A quenched central limit theorem for biased random walks on supercritical Galton-Watson trees

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by  Adam Bowditch

A quenched central limit theorem for biased random walks on
supercritical Galton-Watson trees
Adam Bowditch, University of Warwick
Abstract
In this note, we prove a quenched functional central limit theorem for a biased random walk on
a supercritical Galton-Watson tree with leaves. This extends a result of Peres and Zeitouni (2008)
7
wherethecasewithoutleavesisconsidered. AconjectureofBenArousandFribergh(2016)suggests
1
0 an upperbound on thebias which we observe to besharp.
2

n
a
J 1 Introduction

6
1 We investigate biased random walks on supercritical Galton-Watson trees with leaves. A GW-tree

conditioned to survive consists of an infinite backbone structure with finite trees attached as branches.
]
R ThebackbonestructureisaGW-treewhoseoffspringlawdoesnothavedeaths. Thebranchesareformed
P by attaching a random number of independent GW-trees (conditioned to die out) to each vertex of the
. backbone. This forms dead-ends in the environment which makes it a natural setting for observing
h
trapping as the walk is slowed by taking excursions in the branches of the tree. The number of trees
t
a attached to a given vertex on the backbone has a distribution depending on the backbone locally. This
m
means that the branches are not i.i.d. and therefore the trapping incurred by the addition of the leaves
[ demonstrates a significant complication to the model without leaves studied in [8].

Theinfluenceofthebiasonthetrappingisanimportantfeatureofthemodel. Asthebiasisincreased,
1
v the local drift away from the root will increase but this does not necessarily speed up the walk. This
4 is because it increases the time trapped in the finite leaves from which the walk cannot escape without
9 takinglongsequencesofmovementsagainstthebias. In[7]itisshownthat,forasuitablylargebias,the
2
trapping is sufficient to slow the walk to zero speed whereas, for small bias, the expected trapping time
4
is finite and the walk has a positive limiting speed. Under a further restriction on the bias the trapping
0
. times have finite variance; we use this to prove a quenched invariance principle for the walk.
1
We next describe the supercriticalGW-tree in greater detail. Let p denote the offspring distribu-
0 { k}
tion of a GW-process Z with a single progenitor, mean µ > 1 and probability generating function f.
7 n
1 The process Zn gives rise to a random tree f where individuals are represented by vertices and edges
T
: connect individuals with their offspring. We denote by ρ the root, which is the vertex representing the
v
i unique progenitor. Letq denote the extinctionprobability ofZn which is non-zeroonly whenp0 >0. In
X
this case we then define
r
a f(s) f(qs) f(qs)
g(s):= − and h(s):=
1 q q
−
which are generating functions of GW-processes. In particular, g is the generating function of a GW-
process without deaths and h is the generating function of a subcritical GW-process. An f-GW-tree
conditioned on nonextinction can be generated by first generating a g-GW-tree g and then, to each
T T
vertex x of g, appending a random number M of independent h-GW-trees. We refer to g as the
x
T T
backbone of and the finite trees appended to g as the traps.
T T
We now introduce the biased random walk on a fixed tree. For fixed with x let ←x− denote the
T ∈T
parent of x and c(x) the set of children of x. A β-biased random walk on is a random walk (X )
n n 0
T ≥
MSC2010 subject classifications: Primary60K37,60F05, 60F17;secondary60J80.
Keywords: Randomwalk,randomenvironment,Galton-Watsontree,quenched,functionalcentrallimittheorem,invariance
principle.
1
on the vertices of which is β-times more likely to make a transition to a given child of the current
T
vertex than the parent (which are the only options). More specifically, the random walk started from
z is the Markov chain defined by P (X =z)=1 and the transition probabilities
∈T zT 0
1+β1c(x) , if y =←x−,
 β| |, if y c(x), x=ρ,
P (X =y X =x)= 1+βc(x) ∈ 6
zT n+1 | n d1ρ,| | if y ∈c(x), x=ρ,
0, otherwise.
We use P () := P ()P(d ) for the annealedlaw obtained by averaging the quenched law P over
ρ · ρT · T ρT
the law P on f-GW-trees conditioned to survive. Unless indicated otherwise, we start the walk at ρ.
R
For x , let x := d(ρ,x) denote the distance between x and the root of the tree. It has been
shown in [∈7]Tthat if|β| (µ 1,f (q) 1) then X n 1 converges P-a.s. to a deterministic constant ν > 0
− ′ − n −
called the speed of the∈walk. When β < µ |1 th|e walk is recurrent and X n 1 converges P-a.s. to 0.
− n −
| |
When the bias is large the walk is transientbut slowedby having to make long sequences of movements
against the bias in order to escape the traps; in particular, if β f (q) 1 then the slowing affect is
′ −
strongenoughto cause X n 1 to convergeP-a.s.to 0. This regim≥ehas been studied further by [2] and
n −
| |
[5] where polynomial scaling results are shown.
For σ,t>0 and n=1,2,... define
X nνt
Bn := | ⌊nt⌋|− .
t σ√n
Our main result, Theorem 1, is a quenched invariance principle for Bn.
t
Theorem 1. Suppose p >0, µ>1, β (µ 1,f (q) 1/2) and that there exists λ>1 such that
0 − ′ −
∈
λkp < . (1.1)
k
∞
k 0
X≥
Thereexistsσ >0suchthat,for P-a.e. , wehavethattheprocess(Bn) convergesinP -distribution
on D([0, ),R) endowed with the SkoroThod J topology to a standardtBtr≥o0wnian motion. T
1
∞
The condition p >0 is so that the tree has leaves; the case without leaves, p =0, is considered in
0 0
[8]. This no longer requires the condition β < f (q) 1/2 which is due to the trapping in the dead-ends
′ −
caused by the leaves. Indeed, when p = 0 we have that q = 0 and f (0) = 0 therefore the condition
0 ′
becomes irrelevant. This regime is studied further in [8] to the case where β = µ 1 and p = 0; in this
− 0
setting ν = 0 and it is shown that Bn converges in distribution to the absolute value of a Brownian
t
motion. This result is extended in [6] to random walks on multi-type GW-trees with leaves. Although
the dead-ends in this model trap the walk, the bias is small and therefore the slowing is weak.
Bychoosingp >0we allowthe treetohaveleaves;this createstrapsinthe environmentwhichslow
0
thewalk. Akeyinputforthisworkisthesecondmomentestimatefortrappingtimesintreesdetermined
in [4]. Fromthis we infer that β <f (q) 1/2 is the correctupper bound onthe biasso that the trapping
′ −
times inourmodelhavefinite variance. We conclude,inRemark2.4,thatthis upper boundisnecessary
which confirms [1, Conjecture 3.1].
Weassumethattheexponentialmomentscondition(1.1)holdsthroughout. Thisisapurelytechnical
assumptionwhichwe expectcouldbe relaxedto asufficiently largemomentconditionhoweverthe main
focus of this note has been to obtain the optimal upper bound on the bias.
In [4], the second moment condition for trapping times in finite trees is used to prove an annealed
invariance principle and quenched central limit theorem for a biased random walk on a subcritical GW-
tree conditioned to survive. In that model the backbone is one-dimensional and the fluctuations are
significantly influenced by the specific instance of the environment. This results in an environment
dependent centring for the walk in the quenched central limit theorem. In the supercritical case the
walkwillrandomlychooseoneofinfinitely manyescaperoutes;this createsamixing ofthe environment
which yields a deterministic centring in the quenched result Theorem 1.
We begin, in Section 2, by proving an annealed functional central limit theorem for the walk by
adapting the renewal argument used in [9, Theorem 4.1]. This is then extended to the quenched result
2
Theorem1inSection3byapplyingtheargumentusedin[3]whichlargelyinvolvesshowingthatmultiple
copies of the walk see sufficiently different areas of the tree.
2 An annealed invariance principle
h
We begin this sectionbyshowingthatthe time spent ina branchhasfinite variance. Let be the tree
T
formedbyattachinganadditionalvertexρ(astheparentoftherootρ)toanh-GW-tree h. Forafixed
T
tree T and vertex x T let τ+ := inf k > 0 : X = x denote the first return time to x. Let ξf,ξg,ξh
∈ x { k }
be random variables with probability generating functions f,g and h respectively then let ξ be equal in
distribution to the number of vertices in the first generation of . Since the generation sizes of g are
T T
dominated by those of we have that ξg is stochastically dominated by ξ. Using Bayes’ law we have
T
that P(ξ =k)=p (1 qk)(1 q) 1 cp therefore both ξ and ξg inherit the exponential moments of
k − k
− − ≤
ξf. Furthermore P(ξh =k)=p qk therefore ξh automatically has exponential moments.
k
Lemma 2.1. Suppose that p >0, µ>1 and β (µ 1,f (q) 1/2), then we have that
0 − ′ −
∈
E E h τ+ 2 < .
ρT ρ ∞
(cid:20) (cid:20)(cid:16) (cid:17) (cid:21)(cid:21)
Proof. We can write
τ+
ρ
τ+ = v where v = 1
ρ x x {Xk=x}
xXh Xk=1
∈T
is the number of visits to x before returning to ρ. Recall that c(x) denotes the set of children of x. It
then follows that
E E h τ+ 2 =E E h[v v ]
ρT ρ  ρT x y 
(cid:20) (cid:20)(cid:16) (cid:17) (cid:21)(cid:21) x,Xy h
∈T
 
C E (c(x)β+1)(c(y)β+1)βx+y
β | | | |
≤  | | | | 
x,Xy h
∈T
 
=C E (c(x)β+1)βx (c(y)β+1)βy (2.1)
β | | | |
 | | | | 
xXh yXh
∈T ∈T
 
where the inequality follows from [4, Lemma 4.8]. Letting Zh denote the size of the kth generation of
k
h
and collecting terms in each generation we have that
T
(|c(x)|β+1)β|x| = 1+ Zkh(βk+βk−1) ≤ (1+β−1) Zkhβk.
xX∈Th kX≥1 kX≥0
By[4,Lemma4.1],sinceξh hasexponentialmoments,wehavethatE[ZhZh] Cf (q)j wheneverj k.
k j ≤ ′ ≥
Substituting this and the above inequality into (2.1) we have that
E E h τ+ 2 C βk E[ZhZh]βj
ρT ρ ≤ β k j
(cid:20) (cid:20)(cid:16) (cid:17) (cid:21)(cid:21) kX≥0 Xj≥k
C βk (f (q)β)j
β ′
≤
k 0 j k
X≥ X≥
Cβ,f′(q) (f′(q)β2)k
≤
k 0
X≥
which is finite by the assumption that β <f (q) 1/2.
′ −
3
Let r(0) := 0, r(n) := inf k > r(n 1) : X ,X g for n 1 and Y := X , then Y is a
k k 1 n r(n) n
β-biased random walk on g{coupled t−o X . Write ζ−Y :∈=T0 a}nd for m≥=1,2,... let
T n 0
ζY :=inf k>ζY : Y < Y Y for all j <k l
m { m−1 | j| | k|≤| l| ≤ }
be regeneration times for the backbone walk. We can then define ζX := inf m 0 : X = Y to be
k { ≥ m ζkY}
the correspondingregenerationtimes forX. By [7, Proposition3.4]we havethatthere exists, P-a.s.,an
infinite sequence of regeneration times ζX and the sequence
{ k }k≥1
ζX ζX , X X
k+1− k ζkX+1 − ζkX k 1
n(cid:0) (cid:1) (cid:16)(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)(cid:17)o ≥
is i.i.d. (as is the corresponding sequence for Y).(cid:12)Furthe(cid:12)rmo(cid:12)re, le(cid:12)tting m :=sup j 0:ζX t be the
number of regenerations by time t, we have that m is non-decreasing antd diverg{es≥P-a.s.j ≤ }
t
By [7, Theorems 3.1 & 4.1], whenever µ > 1 and µ 1 < β < f (q) 1 we have that there exists
− ′ −
ν (0,1) such that X n 1 converges P-a.s. to ν. Moreover,combined with [7, Corollary 3.5], we have
n −
th∈at the time and di|stan|ce between regenerations of X both have finite means with respect to P. Let
χ := X X ν(ζX ζX ) = Y Y ν(ζX ζX ).
j ζjX − ζjX−1 − j − j−1 ζjY − ζjY−1 − j − j−1
By the previous remarkwe have that χ are i.i.d. with respect to P. By the stronglaw of largenumbers
j
and the definition of ν we have that χ are centred (see [7, Theorems 3.1 & 4.1]). We will show that χ
j j
have finite second moment and that their sum
m
Σ := χ = X νζX X νζX
m j ζmX − m − ζ1X − 1
Xj=2 (cid:0) (cid:1) (cid:16) (cid:17)
can be used to approximate Bn.
t
By the remark preceding Lemma 2.1, the offspring distribution ξg of g has exponential moments.
Since Y is a random walk on g, by [8, Proposition 3] we have that E[(ζTY ζY)k] < for all k Z
T 2 − 1 ∞ ∈
whenever β >µ 1.
−
Let η := r(k +1) r(k) denote the total time taken between X making the kth and (k +1)th
k
−
transition along the backbone. This time consists of
r(k+1)
N := 1
k {Xj=Yk}
j=rX(k)+1
excursions into the finite trees appended to the backbone at this vertex and one additional step to the
next backbone vertex. Therefore, we can write
Nk r(k+1)
η :=1+ γ where γ := 1 (2.2)
k k,j k,j {Pil=r(k)1{Xl=Yk}=j}
Xj=1 i=Xr(k)
is the duration of the jth such excursion.
Proposition 2.2. Under the assumptions of Theorem 1 we have that
E[(ζX ζX)2]< .
2 − 1 ∞
Proof. ThelawofζX ζX underPisidenticaltoitslawunderP ( ζY =1). Thatis,bytheindependence
2 − 1 ρ ·| 1
structure, we can condition the first regeneration vertex to be the first vertex reached by Y without
changing the law of ζX ζX. We therefore have that E (ζX ζX)2 can be written as
2 − 1 2 − 1
2 (cid:2) (cid:3)
ζY ζY ζY ζY
2 − 1 2 − 1
E (ζX ζX)2 ζY =1 = E η ζY =1 E (ζY ζY) η2 ζY =1
2 − 1 1  k 1  ≤  2 − 1 k 1 
(cid:2) (cid:12)(cid:12) (cid:3)  Xk=1  (cid:12)(cid:12)(cid:12)   kX=1 (cid:12)(cid:12)(cid:12) 
4
by convexity. Using convexity again with the decomposition (2.2) we can write this as
ζ2Y−ζ1Y Nk 2 ζ2Y−ζ1Y Nk
E (ζY ζY) 1+ γ ζY =1 E (ζY ζY) (N +1) 1+ γ2 ζY =1 .
 2 − 1  k,j 1 ≤  2 − 1 k  k,j 1 
Xk=1 Xj=1 (cid:12) kX=1 Xj=1 (cid:12)
 (cid:12)  (cid:12)
   (cid:12)    (cid:12) 
By conditioning on the backbone, buds and the walk on the backbone and buds we have that the
individualexcursiontimesareindependentofthe regenerationtimes ofY andthe numberofexcursions.
h
The excursion times are also distributed as the first return time to ρ for a walk started from ρ on .
T
We therefore have that the above expectation can be bounded above by
ζY ζY
E E h (τ+)2 E (ζY ζY) 2 − 1 (N +1)2 ζY =1 .
ρT ρ  2 − 1 k 1 
h h ii Xk=1 (cid:12)
(cid:12)
 (cid:12) 
Where, by Lemma 2.1, we have that E E h (τ+)2 < .
ρT ρ ∞
Let (zj)∞j=0 denote the ordered distinhct vehrtices viiisited by Y and
j
L(z,j):= 1 , L(z):=L(z, )
{Yj=z} ∞
i=0
X
the local times of the vertex z. Write
∞
W := 1
z,l {Xj=z,Xj+1∈/Tg,L(z,j)=l}
j=0
X
to be the number of excursions from z (by X) on the lth visit to z (by Y) for l = 1,...,L(z) and
M :=|{Yk}ζk2Y=−11| the number of distinct vertices visited by Y between time 1 and time ζ2Y −1 then
ζY ζY
2 − 1
E (ζY ζY) (N +1)2 ζY =1
 2 − 1 k 1 
kX=1 (cid:12)
(cid:12)
 M L(z(cid:12)k) 
=E (ζY ζY) (W +1)2 ζY =1
 2 − 1 zk,l 1 
kX=1 Xl=1 (cid:12)
(cid:12)
 (cid:12) 
= ∞ ∞ E (ζY ζY)1 (W +1)2 ζY =1
2 − 1 {k≤M,l≤L(zk)} zk,l | 1
k=1l=1
XX (cid:2) (cid:3)
∞ ∞ 1/2
E (ζY ζY)21 ζY =1 E (W +1)4 ζY =1 (2.3)
≤ 2 − 1 {k≤M,l≤L(zk)}| 1 zk,l | 1
kX=1Xl=1(cid:16) (cid:2) (cid:3) (cid:2) (cid:3)(cid:17)
byCauchy-Schwarz. ConditionalonζY =1,forall1 k M wehavethatL(z ) ζY ζY;moreover,
1 ≤ ≤ k ≤ 2 − 1
M ζY ζY therefore
≤ 2 − 1
1 1 .
{k≤M,l≤L(zk)} ≤ {k,l≤ζ2Y−ζ1Y}
Since the root does not have a parent, without any further information concerning the number
of children from a given vertex, we have that the walk is more likely to take an excursion into one
of the neighbouring traps when at the root than from this vertex. We can, therefore, stochastically
dominate the number of excursions from a vertex by the number of excursions from the root to see that
E (W +1)4 E (W +1)4 . Usingthis,Cauchy-SchwarzandthatP(ζY =1)>0,theexpression
zk,l ≤ z0,1 1
(2.3) can be bounded above by
(cid:2) (cid:3) (cid:2) (cid:3)
∞ ∞ 1/2
P(ζY =1) 1 E (ζY ζY)21 E (W +1)4
1 − 2 − 1 {k,l≤ζ2Y−ζ1Y} zk,l
Xk=1Xl=1(cid:16) h i (cid:2) (cid:3)(cid:17)
5
CE (ζY ζY)4 1/4E (W +1)4 1/2 ∞ ∞ P k,l ζY ζY 1/4.
≤ 2 − 1 z0,1 ≤ 2 − 1
k=1l=1
(cid:2) (cid:3) (cid:2) (cid:3) XX (cid:0) (cid:1)
Sincetheoffspringdistributionξg hasexponentialmomentswehavethatthetimebetweenregenerations
has finite fourth moments by [8, Proposition 3]. That is, E (ζY ζY)4 < .
2 − 1 ∞
WriteZ andZg to be the GW-processesassociatedwith and g. Thenumber ofexcursionsfrom
n n (cid:2) T T (cid:3)
the root is geometrically distributed with termination probability 1 p where
ex
−
Z Zg
p := 1− 1.
ex
Z
1
Using properties of geometric random variables we therefore have that
E (Wz0,1+1)4 ≤ CE[(1−pex)−4] ≤ CE[Z14] < ∞
(cid:2) (cid:3)
since Z =d ξ which has exponential moments.
1
It remains to show that
∞ ∞ P k,l ζY ζY 1/4 (2.4)
≤ 2 − 1
k=1l=1
XX (cid:0) (cid:1)
is finite. Note that P k,l ζY ζY =P ζY ζY l whenever l k. Using Chebyshev’s inequality
≤ 2 − 1 2 − 1 ≥ ≥
we can then bound (2.4) above by
(cid:0) (cid:1) (cid:0) (cid:1)
E ζY ζY j 1/4
2 ∞ ∞ P ζY ζY l 1/4 2 ∞ ∞ 2 − 1
2 − 1 ≥ ≤  h(cid:0) lj (cid:1) i
k=1l=k k=1l=k
XX (cid:0) (cid:1) XX
 
foranyintegerj. Inparticular,by[8,Proposition3]wehavethatE ζY ζY j isfiniteforanyinteger
2 − 1
j. Choosing j >8 we then have that this sum is finite which complhe(cid:0)tes the pr(cid:1)ooif.
Forx let denote the subtree consistingofalldescendants ofx in . Then, fory g, let
∈T Tx T ∈T Ty−
be the branch at y; that is, the subtree rooted at y consisting only of y, the children of y not on g and
T
their descendants. The tree then has the law of a tree rooted at y with some random number M
Ty− y−
of h-GW-trees attached to y. Since M is dominated by ξ, by (1.1) we have that M has exponential
y− y−
moments. It therefore follows from [7, Theorem B] that there exists some constant C such that
P(H(Ty−)≥n)≤Cf′(q)n (2.5)
where, for a fixed rooted tree T, (T) := sup d(ρ,x) : x T is the height of T. Let :=
n
H { ∈ } H
max ( ):y Y n denote the largest branch seen by Y by time n. It follows that
{H Ty− ∈{ k}k=0}
sup Bn Σmtn |Xζ1X|+νζ1X +HnT + sup |YζjY+1|−|YζjY|+ν(ζjX+1−ζjX). (2.6)
t − σ√n ≤ σ√n σ√n
t [0,T](cid:12) (cid:12) j=1,...,mnT
∈ (cid:12) (cid:12)
(cid:12) (cid:12)
Upto tim(cid:12)e nT,the w(cid:12)alkY canhavevisitedatmostnT verticeson g thereforethe probabilitythat
T
X has visited a branchof heightat least Clog(n) by time nT is at mostC nf (q)Clog(n). In particular,
T ′
by Borel-Cantelli, choosing C suitably large we have that there are almost surely only finitely many n
such that Y has visited the root of a branch of height at least Clog(n) by time nT. Since X and ζX
ζX 1
do not depend on n and have finite mean, we have that the first term in (2.6) converges P-a.s.1to 0.
By [8, Proposition 3], for any k Z+ we have that E[(Y Y )k] < therefore the distance
∈ | ζ1Y|−| ζ1Y| ∞
between regeneration points is small. In particular, bounding m above by nT, using a union bound
nT
and Markov’s inequality we have that for any ε>0,
P sup |YζjY+1|−|YζjY| >ε C E Y Y 21
j=1,...,mnT σ√n !≤ T,ε (cid:20)(cid:16)| ζ2Y|−| ζ1Y|(cid:17) n|Yζ2Y|−|Yζ1Y|>ε√no(cid:21)
6
which converges to 0 as n by dominated convergence. Similarly, using Proposition 2.2, we have
→ ∞
that the same holds for the supremum of ζX ζX; therefore, we have that
j+1− j
Σ
P sup Bn mtn >ε
t [0,T](cid:12) t − σ√n(cid:12) !
∈ (cid:12) (cid:12)
(cid:12) (cid:12)
converges to 0 as n . (cid:12) (cid:12)
By the law of lar→ge∞numbers and that ζX/n converges P-a.s. to 0 we have that
1
n
ζX =ζX + (ζX ζX )
n 1 k − k−1
k=2
X
converges P-a.s. It therefore follows by continuity of the inverse at strictly increasing functions, [10,
Corollary 13.6.4], that m /n converges P-a.s. to a deterministic linear process.
nt
By Proposition2.2 and the remarkleading to it we havethat Σ is the sum of i.i.d. centred random
m
variables with finite second moment. By Donsker’s theorem we therefore have that (Σ /√n) con-
nt t 0
≥
verges to a scaled Brownian motion. By continuity of composition at continuous limits, [10, Theorem
13.2.1], and the previous remarks we therefore have the following annealed central limit theorem.
Corollary 2.3. Under the assumptions of Theorem 1, there exists a constant σ2 > 0 such that the
process
X nνt
Bn := | ⌊nt⌋|−
t σ√n
converges in P-distribution on D([0, ),R) endowed with the Skorohod J topology to a standard Brow-
1
∞
nian motion.
Remark2.4. The branchofasubcriticalGW-treeconditioned tosurvivecanbeconstructedbyattaching
arandomnumberofsubcriticalGW-treestoarootvertex. In[4,Lemma4.12]itisshownthat,conditional
on having a single vertex in the first generation of the branch, the second moment of the first return time
to the root is infinite whenever β2µ˜ 1 where µ˜ is the mean of the subcritical GW-law. It therefore
≥
follows from this that
E E h (τ+)2 =
ρT ρ ∞
h h ii
whenever β2f (q) 1 and µ > 1. In particular, if we have that β2f (q) 1 then χ have infinite
′ ′ j
≥ ≥
second moments since Proposition 2.2 fails. In this case, we do not have a central limit theorem for
Σ from which it follows that Bn does not converge in distribution. This shows that the condition
m t
β2f (q)<1 is necessary for the annealed central limit theorem. We note here that when p =0 we have
′ 0
that q =0=f (q) and, therefore, this condition is necessarily satisfied.
′
3 A quenched invariance principle
We now extend Corollary 2.3 to a quenched functional central limit theorem. For each n N write
Bn(X) to be the linear interpolation satisfying ∈
t
X kν
Bn (X)= | k|−
k/n σ√n
for k N. We then have that Bn = Bn for t > 0 such that nt N and Bn Bn n 1/2(ν +1)/σ
∈ t t ∈ | t − t| ≤ −
therefore it suffices to consider the interpolation. To begin, we prove the following lemma which is the
analogue of [3, Lemma 4.1] and follows by the same method.
Lemma 3.1. Suppose that the assumptions of Theorem 1 hold and that for any bounded Lipschitz
7
function F :C([0,T],R) R and b (1,2) we have that
→ ∈
VarP ET F B⌊bk⌋ < . (3.1)
∞
kX≥1 (cid:16) h (cid:16) (cid:17)i(cid:17)
Then, for P-a.e. , the process (Bn) converges in P -distribution on D([0, ),R) endowed with the
T t t≥0 T ∞
Skorohod J topology to a standard Brownian motion.
1
Proof. SupposethatforanyboundedLipschitzfunctionF :C([0,T],R) Randb (1,2)wehavethat
→ ∈
P-a.s.
ET[F(B⌊bk⌋)] E[F(B)] (3.2)
→
whereB isastandardBrownianmotion. Foranyδ,T >0,thefunctionF (ω):=sup ω(s) ω(t) 1:
T,δ
{| − |∧
s,t T, t s δ is bounded and Lipschitz; furthermore, for P-a.e.
≤ | − |≤ } T
limlimsupE [F (B bk )]=0 (3.3)
T T,δ ⌊ ⌋
δ→0 k→∞
since,bypropertiesofBrownianmotion,E[F (B)] 0asδ 0. Inparticular,byMarkov’sinequality
T,δ
→ →
we then have that for any ε>0
δli→m0likm→s∞upPT ss,stu≤tpT:δ|B⌊sbk⌋−B⌊tbk⌋|>ε ≤ δli→m0likm→s∞upε−1ET hFT,δ(B⌊bk⌋)i = 0
| − |≤ 
 
which gives tightness of (B⌊bk⌋)∞k=1 under PT.
For n N let kn deno·te the unique integer such that bkn n < bkn+1 then, by Markov’s
∈ ⌊ ⌋ ≤ ⌊ ⌋
inequality and the definition of the interpolation, we have that for any ε (0,1)
∈
limlimsupP sup Bn Bn >ε limlimsupE sup Bn Bn 1 ε 1
δ→0 k→∞ T ss,t≤tT:δ| s − t| ≤δ→0 n→∞ T ss,t≤tT:δ| s − t|∧  −
| − |≤  | − |≤ 
   
≤δli→m0linm→s∞upET |ss,−stu≤t|pT≤:δ(cid:12)(cid:12)(cid:12)B⌊sb⌊kbnknn⌋⌋ −B⌊tb⌊kbknnn⌋⌋(cid:12)(cid:12)(cid:12)∧1ε−1
 (cid:12) (cid:12) 
limlimsupE sup B bk B bk 1 ε 1
≤δ→0 k→∞ T uu,vv≤2T2:δ| ⌊u ⌋− ⌊v ⌋|∧  −
| − |≤ 
 
since b<2 implies that s n t n <2δ whenever s t <δ. In particular, the above expression
| bkn − bkn | | − |
is equal to 0 by (3.3) there⌊fore⌋the ⌊laws⌋of Bn are tight under P .
T
Let F be bounded and Lipschitz; without· loss of generality we may assume F , F 1. For
Lip
n N we have that || ||∞ || || ≤
∈
limsup E [F(Bn)] E [F(B bkn )]
T T ⌊ ⌋
| − |
n
→∞
≤||F||LiplimsupET sup|Bns −B⌊sbkn⌋|∧1
n→∞ (cid:20)s≤T (cid:21)
≤linmsupET "ssupT(cid:12)r⌊bnkn⌋B⌊sb⌊kbnknn⌋⌋ −B⌊sbkn⌋(cid:12)∧1#
→∞ ≤ (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
8
≤linmsupET "ssupT(cid:12) r⌊bnkn⌋ −1!B⌊sb⌊kbnknn⌋⌋(cid:12)∧1#+ET  s,stupT: B⌊sbkn⌋−B⌊tbkn⌋ ∧1
→∞ ≤ (cid:12) (cid:12) s t ≤T(b 1)(cid:12) (cid:12)
(cid:12) (cid:12) | − |≤ − (cid:12) (cid:12) 
(cid:12) (cid:12)  (cid:12) (cid:12) 
1 b−1/2 E su(cid:12)p Bs +limsupET FT(cid:12),T(b 1)(B⌊bk⌋)
≤| − | (cid:20)s≤bT| |(cid:21) k→∞ h − i
which converges to 0 as b 1. In particular, when (3.2) holds for any F : C([0,T],R) R with
→ →
F , F 1 and b (1,2) P-a.s. we have that
Lip
|| ||∞ || || ≤ ∈
E [F(Bn)] E[F(B)]. (3.4)
T
→
Bounded Lipschitz functions are separable and dense in the space of continuous bounded functions
thereforewehavethat(3.4)holdsP-a.s. F C (C([0,T],R))whichcompletesthe quenchedfunctional
b
∀ ∈
CLT. It therefore remains to show that (3.1) implies (3.2).
By Corollary 2.3 we have that for any bounded Lipschitz function,
E E F(B bk ) =E F(B bk ) E[F(B)]
T ⌊ ⌋ ⌊ ⌋
→
h h ii h i
as k therefore, it suffices to show that for P-a.e. tree we have that E[E [F(B bk )]]
T ⌊ ⌋
→ ∞ T | −
E [F(B bk )] converges to 0. By Chebyshev’s inequality, for ε>0,
T ⌊ ⌋
|
P E[ET[F(B⌊bk⌋)]] ET[F(B⌊bk⌋)] >ε ε−2VarP ET F B⌊bk⌋ .
| − | ≤
(cid:16) (cid:17) (cid:16) h (cid:16) (cid:17)i(cid:17)
The result follows from Borel-Cantelli and (3.1).
We now complete the proof of the quenched functional CLT by following the method used in [8] to
show that condition (3.1) holds for any bounded Lipschitz function F : C([0,T],R) R and b (1,2)
→ ∈
under the assumptions of the theorem.
Proof of Theorem 1. For a fixed tree , let X1,X2 be independent β-biasedwalks on and Y1,Y2 the
corresponding backbone walks. For iT=1,2, k N and t,s 0 let T
∈ ≥
Bkt,,si =B⌊tbk⌋(X·i+s)−B⌊tbk⌋(Xsi)
be a random variable with law of the interpolation B bk started from the vertex Xi. Define
⌊ ⌋ s
ϑYi :=min m> bk/4 :m ζYi and ϑXi =min m 0:Xi =Yi
k { ⌊ ⌋ ∈{ j }j≥1} k ≥ m ϑYki
n o
to be the first regenerationtime of Yi after time bk/4 and the corresponding time for Xi.
⌊ ⌋
Let
1 := Y1 : s ϑY1 Y2 =φ = X1 : s ϑX1 X2 =φ ,
Ak { s ≤ k }∩{ ϑY2} { s ≤ k }∩{ ϑX2}
k k
n o n o
2 := Y2 : s ϑY2 Y1 =φ = X2 : s ϑX2 X1 =φ
Ak { s ≤ k }∩{ ϑY1} { s ≤ k }∩{ ϑX1}
k k
n o n o
and := 1 2 be the event that, after the first regeneration times after time bk/4 , the paths of
Ak Ak ∩Ak ⌊ ⌋
Y1,Y2 donotintersect. Write k,i := ϑYi bk/3 tobethe eventthatthefirstregenerationaftertime
B { k ≤ }
bk/4 happens before time bk/3.
Recallthatforx g we denoteby ( ) the heightofthe branchattachedtothe vertexx. Using
Lipschitz properties ∈ofTBk,i we have thatH Tx−
sup Bk,i Bk,i sup b k/2 Xi mν Xi +(m+ϑXi)ν+ Xi ϑXiν
t≤T(cid:12)(cid:12)(cid:12)(cid:12) t,0− t,ϑXki(cid:12)(cid:12)(cid:12)(cid:12)=≤ms≤uTpbkb−−k/2(cid:12)(cid:12)(cid:12)||Xmmi ||−−|Xmi−+ϑ|Xim|++ϑ|XkXiϑ|iXi| k | ϑXki|− k (cid:12)(cid:12)(cid:12)
m Tbk k k
≤ (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
9
b k/2 max Yi Yi + Yi +b k/2 i
≤ − m Tbk | m|−| m+ϑYki| | ϑYki| − HTbk
≤ (cid:12)(cid:12) (cid:12) (cid:12)
(cid:12)(cid:12) (cid:12) (cid:12)
where i istheheightofthetallestbra(cid:12)(cid:12)nchseenbytime(cid:12)Tbk byY(cid:12)i. BytimeTbk thewalkYi canvisit
HTbk
atmostTbk+1 unique vertices. Atthe firsthitting time ofa vertex,the budandbackbone distribution
from this vertex are independent of the past; therefore, by (2.5)
P i Clog(bk) C bkP( ( ) Clog(bk)) C bkf (q)Clog(bk) C b k (3.5)
HTbk ≥ ≤ T H Tρ− ≥ ≤ T ′ ≤ T −
for C su(cid:0)fficiently large. Fu(cid:1)rthermore, by the Lipschitz property of Yi we have that
b k/2 max Yi Yi + Yi 2ϑYib k/2
− m Tbk | m|−| m+ϑYki| | ϑYki| ≤ k −
≤ (cid:12)(cid:12) (cid:12) (cid:12)
(cid:12)(cid:12) (cid:12) (cid:12)
which is bounded above by 2b k/6 on(cid:12)(cid:12)the event k,i. L(cid:12)etting k,(cid:12)i := i <Clog(bk) , we then have
− B C {HTbk }
that, on the event k,i k,i,
B ∩C
(cid:12)(cid:12)F (cid:16)Bk·,,0i(cid:17)−F (cid:18)Bk·,,ϑiXki(cid:19)(cid:12)(cid:12)≤Cb−k/6
for any bounded Lipschitz function(cid:12) F :C([0,T],R) R. (cid:12)
(cid:12) → (cid:12)
Using the Lipschitz and boundedness properties of F, we then have that
VarP ET F B⌊bk⌋
(cid:16) h (cid:16) (cid:17)2i(cid:17) 2
=E E F(B bk ) E E F(B bk )
T ⌊ ⌋ T ⌊ ⌋
−
(cid:20) h i (cid:21) h h ii
=E F(Bk,1)F(Bk,2) E F(Bk,1) E F(Bk,2)
−
C(cid:2) P ( k,1)c +P(cid:3)( k,1(cid:2))c +b k(cid:3)/6 (cid:2)+E F (cid:3)Bk,1 F Bk,2 E F Bk,1 E F Bk,2 .
≤ B C − ,ϑX1 ,ϑX2 − ,ϑX1 ,ϑX2
(cid:16) (cid:0) (cid:1) (cid:0) (cid:1) (cid:17) (cid:20) (cid:18) · k (cid:19) (cid:18) · k (cid:19)(cid:21) (cid:20) (cid:18) · k (cid:19)(cid:21) (cid:20) (cid:18) · k (cid:19)(cid:21)
On the event we have that Bk,1 , Bk,2 are independent therefore
Ak ,ϑX1 ,ϑX2
· k · k
E F Bk,1 F Bk,2 E F Bk,1 E F Bk,2 =0.
,ϑX1 ,ϑX2 |Ak − ,ϑX1 |Ak ,ϑX2 |Ak
(cid:20) (cid:18) · k (cid:19) (cid:18) · k (cid:19) (cid:21) (cid:20) (cid:18) · k (cid:19) (cid:21) (cid:20) (cid:18) · k (cid:19) (cid:21)
Using the Lipschitz property of F we then have that
VarP ET F B⌊bk⌋ C P ( k,1)c +P ( k,1)c +P ( k,1)c +b−k/6 .
≤ A B C
(cid:16) h (cid:16) (cid:17)i(cid:17) (cid:16) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:17)
For i=1,2 we have that Yi are biased random walks on a supercriticalGW-tree without leaves g,
whoseoffspringlawhasexponentialmoments. ItfollowsthattheestimatesP(( k,1)c),P(( k,1)c) bTc˜k
−
A B ≤
given in the proof of [8, Theorem 3] still hold. Combining these with (3.5) we have that there exists
c>0 such that for k sufficiently large
VarP ET F B⌊bk⌋ Cb−ck
≤
(cid:16) h (cid:16) (cid:17)i(cid:17)
which shows (3.1) and therefore the result follows from Lemma 3.1.
Acknowledgements
I would like to thank my supervisor David Croydon for suggesting the problem, his support and many
useful discussions. This work is supported by EPSRC as part of the MASDOC DTC at the University
of Warwick. Grant No. EP/H023364/1.
10