Investigación de Mercado

Define the shift function σ : Ω→Ω by

(σ(ω))v :=ωγ1tv. (4.4.1)

Informally, σ shifts the values of random variables at nodes γ1tv inT(γ1) back to

nodev; these values populate the whole Ulam tree; values of variables not inT(γ1)

are discarded; this is a tree-indexed version of the shift for an ordinary Markov chain. The n-fold shift σn _{shifts n steps down the backbone.}

The main purpose of this section will be to understand the shift functionθ, and thereby understand the behavior of the pivots. While this section contains many intermediate results—a fair number of which may be of independent interest— only a handful will be directly of use in the proof of Theorem 4.1.1: the pair of Propositions 4.4.4(i) and 4.4.5 demonstrating that shifting down the backbone is the same as conditioning on the pivot being at most a certain value (this is step 2 in the outline in the introduction); also of use will be Theorem 4.4.8, which accomplishes step 3 of the outline by showing that β_n∗−pc approaches 0 rapidly.

Before showing the necessary Markov properties, a fair bit of notation is nec- essary. We begin with the definition of the dual pivots β_n∗; these variables will be central to the proof of Theorem 4.1.1, primarily due to their appearance in Propo- sition 4.4.4.

from v, moved to the root. Let T∗(v) denote the rooted subtree induced on all vertices w /∈T(v), and let β_v,w∗ represent the pivot of the vertex w on T∗(v), that is, the least xsuch that wis connected to infinity by a path with weights ≤xthat avoids going through v. The dual pivot β_v∗ is defined to be min

w<v β

∗

v,w. In keeping

with the notation for pivots, we denote β_n∗ :=β_γ∗_n.

Definition 4.4.2. We define the following σ-fields.

(i) For fixed v 6= 0, define Cv to be the σ-field generated by degw and Uw for all w 6= v in T(v) along with deg_v. Define B∗

v to be the σ-field generated by all

the other data: Uw and degw for all w∈ T

∗_{(v), along withU}

v.

(ii) For n ≥ 1, let B∗

n denote the σ-field containing γn and all sets of the form

{γn=v} ∩B where B ∈ B∗v. Informally, B

∗

n is generated by γn and Bγ∗n.

(iii) Let Cn be the σ-field generated by σnω; in other words it contains deg(γn)

and all pairs (deg_γntx, Uγntx). It is not important, but this definition does not allowCn to know the identity ofγn.

It is elementary that{B∗

n}is a filtration, thatB∗n∩Cnis trivial, and thatBn∗∨Cn =

F.

Definition 4.4.3. We define the following conditioned measures.

(i) For x ∈ (pc,1), let Qx := (P|β0 ≤ x) denote the conditional law given

0↔x ∞, in other words, Qx[A] = θ_θA₍(_xx₎) where θA(x) := P[A∩ {β0 ≤x}].

(ii) LetL denote the law ofβ0, the pivot at the root. By [Dur10, Theorem 5.1.9],

one may define regular conditional distributionsPx := (P|β0 =x). These sat-

isfyPx[β0 =x] = 1 and

R

Px dL(x) = P. Also,Qy = (1/θ(y))

R

PxdL|[0,y](x).

A common null set for all the conditioned measures is the set where either the invasion ray is not well defined or β(v) = β_v∗ for somev. Equalities below are always interpreted as holding modulo this null set.

Proposition 4.4.4 (Markov property for dual pivots).

(i) For any A∈ F,

P[σnω ∈A| B∗_n] =Qβ∗

n[A].

(ii) More generally, if 0< y ≤1 then for any A∈ F,

Qy[σnω ∈A| B∗n] =Qβ∗

n∧y[A].

(iii) Under P, the sequence {β_n∗} is a time homogeneous Markov chain adapted to B∗

n with transition kernel p(x, S) = Qx[β1∗∧x∈S] and initial distribution δ1.

It is immediate that Qx P for all x. The following more quantitative state-

ment will be useful, especially when used in conjunction with Proposition 4.4.4(i).

Proposition 4.4.5. Let q > 1 and suppose that the offspring distribution has a finite q-moment. Then there exists a constant Cq such that for all A ∈ T and for

all δ >0 and all x∈(pc,1),

Proof. On T, the density of Qx with respect to P is given by dQx

dP (T) = θT(x)

θ(x) .

Combining Proposition 4.3.1, which impliesθ(x)∼K(x−pc), with Proposition 4.3.4,

which shows R

θT(pc+ε)qdGW(T)≤ cqεq provided pc+ is bounded away from 1,

we see that Z dQx dP (T) q dGW(T)≤c0_q

for some constant c0_q and all x ∈ (pc,1). Applying H¨older’s inequality with 1/p+

1/q= 1 then gives Qx[A] = Z 1A dQx dP dP≤ Z 1AdP 1/pZ dQx dP q dP 1/q ≤Cqδ1−1/q when Cq = (c0q)1/q.

The measuresPx are in some sense more difficult to compute with thanQx be-

cause of the conditioning on measure zero sets. Relations such as the Markov prop- erty, however, are conceptually somewhat simpler. The following statement of the Markov property generalizes what was proved in [AGdHS08, Theorem 1.2 and Propo- sition 3.1], withB+

n representing theσ-field generated byB

∗

ntogether withβn. Note,

however, that the only role Propositions 4.4.6 and 4.4.7 play in the proof of Theo- rem 4.1.1 is that they are utilized to prove Theorem 4.4.8. The proposition below is also of independent interest, and will be crucial for studying the forward maximal weight process in Section 4.6.

Proposition 4.4.6 (Markov property for pivots). For any A∈ F,

Pσnω ∈A| B_n+

=Pβn[A]

on (Ω,F,Px).

In fact, the pair {βn, βn∗} is Markov:

Proposition 4.4.7. The sequence {βn, βn∗} is a time-homogeneous Markov chain

adapted to {B+

n} with initial distribution L ×δ1. Further, if we defineh∗n :=βn∗−pc

and f(x) :=φ0(1−(pc+x)θ(pc+x)), then{hn, h∗n} has transition probabilities given

by p({a, b},·) = νa×ν˜a,b where dνa dx = f(a)θ0(pc+x) θ0_(p c+a) 1x<a+Caδa and dν˜a,b dx =− f0(x) f(a)1a<x<b+ ˜Ca,bδb with Ca =f(a)(pc+a) and C˜a,b = _ff₍(_ab)₎.

The decay rate of β_n∗−pc = h∗n follows from analyzing this Markov chain; the

following Theorem accomplishes Step 3 of the outline.

Theorem 4.4.8. There exists C > 0 such that for any t∈(1/2,1), P[h∗_n> n−t_{] is} O(e−Cn1−t