getdoc3730. 270KB Jun 04 2011 12:04:15 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y}

Vol. 14 (2009), Paper no. 66, pages 1936–1962. Journal URL

http://www.math.washington.edu/~ejpecp/

Limit theorems for vertex-reinforced jump processes on

regular trees

Andrea Collevecchio

∗ Dipartimento di Matematica applicata†

Università Ca’ Foscari –Venice, Italy.

collevec@unive.it

Abstract

Consider a vertex-reinforced jump process defined on a regular tree, where each vertex has exactlybchildren, with b≥3. We prove the strong law of large numbers and the central limit theorem for the distance of the process from the root. Notice that it is still unknown if vertex-reinforced jump process is transient on the binary tree.

Key words:Reinforced random walks; strong law of large numbers; central limit theorem. AMS 2000 Subject Classification:Primary 60K35, 60F15, 60F05.

Submitted to EJP on June 6, 2008, final version accepted July 13, 2009.

∗_{The author was supported by the DFG-Forschergruppe 718 ”Analysis and stochastics in complex physical systems”, and}

by the Italian PRIN 2007 grant 2007TKLTSR "Computational markets design and agent-based models of trading behavior". The author would like to thank Burgess Davis, Fabiola del Greco and an anonymous referee for helpful suggestions.

1

Introduction

Let_D be any graph with the property that each vertex is the end point of only a finite number of edges. Denote by Vert(_D)the set of vertices ofD. The following, together with the vertex occupied at time 0 and the set of positive numbers _{a_ν:ν ∈ Vert(_D)_}, defines a right-continuous process X=_{Xs, s≥0}. This process takes as values the vertices ofDand jumps only to nearest neighbors,

i.e. vertices one edge away from the occupied one. Given X_s, 0 ≤ s ≤ t, and {X_t = x}, the conditional probability that, in the interval(t,t+dt), the process jumps to the nearest neighbor y

ofx is L(y,t)dt, with

L(y,t):=a_y+

Z t

0

1l_{X

s=y}ds, ay>0,

where 1l_Astands for the indicator function of the setA. The positive numbers_{aν:ν∈Vert(D)}are called initial weights, and we supposea_{ν ≡}1, unless specified otherwise. Such a process is said to be a Vertex Reinforced Jump Process (VRJP) onD.

Consider VRJP defined on the integers, which starts from 0. With probability 1/2 it will jump either to 1 or ₋1. The time of the first jump is an exponential random variable with mean 1/2, and is independent on the direction of the jump. Suppose the walk jumps towards 1 at timez. Given this, it will wait at 1 an exponential amount of time with mean 1/(2+z). Independently of this time, the jump will be towards 0 with probability(1+z)/(2+z).

In this paper we define a process to be recurrent if it visits each vertex infinitely many times a.s., and to be transient otherwise. VRJP was introduced by Wendelin Werner, and its properties were first studied by Davis and Volkov (see[8]and[9]). This reinforced walk defined on the integer lattice is studied in[8]where recurrence is proved. For fixed b _∈N_{:= {1, 2, . . .}, the b-ary tree, which}

we denote by_Gb, is the infinite tree where each vertex has b+1 neighbors with the exception of a single vertex, called the root and designated byρ, that is connected to b vertices. In[9]is shown that VRJP on the b-ary tree is transient if b ≥ 4. The case b = 3 was dealt in[4], where it was proved that the process is still transient. The caseb=2 is still open.

Another process which reinforces the vertices, the so called Vertex-Reinforced Random Walk (VRRW), shows a completely different behaviour. VRRW was introduced by Pemantle (see[17]). Pemantle and Volkov (see[19]) proved that this process, defined on the integers, gets stuck in at most five points. Tarrès (see[23]) proved that it gets stuck in exactly 5 points. Volkov (in [24]) studied this process on arbitrary trees.

The reader can find in[18]a survey on reinforced processes. In particular, we would like to mention that little is known regarding the behaviour of these processes on infinite graphs with loops. Merkl and Rolles (see[14]) studied the recurrence of the original reinforced random walk, the so-called linearly bond-reinforced random walk, on two-dimensional graphs. Sellke (see[21]) proved than once-reinforced random walk is recurrent on the ladder.

We define the distance between two vertices as the number of edges in the unique self-avoiding path connecting them. For any vertex ν, denote by _|ν| its distance from the root. Level i is the set of verticesν such that_|ν|=i. The main result of this paper is the following.

such that

lim

t→∞

|X_t|

t =K

(1)

b a.s., (1.1)

|X_{t| −}K(1)

b t p

t =⇒Normal(0,K

(2)

b ), (1.2)

where we took the limit as t _{→ ∞}, _⇒stands for weak convergence and Normal(0, 0) stands for the Dirac mass at0.

Durrett, Kesten and Limic have proved in[11]an analogous result for a bond-reinforced random walk, called one-time bond-reinforced random walk, on_Gb, b _≥2. To prove this, they break the path into independent identically distributed blocks, using the classical method of cut points. We also use this approach. Our implementation of the cut point method is a strong improvement of the one used in[3]to prove the strong law of large numbers for the original reinforced random walk, the so-called linearly bond-reinforced random walk, on_Gb, with b_≥ 70. Aidékon, in[1]gives a sharp criteria for random walk in a random environment, defined on Galton-Watson tree, to have positive speed. He proves the strong law of large numbers for linearly bond-reinforced random walk on_Gb, with b_≥2.

2

Preliminary definitions and properties

From now on, we consider VRJPX defined on the regular tree_Gb, with b_≥3. For ν 6=ρ, define par(ν), called the parent ofν, to be the unique vertex at level|ν| −1 connected to ν. A vertexν0

is a child ofν if ν =par(ν0). We say that a vertexν0 is a descendant of the vertexν if the latter

lies on the unique self-avoiding path connectingν0toρ, andν06=ν. In this case,ν is said to be an

ancestor ofν0. For any vertexµ, letΛµbe the subtree consisting ofµ, its descendants and the edges connecting them, i.e. the subtree rooted atµ. Define

T_i := inf_{t_≥0:_|X_t|=i_}.

We give the so-called Poisson construction of VRJP on a graph_D(see[20]). For each ordered pair of neighbors(u,v)assign a Poisson processP(u,v)of rate 1, the processes being independent. Call

h_i(u,v), withi_≥1, the inter-arrival times ofP(u,v)and letξ1:=inf{t≥0:Xt=u}. The first jump

afterξ1is at timec1:=ξ1+minvh1(u,v) L(v,ξ1)

−1

, where the minimum is taken over the set of neighbors ofu. The jump is towards the neighbor vfor which that minimum is attained. Suppose we defined_{(ξj,cj), 1≤j≤i−1}, and let

ξi :=inf

t>c_i−1:X_t=u , and

j_v−1= j_u,v−1 := number of timesXjumped fromutovby timeξi.

The first jump afterξihappens at timeci:=ξi+minvhjv(u,v) L(v,ξi)

−1

, and the jump is towards the neighborv which attains that minimum.

Definition 2.1. A vertexµ, with_|µ| ≥2, isgoodif it satisfies the following

h₁(µ0,µ)<

h1 µ0, par(µ0)

1+h₁ par(µ0),µ0

By virtue of our construction of VRJP, (2.3) can be interpreted as follows. When the process X visits the vertex µ0 for the first time, if this ever happens, the weight at its parent is exactly 1+ h1 par(µ0),µ0

while the weight at µ is 1. Hence condition (2.3) implies that when the process visitsµ0(if this ever happens) then it will visitµbefore it returns to par(µ0), if this ever happens.

The next Lemma gives bounds for the probability that VRJP returns to the root after the first jump.

Lemma 2.2. Let

αb:=P Xt=ρfor some t≥T1

,

and letβbbe the smallest among the positive solutions of the equation

x =

b

X

k=0

xkp_k, (2.4)

where, for k_{∈ {}0, 1, . . . ,b_},

p_k:=

k

X

j=0

_b

k

_k

j

(₋1)j

Z ∞

0

1+z j+b−k+1+ze

−z_{dz. (2.5)}

We have _Z

∞

0

1+z b+1+zbe

−bz_dz

≤αb≤βb. (2.6)

Proof. First we prove the lower bound in (2.6). The left-hand side of this inequality is the prob-ability that the process returns to the root with exactly two jumps. To see this, notice that L(ρ,T1)

is equal 1+minν:|ν|=1h1(ρ,ν). Hence T1 = L(ρ,T1)−1 is distributed like an exponential with

mean 1/b. Given that T₁ = z, the probability that the second jump is from X_T

1 toρ is equal to

(1+z)/(b+1+z). Hence the probability that the process returns to the root with exactly two jumps

is _Z

∞

0

1+z b+1+zbe

−bz_dz.

As for the upper bound in (2.6) we reason as follows. We give an upper bound for the probability that there exists an infinite random tree which is composed only of good vertices and which has root at one of the children ofX_T1. If this event holds, then the process does not return to the root after timeT₁(see the proof of Theorem 3 in[4]). We prove that a particular cluster of good vertices is stochastically larger than a branching process which is supercritical. We introduce the following color scheme. The only vertex at level 1 to begreenisXT1. A vertexν, with|ν| ≥2, isgreenif and

only if it is good and its parent is green. All the other vertices are uncolored. Fix a vertexµ. LetC

be any event in

Hµ:=σ(hi(η0,η1):i≥1, withη0∼η1and bothη0 andη1∈/Λµ), (2.7)

that is theσ-algebra that contains the information aboutXtobserved outsideΛµ. Next we show that givenC_{∩ {}µis green_}, the distribution ofh₁(par(µ),µ)is stochastically dominated by an exponen-tial(1). To see this, first notice thath₁(par(µ),µ)is independent ofC. LetD:=_{par(µ)is green_{} ∈}

Hµ and set

W := h1 µ0, par(µ0)

1+h1 par(µ0),µ0

The random variableW is independent ofh₁(par(µ),µ) and is absolutely continuous with respect the Lebesgue measure. By the definition of good vertices we have

{µis green_}=_{h₁(par(µ),µ)<W_{} ∩}D.

we get that the expression in (2.9) is less or equal toP

Hence the cluster of green vertices is stochastically larger than a Galton–Watson tree where each vertex haskoffspring,k∈ {0, 1, . . . ,b}, with probabilitypk defined in (2.5). To see this, fix a vertex

with parameter one, we have

From the basic theory of branching processes we know that the probability that this Galton–Watson tree is finite (i.e. extinction) equals the smallest positive solution of the equation

x₋ b

X

k=0

xkp_k=0. (2.13)

The proof of (2.6) follows from the fact that 1₋βb≤1−αb. This latter inequality is a consequence of

the fact that the cluster of green vertices is stochastically larger than the Galton-Watson tree, hence its probability of non-extinction is not smaller. As b≥3, the Galton-Watson tree is supercritical (see [4]),henceβb<1.

For example, if we consider VRJP on_G3, Lemma 2.2 yields

0.3809≤α3≤0.8545.

Definition 2.3. Level j_≥1is acut levelif the first jump after T_jis towards level j+1, and after time Tj+1 the process never goes back to XTj, and

L(X_Tj,_∞)<2 and L(par(X_Tj),_∞)<2.

Define l1to be the cut level with minimum distance from the root, and for i>1, l_i:=min_{j>l_i−1: j is a cut level_}.

Define the i-thcut timeto beτi:=Tli. Notice that li=|Xτi|.

3

l

₁

has an exponential tail

For any vertexν∈Vert(_Gb), we define fc(ν), which stands forfirst childofν, to be the (a.s.) unique vertex connected toν satisfying

h1(ν, fc(ν)) =min

h1(ν,µ): par(µ) =ν . (3.14)

For definiteness, the rootρis not a first child. Notice that condition (3.14) does not imply that the vertex fc(ν) is visited by the process. IfXvisits it, then it is the first among the children ofν to be visited.

For any pair of distributions f and g, denote by f _∗gthe distribution ofPV_k=1M_k, where

• V has distribution f, and

• {M_k,k_∈N_}is a sequence of i.i.d random variables, independent ofV, each with distribution

g.

Recall the definition ofp_i,i_{∈ {}0, . . . ,b_}, given in (2.5). Denote byp(1)_{the distribution which assigns}

toi_{∈ {}0, . . . ,b_}probability p_i. Define, by recursion,p(j) _:=p(j−1)

∗p(1)_{, with j}

≥2. The distribution p(j) _{describes the number of elements, at time}

j, in a population which evolves like a branching process generated by one ancestor and with offspring distributionp(1)_{. If we let}

m:=

b

X

then the mean ofp(j)_ismj_{. The probability that a given vertexµis good is, by definition,}

P

h₁(µ0,µ)<

h1 µ0, par(µ0)

1+h₁ par(µ0),µ0

whereµ0=par(µ).

As the h1 par(µ0),µ0

is exponential with parameter 1, conditioning on its value and using inde-pendence between different Poisson processes, we have that the probability above equals

P

h1(µ0,µ)<

1

1+zh1 µ0, par(µ0)

e−zdz=

Z ∞

0

1 2+ze

−z_{dz=0.36133 . . . . (3.15)}

Hence

m=b_·0.36133>1, because we assumedb≥3.

Letq₀=p₀+p₁, and fork_{∈ {}1, 2, . . . ,b₋1_}setq_k=p_k+1. Setqto be the distribution which assigns toi∈ {0, . . . ,b−1}probabilityqi. For j≥2, letq(j):=p(j−1)∗q. Denote byq

(j)

i the probability that

the distributionq(j) _{assigns toi}

∈ {0, . . . ,(b₋1)bj−1_}. The mean ofq(j) _ismj−1_(m

−1). From now on,ζdenotes the smallest positive integer in_{2, 3, . . . ,_}such that

mζ−1(m₋1)>1. (3.16)

Next we want to define a sequence of events which are independent and which are closely related to the event that a given level is a cut level. For any vertexν of_Gb letΘ_ν be the set of verticesµ

such that

• µis a descendant ofν,

• the difference_|µ|-_|ν|is a multiple ofζ,

• µis a first child.

By subtree rooted atν we mean a subtree ofΛ_ν that containsν. Setν_e=fc(ν)and let

A(ν):=_∃an infinite subtree of_Gb root at a child ofν_e, which is composed only by

good vertices and which contains none of the vertices inΘ_ν} (3.17)

Fori_∈N_{, letAi :=A XT}

i

. Notice that if the process reaches the first child ofν and ifA(ν)holds, then the process will never return toν. Hence ifAi holds, and ifXTi+1=XTi+1, theniis a cut level, provided that the total weights atX_Ti and its parent are less than 2.

Proposition 3.1. The events A_iζ, with i_∈N_{, are independent.}

Proof. We recall that ζ ≥ 2. We proceed by backward recursion and show that the events A_iζ

depend on disjoint Poisson processes collections. Choose integers 0 < i1 < i2 < . . . < ik, with i_j∈ζN_{:={ζ, 2ζ, 3ζ, . . .}for all j∈ {1, 2, . . . ,k}. It is enough to prove that}

P

_\k

j=1 A_i

j

=

k

Y

j=1

P A_i

j

Fix a vertexν at leveli_k. The setA(ν)belongs to the sigma-algebra generated byP(u,w): u,w _∈

Vert(Λ_ν) . On the other hand, the setTk_j=−₁1Aij∩ {XT_ik =ν}belongs to

P(u,w): u∈/Vert(Λ_ν) . As the two events belong to disjoint collections of independent Poisson processes, they are independent. AsP_{(A(ν)) =}P_{(A(ρ)), we have}

Reiterating (3.20) we get (3.18).

Lemma 3.2. Defineγbto be the smallest positive solution of the equation

x=

k )have been defined at the beginning of this section. We have

P_{(Ai)≥1−γb>0, ∀i∈}N_{. (3.22)}

Proof. Fix i∈N _{and letν}∗ = X_Ti. We adopt the following color scheme. The vertex fc X_Ti is coloredblue. A descendantµofν∗is colored blue if it is good, its parent isblue, and either

• |µ| − |ν∗|is not a multiple ofζ, or

• 1_ζ |µ| − |ν∗|∈N_{andµis not a first child.}

Vertices which are not descendants ofν∗are not colored. Following the reasoning given in the proof of Lemma 2.2, we can conclude that the number of blue vertices at levels_|ν∗| + jζ, with j_≥1, is stochastically larger than the number of individuals in a population which evolves like a branching process with offspring distributionq(ζ)

, introduced at the beginning of this section. Again, from the basic theory of branching processes we know that the probability that this tree is finite equals the smallest positive solution of the equation (3.21). By virtue of (3.16) we have thatγb<1.

Lemma 3.3. Suppose U_n is Bin(n,p). For x _∈(0, 1)consider the entropy

H(x_|p):=xln x

p + (1−x)ln

1₋x

1₋p.

We have the following large deviations estimate, for s∈[0, 1],

P _{Un≤sn≤exp{−n inf}

x∈[0,s]H(x|p)}.

Proposition 3.4.

i) Letν be a vertex with|ν| ≥1. The quantity

P _A(ν)_|h1(ν, fc(ν)) =x

is a decreasing function of x, with x_≥0.

ii) P _A(ν)_|h1(ν, fc(ν))≤x

≥P _A(ν), for any x ≥0.

Proof. Suppose {fc(ν) =ν}. Givenh1(ν,ν) =x , the set of good vertices inΛν is a function of

x. Denote this function by_T :R+_{→ {subset of vertices of Λν}. A child ofν, sayν1, is good if and}

only if

h₁(ν,ν1)<

h1(ν,ν)

1+x .

Hence the smallerx is, the more likelyν1 is good. This is true for any child ofν. As for descendants

ofν at level strictly greater than_|ν|+2, their status of being good is independent ofh₁(ν, fc(ν)). Hence _T(x) _{⊃ T}(y) for x < y. This implies that the connected component of good vertices continingν is larger if{h₁(ν,ν) =x}rather than{h₁(ν,ν) = y}, for x< y. Hence

P _{A(ν)|h1(ν, fc(ν)) =x, fc(ν) =ν≥}P _{A(ν)|h1(ν, fc(ν)) = y, fc(ν) =ν, for x< y.}

Using symmetry we get i). In order to prove ii), use i) and the fact that the distribution ofh₁(ν, fc(ν))

is stochastically larger that the conditional distribution ofh1(ν, fc(ν))given{h1(ν, fc(ν))≤x}.

Denote by[x]the largest integer smaller than x.

Theorem 3.5. For VRJP defined onGb, with b≥3, and s∈(0, 1), we have

P _l[sn]≥n≤expn₋[n/ζ] inf

x∈[0,s]H

x(1₋γb)ϕb

o

, (3.23)

whereγ_b was defined in Lemma 3.2, and

ϕb:=

€

1₋e−bŠ €1₋e−(b+1)Š b

b+2. (3.24)

Proof. By virtue of Proposition 3.1 the sequence 1lAkζ, with k ∈ N, consists of i.i.d.

ran-dom variables. The random variable P[_jn₌/ζ₁]1lAjζ has binomial distribution with parameters

P _A(ρ),[n/ζ]. We define the event

Bj:={the first jump afterTj is towards level j+1 and L XTj,Tj+1

<2, and L par(X_Tj),T_j+1

Let _Ft be the smallest sigma-algebra defined by the collection_{X_s, 0 _≤ s_≤ t_}. For any stopping timeSdefine_FS :=A:A∩ {S≤t} ∈ Ft . Now we show

P€_{Bj | FT}

i−1

Š

≥€1−e−bŠ €1−e−(b+1)Š b

b+2=ϕb, (3.25) where the inequality holds a.s.. In fact, by time Ti the total weight of the parent ofXTi is stochasti-cally smaller than 1+ an exponential of parameter b, independent ofFTi−1. Hence the probability

that this total weight is less than 2 is larger than 1₋e−b. Given this, the probability that the first jump after T_i is towards level i+1 is larger than b/(b+2). Finally, the conditional probability that T_i+1−T_i < 1 is larger than 1₋e−(b+1). This implies, together withζ ≥ 2, that the random variableP[_j=n/ζ₁]1l_B

j is stochastically larger than a binomial(n,ϕb). For anyi∈N, and any vertexν with|ν|=iζ, set

Z := min1, h1 ν, par(ν)

1+h1 par(ν),ν

E := _{X_T

iζ =ν} ∩ {L(par(ν),Tiζ)<2}. We have

B_iζ∩ {XTiζ=ν}={h1(ν, fc(ν))<Z} ∩E.

Moreover, the random variable Z and the event E are both measurable with respect the sigma-algebra

f

Hν :=σ

n

P(par(ν),ν),P(u,w): u,w∈/Vert(Λ_ν) o.

Let f_Z be the density of Zgiven_{h₁(ν, fc(ν))<Z_}∩E. Using 3.4, ii), and the independence between

h1(ν, fc(ν))andHfν, we get

P _{AiζBiζ∩ {XT}

iζ =ν}

=P _A(ν)_{h1(ν, fc(ν))<Z} ∩E

=

Z _∞

0

P _{A(ν){h1(ν, fc(ν))<z}fZ(z)dz≥}P _A(ν)

= X

ν:|ν|=iζ

P _A(ν)_{∩ {}XTiζ=ν}

=P(Aiζ).

(3.26)

The first equality in the last line of (3.26) is due to symmetry. Hence

P(A_iζ

B_iζ)≥P(Aiζ). (3.27)

If A_k∩B_k holds then k is a cut level. In fact, on this event, when the walk visits level k for the first time it jumps right away to level k+1 and never visits level k again. This happens because

X_Tk+1 = fc(X_Tk) has a child which is the root of an infinite subtree of good vertices. Moreover the total weights atX_T

k and its parent are less than 2. Define

en:=

[_Xn/ζ]

i=1

1l_A iζ∩Biζ.

By virtue of (3.22), (3.25), (3.27) and Proposition 3.1 we have thate_nis stochastically larger than a bin([n/ζ],(1−γb)ϕb). Applying Lemma 3.3, we have

P _l[sn]≥n≤P _en≤[sn]_≤exp n

−[n/ζ] inf

x∈[0,s]H

The functionH x(1₋γb)ϕb

is decreasing in the interval(0,(1₋γb)ϕb). Hence forn>1/ (1−

γb)ϕb

, we have inf_x∈[0,1/n]H x(1₋γb)ϕb

=H 1/n(1₋γb)ϕb

.

Corollary 3.6. For n>1/ (1₋γb)φb

, by choosing s=1/n in Theorem 3.5, we have

P _l1≥n≤exp

n

−[n/ζ] inf

x∈[0,1/n]H

x(1₋γb)ϕb

o

=exp n

−[n/ζ]H

₁

n

(1−γb)ϕb

o ,

(3.28)

where, from the definition of H we have

lim

n→∞H

₁

n

(1₋γb)ϕb

=ln 1 1₋(1₋γb)ϕb

>0.

4

τ

₁

has finite

(

2

+

δ

)

-moment

The goal of this section is to prove the finiteness of the 11/5 moment of the first cut time. We adopt the following strategy

• first we prove the finiteness of all moments for the number of vertices visited by timeτ1, then • we prove that the total time spent at each of these sites has finite 12/5-moment.

Fixn_∈N_{and let}

Π_n:= number of distinct vertices thatXvisits by timeT_n,

Π_n,k:=number of distinct vertices thatXvisits at levelkby timeTn.

LetT(ν):=inf_{t≥0:Xt=ν}. For any subtree EofGb, b≥1, define

δ(a,E):=sup ¨

t: Z t

0

1l_{X

s∈E}ds≤a

« .

The processX_δ(t,E)is called the restriction ofXtoE.

Proposition 4.1(Restriction principle (see [8])). Consider VRJPXdefined on a tree_J rooted at

ρ. Assume this process is recurrent, i.e. visits each vertex infinitely often, a.s.. Consider a subtree Je rooted atν. Then the process X_δ(t,Je)is VRJP defined onJe. Moreover, for any subtreeJ∗disjoint from

e

J, we have that X_δ(t,Je)and Xδ(t,J∗₎are independent.

Proof. This principle follows directly from the Poisson construction and the memoryless property of the exponential distribution.

Definition 4.2. Recall that P(x,y), with x,y∈Vert Gbare the Poisson processes used to generateX

onGb. LetJ be a subtree ofGb. Consider VRJPVonJ which is generated by usingP(u,v):u,v ∈ Vert(_J) , which is the same collection of Poisson processes used to generate the jumps of X from the vertices ofJ. We say thatVis theextensionofXinJ. The processes Vt and Xδ(t,J)coincide up to a

We construct an upper bound for Π_n,k, with 2 _≤ k _≤ n₋1. . Let G(k) be the finite subtree of

Gb composed by all the vertices at level i with i ≤ k−1, and the edges connecting them. Let V be the extension ofX toG(k). This process is recurrent, because is defined on a finite graph. The total number of first children at levelk₋1 is bk−2, and we order them according to when they are visited by V, as follows. Letη1 be the first vertex at level k−1 to be visited by V. Suppose we

have definedη1, . . . ,ηm−1. Letηm be the first child at levelk−1 which does not belong to the set {η1,η2, . . . ,ηm−1}, to be visited. The vertices ηi, with 1≤ i≤ bk−2 are determined byV. All the

other quantities and events such as T(ν)andA(ν), withν running over the vertices of_Gb, refer to the processX. Define

f_n(k):=1+b2inf_{m_≥1: 1l_A(par(ηm))=1_}.

Let J := inf{n:T(ηn) =∞}, if the infimum is over an empty set, let J =∞. Suppose thatA(ηm)

holds, then X, after time T(ηm), is forced to remain inside Ληm, and never visits fc(ηm) again. This implies that T(ηm+1) = ∞. Hence, if J = m then

Tm−1

i=1 (A(par(ηi)))c holds, and fn(k) ≥

1+b2(m−1). Similarly ifJ =_∞ then fn(k) =1+b2bk−2 =1+bk, which is an obvious upper

bound for the number of vertices at levelkwhich are visited byX. On the other hand, ifJ=mthen the number of vertices at levelkwhich are visited byXis at most 1+(m₋1)b2. In fact, the processes XandVcoincide up to the random time when the former process leaves G(k)and never returns to it. Hence ifT(ηi)<∞thenXvisited exactlyi−1 distinct first children at levelk−1 before time T(ηi). On the event {J = m}we have that {T(ηm−1 < ∞} ∩ {T(ηm) =∞}, hence exactlym−1

first children are visited at levelk−1. This implies that at most 1+ (m−1)b2vertices at levelkare visited.

We conclude that fn(k)overcounts the number of vertices at level kwhich are visited, i.e. Πn,k ≤ f_n(k).

Recall thath₁(ν, fc(ν)), being the minimum over a set ofbindependent exponentials with rate 1, is distributed as an exponential with mean 1/b.

Lemma 4.3. For any m_∈N_{, we have}

P _fn(k) > 1+mb2≤(γb)m.

Proof. Given Tm_i=−₁1(A(par(ηi))c the distribution ofh(par(ηm),ηm) is stochastically smaller than

an exponential with mean 1/b. Fix a set of verticesνiwith 1≤i≤m−1 at levelk−1 and each with

a different parent. Givenη_i =ν_i fori≤m−1, consider the restriction ofVto the finite subgraph obtained from G(k) by removing each of theνi and par(νi), withi ≤ m−1. The restriction ofV

to this subgraph is VRJP, independent ofTm_i=−₁1(A(par(ηi))c, and the total time spent by this process

in levelk₋2 is exponential with mean 1/b. This total time is an upper bound forh(par(ηm),ηm).

This conclusion is independent of our choice of the verticesνi with 1≤ i≤ m−1. Finally, using

Proposition 3.4 i), we have

P _fn(k)>1+mb2_{| fn(k)>1+ (m−1)b}2₌P _{(A(par(ηm)))}c_|

m_\−1

i=1

(A(par(ηi))c

≤P (A(par(ηm)))c

≤γb.

(4.29)

Lemma 4.4. For p_≥1, we haveE”_Πnp—_=O(np_).

where the sum is over the vertices ofGb. In words,Πis the number of vertices visited beforeτ1.

Lemma 4.5. For any p>0we haveE_[Πp_]<∞.

Proof. By virtue of Lemma 4.4, Æ

E_Π2p_{n ≤C}(1)

b,pn

p_{, for some positive constantC}(1)

b,p. Hence using

In the last inequality we used Corollary 3.6.

Next, we want to prove that the 12/5-moment of L(ρ,_∞)is finite. We start with three intermediate results. The first two can be found in[9]. We include the proofs here for the sake of completeness.

Lemma 4.6. Consider VRJP on{0, 1}, which starts at1, and with initial weights a0 =c and a1=1. variablesχ andηare independent. Hence

i.e. E_{[L(0,ξ(t))] is solution of the equation y}′_{(t) = y(t)/t, with initial condition y(1) = c (see}

Finally, reasoning in a similar way, we get thatE

h

A rayσ is a subtree of_Gb containing exactly one vertex of each level of _Gb. Label the vertices of this ray using_{σi,i≥0}, whereσi is the unique vertex at leveliwhich belongs toσ. Denote byS

the collection of all rays ofGb.

Lemma 4.7. For any ray σ, consider VRJP X(σ) _:=

Proof. By the tower property of conditional expectation,

E

using the fact thath₁(σ0,σ1)is exponential with mean 1, we have

The Lemma follows by recursion and restriction principle.

Next, we prove that

L(ρ,T(σ_n))_≤L(σ)₍

σ0,Tn(σ)). (4.36)

In fact, we have equality ifT(σn)<∞, because the restriction and the extension ofXtoσcoincide

during the time interval[0,T(σn)]. IfT(σn) =∞, it means thatXleft the rayσat a times<Tn(σ).

Proof. Recall the definition ofA(ν)from (3.17) and set

D_k:= [

ν:|ν|=k−2 A(ν).

IfA(ν)holds, after the first time the process hits the first child ofν, if this ever happens, it will never visitν again, and will not increase the local time spent at the root. Roughly, our strategy is to use the extensions on paths to give an upper bound of the total time spent at the root by time T_k and show that the probability thatTk_i=1D_icdecreases quite fast ink.

Using the independence between disjoint collections of Poisson processes, we infer thatA(ν), with

|ν|= k₋2 are independent. In fact eachA(ν)is determined by the Poisson processes attached to reachesµit will never return to the root. Hence

Using this fact, combined with

Using (4.39), Holder’s inequality (withp=5/4) and (4.38) we have

Proof. Suppose we relabel the vertices that have been visited by time τ1, using θ1,θ2, . . . ,θΠ, where vertexν is labeledθkif there are exactlyk−1 distinct vertices that have been visited before

ν. Notice that ∆_ν and {θ_k = ν} are independent, because they are determined by disjoint

non-By Jensen’s and Holder’s (with p=12/11) inequalities, Lemma 4.9 i) and ii), and Lemma 4.8, we have

for some positive constantsC_b(3) andC(4)_b . It remains to prove the finiteness of the last sum. We use the fact

lim

k→∞k

48_P(Π

≥k) =0. (4.40)

The previous limit is a consequence of the well-known formula

∞

Proof. Using 4.9, and the fact that∆_X

Using Lyapunov inequality, i.e. E_[Zq_]1/q_≤E_[Zp_]1/p _{whenever 0<q≤p, Lemma 4.7, and the fact}

i. This extension is VRJP on Λµi with initial weights 1, hence we can apply Theorem 4.10 to get

where we used Jensen’s inequality, the independence ofτ(µi)andT1 and Lemma 4.11. In fact, as L ρ,∞≥1 , we have

5

Splitting the path into one-dependent pieces

DefineZi =L(Xτi,∞), withi≥1.

Lemma 5.1. The process Z_i, with i_≥1is a homogenous Markov chain with state space[1, 2].

Proof. Fix n _≥ 1. On _{Z_n = x_{} ∩ {}Xτn = ν} the random variable Zn+1 is determined by the variables_{P(u,v),u,v_∈Λ_ν,u₆=ν}. In fact these Poisson processes, on the set_{Z_n= x_{} ∩ {}X_τ

n=ν}, are the only ones used to generate the jumps of the process{XT(fc(ν)+t}t≥0. LetE1,E2, . . . ,En−1,En+1

be Borel subsets of[0, 1]. Conditionally on_{Z_n=x_{} ∩ {}Xτn=ν}, the two events{Zn+1∈En+1}and

{Z_{1 ∈}E₁,Z_{2 ∈}E₂, . . .Z_{n−1 ∈}E_n−1}are independent because are determined by disjoint collections of Poisson processes. By symmetry

P_{(Zn+1∈En+1| {Zn=x} ∩ {Xτ}

n=ν}) does not depend onν. Hence

P(Z_n+1∈En+1|Z1∈E1,Z2∈E2, . . .Zn−1∈En−1,Zn=x)

=X

ν

P_{(Zn+1∈En+1|Z1∈E1, . . . ,Zn−1∈En−1,Zn= x,Xτ}

n=ν)P(Xτn=ν|Z1∈E1, . . . ,Zn=x)

=P_{(Zn+1∈En+1|Zn=x,Xτ}

n=ν) =P(Zn+1∈En+1|Zn= x).

This implies thatZis a Markov chain. The self-similarity property of_GbandXyields the homegene-ity.

From the previous proof, we can infer that given Z_i = x, the random vectors(τi+1−τi,li+1−li)

and(τi−τi−1,li−li−1), are independent.

Proposition 5.2.

sup

i∈N sup

x∈[1,2]

E

h

τi+1−τi

11/5 Zi=x

i

<∞ (5.46)

sup

i∈N sup

x∈[1,2]

E

h

l_i+1−li

11/5

| Z_i =x

i

<∞. (5.47)

Proof. We only prove (5.46), the proof of (5.47) being similar. Define C :=X_{t 6}=ρ,_∀t > T₁

and fix a vertexν. Notice that by the self-similarity property of_Gb, we have

E

h

(τi+1−τi)11/5 | {Zi=x} ∩ {Xτi =ν}

i

=E

h

(τ1)11/5|{L(ρ,T1) =x} ∩C

i .

By the proof of Lemma 2.2, we have that

inf

1≤x≤2

P _{CL(ρ,T1) =x≥} b

b+xP(A1)≥(1−γb) b

Hence

sup

x:x∈[1,2]

E

h

(τ1)11/5

L(ρ,T1) =x

i

≥ sup

x:x∈[1,2]

E

h

(τ1)11/5

_{L(ρ,T₁) =x_{} ∩}CiP_{(C|L(ρ,T1) =x)}

≥(1₋γb) b

b+2_x:sup_x∈[1,2]E h

(τ1)11/5

_{L(ρ,T₁) =x_{} ∩}C

i

≥(1−γb) b

b+2_x:sup_x∈[1,2]

E

h

(τi+1−τi)11/5| {Zi= x} ∩ {Xτi =ν}

i

Hence

E

h

(τi+1−τi)11/5| {Zi= x} ∩ {Xτi =ν}

i

≤ b+2

b(1₋γb)

sup

1≤x≤2

E

h

(τ1)11/5| {L(ρ,T1= x}

i .

Next we prove thatZsatisfies the Doeblin condition.

Lemma 5.3. There exists a probability measureφ(_·)and0< λ≤1, such that for every Borel subset B of[1, 2], we have

P Z_i+1∈B _|Z_i=z _≥ λ φ(B) _∀z_∈[1, 2]. (5.49)

Proof. AsZ_i is homogeneous, it is enough to prove (5.49) fori=1. In this proof we show that the distribution ofZ2 is absolutely continuous and we compare it to 1+an exponential with parameter

1 conditionated on being less than 1. The analysis is technical becauseZ_i depend on the behaviour of the whole processX. Our goal is to find a lower bound for

P

Z_{2 ∈} (x,y)Z₁=z

, withz_∈[1, 2]. (5.50)

Moreover, we require that this lower bound is independent ofz_∈[1, 2].

Fix ǫ ∈ (0, 1). Our first goal is to find a lower bound for the probability of the event _{Z_{2 ∈}

(x,y), Z1∈Iǫ(z)}, whereIǫ(z):= (z−ǫ,z+ǫ). Fixz∈[1, 2]and consider the function

e−(b+u)(t−1)−(b+1)e−(b+2)e−(t−1). (5.51)

Its derivative with respectt is

(b+1)e−(b+2)−(t−1)−(b+u)e−(b+u)(t−1),

which is non-positive fort∈[1, 2]andu∈[1, 2]. In fact

(b+1)e−(b+2)−(t−1)₋(b+u)e−(b+u)(t−1)_≤(b+1)e−(b+2)−(1−1)₋(b+u)e−(b+u)(2−1)

= (b+1)e−(b+2)₋(b+u)e−(b+u)_≤0.

Hence for fixedu∈[1, 2], the function in (5.51) is non-increasing fort∈[1, 2]. For 1≤x < y≤2, we have

e−(b+u)(x−1)₋e−(b+u)(y−1)_≥(b+1)e−(b+2)€e−(x−1)₋e−(y−1)Š. (5.52)

a) T₁<1, then

b) the process spends atX_T1an amount of time enclosed in(z₋1₋ǫ,z₋1+ǫ), then c) it jumps to a vertex at level 2, spends there an amount of timet where t+1_∈(x,y), and

d) it jumps to level 3 and never returns toX_T2.

In the event just described, levels 1 and 2 are the first two cut levels, and_{Z2∈(x,y), Z1∈Iǫ(z)} holds. The probability that a) holds is exactly e−b. GivenT1= s−1, the time spent in XT1 before

the first jump is exponential with parameter(b+s). Hence b) occurs with probability larger than

inf

s∈[1,2]

e−(b+s)(z−ǫ)−e−(b+s)(z+ǫ)

.

Given a) and b), the process jumps to level 2 and then to level 3 with probability larger than

b/(b+2) b/(b+z+ǫ)). The conditional probability, given a) and b), that the time gap between these two jumps lies in(x₋1,y₋1)is larger than

inf

u∈I_ǫ(z)

e−(b+u)(x−1)−e−(b+u)(y−1)

.

At this point, a lower bound for the conditional probability that the process never returns toX_T

2is

b

b+y(1−αb)≥ b

b+2(1−αb). We have

P

Z_2∈(x,y), Z_1∈I_ǫ(z)

≥e−b b

3

(b+2)2_(b+z+ǫ)s∈inf_[1,2]

e−(b+s)(z−ǫ)₋e−(b+s)(z+ǫ)

inf

u∈Iǫ(z)

e−(b+u)(x−1)₋e−(b+u)(y−1)

(1₋αb)

≥(1₋αb)e−b

b3(b+1) (b+2)2_(b+z+ǫ)e

−(b+2)€_e−(x−1)

−e−(y−1)Š inf

s∈[1,2]

e−(b+s)(z−ǫ)₋e−(b+s)(z+ǫ),

(5.53) where in the last inequality we used (5.52). Notice that there exists a constantC_b(4)>0 such that

inf

ǫ∈(0,1)z,s∈inf[1,2] 1

ǫ

e−(b+s)(z−ǫ)₋e−(b+s)(z+ǫ)_≥C(4)

b . (5.54)

Summarizing, we have

P

Z2∈(x,y), Z1∈Iǫ(z)

≥C(5)_b €e−(x−1)₋e−(y−1)Šǫ, (5.55)

whereC_b(5) depends only onb.

In order to find a lower bound for (5.50) we need to prove that

sup ǫ∈(0,1)

1

ǫP

Z_1∈I_ǫ(z)_≤C(6)

for some positive constantC(6)

b . To see this, recall the definition ofBjfrom the proof of Theorem 3.5,

andζfrom (3.16). The event that level iis not a cut level is subset of (Bi∩Ai)c (see the proof of

Theorem 3.5). Denote bym_i =h1 XTi, fc(XTi)

, which is exponential with mean 1/b. Then

P(Z1∈Iǫ(z))≤

∞

X

i

P

mi∈Iǫ(z)

P

i_\−1

k=1

(Bk∩Ak)c|mi∈Iǫ(z)

≤Cǫ

∞

X

i

P

i_\−1

k=1

(B_k∩A_k)c_|m_i∈Iǫ(z)

,

where the constant C is independent of ǫ and z. It remains to prove that the sum in the right-hand side is bounded by a constant independent of ǫ. Notice that, for i > ζ, Ai−ζ and Bi−ζ are independent ofm_i. Moreover the events

Ai−ζ∩Bi−ζ, Ai−2ζ∩Bi−2ζ, Ai−3ζ∩Bi−3ζ, . . .

are independent by the proof of Proposition 3.1. Hence

P_{(Z1∈Iǫ(z))≤Cǫ} ∞

X

i

P

[(i−_\1)/ζ]

k=1

(B_i−kζ∩A_i−kζ)c_|m_i∈I_ǫ(z)

=Cǫ

∞

X

i

P

[(i−_\1)/ζ]

k=1

(B_i−kζ∩Ai−kζ)c

(by independence)

=Cǫ

∞

X

i

P

(B_i−kζ∩Ai−kζ)c

[(i−1)/ζ]

<∞.

Combining (5.53), (5.54) and (5.56), we get

P

Z_{2 ∈} (x,y)Z₁=z=lim ǫ↓0

1

P

Z_1∈Iǫ(z) P

Z_{2 ∈} (x,y), Z_1∈I_ǫ(z)

≥λ

€

e−(x−1)₋e−(y−1)Š

1−e−1 ,

(5.57)

for someλ >0. . A finite measure defined on fieldA can be extended uniquely to the sigma-field generated by_A, and this extension coincides with the outer measure. We apply this result to prove that (5.57) holds for any Borel setC_⊂[1, 2], using the fact that it holds in the field of finite unions of intervals. For any intervalE, the right-hand side of (5.57) can be written in an integral form as

λ

Z

E

e−x+1

1−e−1dx.

i_≥1, withC _⊂S∞_i=1E_i, such that

The first inequality is true because of the extension theorem, and the fact that the right-hand side is a lower bound for the outer measure, for a suitable choice of theE_is. The inequality (5.49), with

φ(C) =R

Ce

−x+1_/€₁

−e−1Šdx, follows by sendingǫto 0. The proof of the following Proposition can be found in[2].

Proposition 5.4. There exists a constant̺∈(0, 1)and a sequence of random times_{N_k,k_≥0_}, with N₀=0, such that

• the sequence_{Z_N

k,k≥ 1}consists of independent and identically distributed random variables

with distributionφ(_·)

∞. By virtue of Jensen’s inequality, we have that

where we used the fact that 0< ̺ <1.

With a similar proof we get the following result.

Lemma 5.6. sup_i∈NE

l_Ni+1−l_Ni2<∞.

Definition 5.7. A process {Yk,k ≥ 1}, is said to be one-dependent if Yi+2 is independent of {Y_j, with1_≤j_≤i_}.

Lemma 5.8. Let Υ_i := τ_Ni+1−τ_Ni,l_Ni+1−l_Ni, for i ≥ 1. The process Υ := Υ_i,i ≥ 1 is one-dependent. MoreoverΥ_i, i_≥1, are identically distributed.

Proof. Given Z_Ni

−1,Υi is independent of{Υj, j≤i−2}. Thus, it is sufficient to prove thatΥi is

independent of Z_N

i−1. To see this, it is enough to realize that givenZNi, Υi is independent ofZNi−1,

and combine this with the fact thatZNi andZNi−1are independent. The variablesZNi are i.i.d., hence

{Υ_i,i≥2}, are identically distributed.

The Strong Law of Large Numbers holds for one-dependent sequences of identically distributed variables bounded in _L1. To see this, just consider separately the sequence of random variables with even and odd indices and apply the usual Strong Law of Large Numbers to each of them.

Hence, for some constants 0<C(7)

b ,C

(8)

b <∞, we have

lim

i→∞

τNi

i →C

(7)

b , and _ilim_→∞ l_Ni

i →C

(8)

b , a.s.. (5.59)

Proof of Theorem 1. IfτNi ≤t< τNi+1, then by the definition of cut level, we have

lNi ≤ |Xt|<lNi+1.

Hence

l_N

i

τNi+1

≤ |Xt| t <

lNi+1

τNi .

Let

K(1)_b =

E[l_N2−l_N1]

E_[τN

2−τN1]

, (5.60)

which are the constants in (5.59). Then

lim sup

t→∞

|Xt| t ≤ilim→∞

lNi+1

τNi

= lim

i→∞

lNi+1

i+1

i

τNi

=K(1)_b , a.s..

Similarly, we can prove that

lim inf

t→∞

|X_t|

t ≥K

(1)

b , a.s..

Now we turn to the proof of the central limit theorem. First we prove that there exists a constant

C ≥0 such that

l_Nm−K(1)_b τ_Nm

p

where Normal(0, 0) stands for the Dirac mass at 0. To prove (5.61) we use a theorem from[12]. The reader can find the statement of this theorem in the Appendix, Theorem 6.1, (see also[22]). In order to apply this result we first need to prove that the quantity

1

in (5.62) can be written as

1

mE

h _Xm

i=1 Y_i2i.

The random variablesY_i are identically distributed with the exception ofY₁. From the definition of

K(1)

b given in (5.60), we have

E_{[Yi] =}E_[lN

2−lN1]−E[lN2−lN1] =0.

HenceY_i, withi_≥1, is a zero-mean one-dependent process, and we get

E

options. If the limit is equal to zero, then using Chebishev we get that

lim

If the limit of the quantity in (5.62) is positive, then we can apply Theorem 6.1 and deduce central limit theorem forY_i,i_≥1, yielding (5.61).

Similarly

|X_{t| −}K(1)

b t K(2)_b pt ≤

È

m+1

τNm

Pm+1 i=1 Yi p

m+1 +

Y_m+1K1b p

m

,

and the right-hand side converges to the same limit of the right-hand side of (5.64).

6

Appendix

We include a corollary to a result of Hoeffding and Robbins (see[12]or[22]).

Theorem 6.1(Hoeffding-Robbins). SupposeY:=_{Yi,i≥1}is a one-dependent process whose com-ponents are identically distributed with mean 0. If

• E[Y_i2+δ]<∞,for someδ >0,

• lim_n→∞1

nVar(

Pn

i=1Yi)converges to a positive finite constant K, then

Pn

i=1Yi−nE[Y1]

Kpn =⇒Normal(0, 1).

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