precisely the content of Lemma 4.4. For (B_{), the required}

convergence of multiple left/right paths to left/right Brownian motions is precisely the content of Lemma 6.5.

We now move on to(C_{). If(xn,1, xn,2)→(x1, x2)and}ˆ_{ln,ˆrn are respectively elements}

of_Pen↓,l(Dn),Pen↓,r(Dn)started at(xn,1, xn,2), then it follows by combining Lemmas 4.10 and Remark 4.16 that(ˆln,rˆn)→(ˆl,rˆ)in distribution, where(ˆl,rˆ)are a pair of left/right Brownian motions. That the first meeting time ofˆlnwithˆrnalso converges (jointly) in distribution to the first meeting time ofˆlwithrˆfollows from Remark 4.8.

It remains to verify(D_{). By Lemma 4.4, left-most fowards and left-most backwards}

paths cannot cross, and similarly for right-most paths. Therefore, a path ofXn that enters a wedgeW ofXˆnfrom the outside must enter through the southern-most point of the wedge (i.e. precisely where the two pathsrˆandˆldefiningW meet, in Figure 2). Fix a wedgeW ofXˆn and denote this event byEW.

For the eventEW to occur, some reproduction event must have a potential parent situated at the spatial location of the southern-most point ofW. The distribution of the

spatial location of a pre-parent (in a reproduction event occurring at given time) has no atoms, and thus the distribution of the spatial location of the meeting point ofrˆandˆl

also has no atoms; hence almost surelyEW does not occur.

The set Xˆn contains countably many paths and thus has at most countably many wedges. Hence, almost surely the eventEW does not occur for any wedgeW ofXˆn. Without loss of generality, we may assume this does not occursurely, so as(D_)holds. Lemma 6.7.Asn_{→ ∞}, we have that_Pe↑

n(Dn)tends in distribution toN

Proof. If a left-most and a right-most path of_Pen↑(Dn)cross, then the point at which they cross much be within one of the setsB_{Υ(p)(p)(defined in Lemma 4.1) associated to a}

selective eventp∈Πn. If an interpolated arrow finishes atp, then by definition there are interpolated arrows ending at both potential parents ofp. Consequently, there is a one to one correspondence,f _7→f′ _{between pathsf}

∈ Hcross e P↑,l n (Dn)∪Pen↑,r(Dn) and pathsf′ ∈Pe↑₍ Dn)such that |f(t)₋f′(t)_{| ≤}2_Rn (6.2) for allt_≥σf =σf′, andf(σ_f) =f′(σ_f′).

By Lemma 6.4, we have that_Pen↑(Dn)is an element ofK(M). Combined with (6.2) this means that_Pen↑(Dn)has the same limit (in distribution) asHcross(Pen↑,l(Dn)∪Pen↑,r(Dn))as n_{→ ∞}. Thus, from Lemma 6.6,_Pe↑

n(Dn)tends in distribution toN.

To finish, we must upgrade the result of Lemma 6.7 from_K(Mf)to_K(M), and use càdlàg paths in place of interpolated paths.

By Lemma 6.4 we have that _Pe↑

n(Dn) ∈ K(M)for all n. Combining this fact with Lemma 4.2, and noting that there is a one to one correspondence between paths and interpolated paths, we obtain from Lemma 6.7 that also_Pn↑(Dn)→ N in distribution in

K(M). This completes the proof of Theorem 3.5.