A quenched central limit theorem for biased random walks on supercritical Galton-Watson trees
aa r X i v : . [ m a t h . P R ] J a n A quenched central limit theorem for biased random walks onsupercritical 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 ona supercritical Galton-Watson tree with leaves. This extends a result of Peres and Zeitouni (2008)where the case without leaves is considered. A conjecture of Ben Arous and Fribergh (2016) suggestsan upper bound on the bias which we observe to be sharp.
We investigate biased random walks on supercritical Galton-Watson trees with leaves. A GW-treeconditioned to survive consists of an infinite backbone structure with finite trees attached as branches.The backbone structure is a GW-tree whose offspring law does not have deaths. The branches are formedby attaching a random number of independent GW-trees (conditioned to die out) to each vertex of thebackbone. This forms dead-ends in the environment which makes it a natural setting for observingtrapping as the walk is slowed by taking excursions in the branches of the tree. The number of treesattached to a given vertex on the backbone has a distribution depending on the backbone locally. Thismeans that the branches are not i.i.d. and therefore the trapping incurred by the addition of the leavesdemonstrates a significant complication to the model without leaves studied in [8].The influence of the bias on the trapping is an important feature of the model. As the bias is increased,the local drift away from the root will increase but this does not necessarily speed up the walk. Thisis because it increases the time trapped in the finite leaves from which the walk cannot escape withouttaking long sequences of movements against the bias. In [7] it is shown that, for a suitably large bias, thetrapping is sufficient to slow the walk to zero speed whereas, for small bias, the expected trapping timeis finite and the walk has a positive limiting speed. Under a further restriction on the bias the trappingtimes have finite variance; we use this to prove a quenched invariance principle for the walk.We next describe the supercritical GW-tree in greater detail. Let { p k } denote the offspring distribu-tion of a GW-process Z n with a single progenitor, mean µ > f .The process Z n gives rise to a random tree T f where individuals are represented by vertices and edgesconnect individuals with their offspring. We denote by ρ the root, which is the vertex representing theunique progenitor. Let q denote the extinction probability of Z n which is non-zero only when p >
0. Inthis case we then define g ( s ) := f ( s ) − f ( qs )1 − q and h ( s ) := f ( qs ) 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-treeconditioned on nonextinction T can be generated by first generating a g -GW-tree T g and then, to eachvertex x of T g , appending a random number M x of independent h -GW-trees. We refer to T g as thebackbone of T and the finite trees appended to T g as the traps.We now introduce the biased random walk on a fixed tree. For T fixed with x ∈ T let ←− x denote theparent of x and c ( x ) the set of children of x . A β -biased random walk on T is a random walk ( X n ) n ≥ MSC2010 subject classifications:
Primary 60K37, 60F05, 60F17; secondary 60J80.
Keywords:
Random walk, random environment, Galton-Watson tree, quenched, functional central limit theorem, invarianceprinciple.
1n the vertices of T which is β -times more likely to make a transition to a given child of the currentvertex than the parent (which are the only options). More specifically, the random walk started from z ∈ T is the Markov chain defined by P T z ( X = z ) = 1 and the transition probabilities P T z ( X n +1 = y | X n = x ) = β | c ( x ) | , if y = ←− x , β β | c ( x ) | , if y ∈ c ( x ) , x = ρ, d ρ , if y ∈ c ( x ) , x = ρ, , otherwise.We use P ρ ( · ) := R P T ρ ( · ) P (d T ) for the annealed law obtained by averaging the quenched law P T ρ overthe law P on f -GW-trees conditioned to survive. Unless indicated otherwise, we start the walk at ρ .For x ∈ T , let | x | := d ( ρ, x ) denote the distance between x and the root of the tree. It has beenshown in [7] that if β ∈ ( µ − , f ′ ( q ) − ) then | X n | n − converges P -a.s. to a deterministic constant ν > β < µ − the walk is recurrent and | X n | n − converges P -a.s. to 0.When the bias is large the walk is transient but slowed by having to make long sequences of movementsagainst the bias in order to escape the traps; in particular, if β ≥ f ′ ( q ) − then the slowing affect isstrong enough to cause | X n | n − to converge P -a.s. to 0. This regime has been studied further by [2] and[5] where polynomial scaling results are shown.For σ, t > n = 1 , , ... define B nt := | X ⌊ nt ⌋ | − nνtσ √ n . Our main result, Theorem 1, is a quenched invariance principle for B nt . Theorem 1.
Suppose p > , µ > , β ∈ ( µ − , f ′ ( q ) − / ) and that there exists λ > such that X k ≥ λ k p k < ∞ . (1.1) There exists σ > such that, for P -a.e. T , we have that the process ( B nt ) t ≥ converges in P T -distributionon D ([0 , ∞ ) , R ) endowed with the Skorohod J topology to a standard Brownian motion. The condition p > p = 0, is considered in[8]. This no longer requires the condition β < f ′ ( q ) − / which is due to the trapping in the dead-endscaused by the leaves. Indeed, when p = 0 we have that q = 0 and f ′ (0) = 0 therefore the conditionbecomes irrelevant. This regime is studied further in [8] to the case where β = µ − and p = 0; in thissetting ν = 0 and it is shown that B nt converges in distribution to the absolute value of a Brownianmotion. This result is extended in [6] to random walks on multi-type GW-trees with leaves. Althoughthe dead-ends in this model trap the walk, the bias is small and therefore the slowing is weak.By choosing p > β < f ′ ( q ) − / is the correct upper bound on the bias so that the trappingtimes in our model have finite variance. We conclude, in Remark 2.4, that this upper bound is necessarywhich confirms [1, Conjecture 3.1].We assume that the exponential moments condition (1.1) holds throughout. This is a purely technicalassumption which we expect could be relaxed to a sufficiently large moment condition however the mainfocus 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 annealedinvariance 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 aresignificantly influenced by the specific instance of the environment. This results in an environmentdependent centring for the walk in the quenched central limit theorem. In the supercritical case thewalk will randomly choose one of infinitely many escape routes; this creates a mixing of the environmentwhich 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 byadapting the renewal argument used in [9, Theorem 4.1]. This is then extended to the quenched result2heorem 1 in Section 3 by applying the argument used in [3] which largely involves showing that multiplecopies of the walk see sufficiently different areas of the tree. We begin this section by showing that the time spent in a branch has finite variance. Let T h be the treeformed by attaching an additional vertex ρ (as the parent of the root ρ ) to an h -GW-tree T h . For a fixedtree T and vertex x ∈ T let τ + x := inf { k > X k = x } denote the first return time to x . Let ξ f , ξ g , ξ h be random variables with probability generating functions f, g and h respectively then let ξ be equal indistribution to the number of vertices in the first generation of T . Since the generation sizes of T g aredominated by those of T we have that ξ g is stochastically dominated by ξ . Using Bayes’ law we havethat P ( ξ = k ) = p k (1 − q k )(1 − q ) − ≤ cp k therefore both ξ and ξ g inherit the exponential moments of ξ f . Furthermore P ( ξ h = k ) = p k q k therefore ξ h automatically has exponential moments. Lemma 2.1.
Suppose that p > , µ > and β ∈ ( µ − , f ′ ( q ) − / ) , then we have that E (cid:20) E T h ρ (cid:20)(cid:16) τ + ρ (cid:17) (cid:21)(cid:21) < ∞ . Proof.
We can write τ + ρ = X x ∈T h v x where v x = τ + ρ X k =1 { X k = x } is the number of visits to x before returning to ρ . Recall that c ( x ) denotes the set of children of x . Itthen follows that E (cid:20) E T h ρ (cid:20)(cid:16) τ + ρ (cid:17) (cid:21)(cid:21) = E X x,y ∈T h E T h ρ [ v x v y ] ≤ C β E X x,y ∈T h ( | c ( x ) | β + 1)( | c ( y ) | β + 1) β | x | + | y | = C β E X x ∈T h ( | c ( x ) | β + 1) β | x | X y ∈T h ( | c ( y ) | β + 1) β | y | (2.1)where the inequality follows from [4, Lemma 4.8]. Letting Z hk denote the size of the k th generation of T h and collecting terms in each generation we have that X x ∈T h ( | c ( x ) | β + 1) β | x | = 1 + X k ≥ Z hk ( β k + β k − ) ≤ (1 + β − ) X k ≥ Z hk β k . By [4, Lemma 4.1], since ξ h has exponential moments, we have that E [ Z hk Z hj ] ≤ Cf ′ ( q ) j whenever j ≥ k .Substituting this and the above inequality into (2.1) we have that E (cid:20) E T h ρ (cid:20)(cid:16) τ + ρ (cid:17) (cid:21)(cid:21) ≤ C β X k ≥ β k X j ≥ k E [ Z hk Z hj ] β j ≤ C β X k ≥ β k X j ≥ k ( f ′ ( q ) β ) j ≤ C β,f ′ ( q ) X k ≥ ( f ′ ( q ) β ) k which is finite by the assumption that β < f ′ ( q ) − / .3et r (0) := 0, r ( n ) := inf { k > r ( n −
1) : X k , X k − ∈ T g } for n ≥ Y n := X r ( n ) , then Y n is a β -biased random walk on T g coupled to X n . Write ζ Y := 0 and for m = 1 , , ... let ζ Ym := inf { k > ζ Ym − : | Y j | < | Y k | ≤ | Y l | for all j < k ≤ l } be regeneration times for the backbone walk. We can then define ζ Xk := inf { m ≥ X m = Y ζ Yk } to bethe corresponding regeneration times for X . By [7, Proposition 3.4] we have that there exists, P -a.s., aninfinite sequence of regeneration times { ζ Xk } k ≥ and the sequence n(cid:0) ζ Xk +1 − ζ Xk (cid:1) , (cid:16)(cid:12)(cid:12)(cid:12) X ζ Xk +1 (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) X ζ Xk (cid:12)(cid:12)(cid:12)(cid:17)o k ≥ is i.i.d. (as is the corresponding sequence for Y ). Furthermore, letting m t := sup { j ≥ ζ Xj ≤ t } be thenumber of regenerations by time t , we have that m t is non-decreasing and diverges P -a.s.By [7, Theorems 3.1 & 4.1], whenever µ > µ − < β < f ′ ( q ) − we have that there exists ν ∈ (0 ,
1) such that | X n | n − converges P -a.s. to ν . Moreover, combined with [7, Corollary 3.5], we havethat the time and distance between regenerations of X both have finite means with respect to P . Let χ j := X ζ Xj − X ζ Xj − − ν ( ζ Xj − ζ Xj − ) = Y ζ Yj − Y ζ Yj − − ν ( ζ Xj − ζ Xj − ) . By the previous remark we have that χ j are i.i.d. with respect to P . By the strong law of large numbersand the definition of ν we have that χ j are centred (see [7, Theorems 3.1 & 4.1]). We will show that χ j have finite second moment and that their sumΣ m := m X j =2 χ j = (cid:0) X ζ Xm − νζ Xm (cid:1) − (cid:16) X ζ X − νζ X (cid:17) can be used to approximate B nt .By the remark preceding Lemma 2.1, the offspring distribution ξ g of T g has exponential moments.Since Y is a random walk on T g , by [8, Proposition 3] we have that E [( ζ Y − ζ Y ) k ] < ∞ for all k ∈ Z whenever β > µ − .Let η k := r ( k + 1) − r ( k ) denote the total time taken between X making the k th and ( k + 1) th transition along the backbone. This time consists of N k := r ( k +1) X j = r ( k )+1 { X j = Y k } excursions into the finite trees appended to the backbone at this vertex and one additional step to thenext backbone vertex. Therefore, we can write η k := 1 + N k X j =1 γ k,j where γ k,j := r ( k +1) X i = r ( k ) { P il = r ( k ) { Xl = Yk } = j } (2.2)is the duration of the j th such excursion. Proposition 2.2.
Under the assumptions of Theorem 1 we have that E [( ζ X − ζ X ) ] < ∞ . Proof.
The law of ζ X − ζ X under P is identical to its law under P ρ ( ·| ζ Y = 1). That is, by the independencestructure, we can condition the first regeneration vertex to be the first vertex reached by Y withoutchanging the law of ζ X − ζ X . We therefore have that E (cid:2) ( ζ X − ζ X ) (cid:3) can be written as E (cid:2) ( ζ X − ζ X ) (cid:12)(cid:12) ζ Y = 1 (cid:3) = E ζ Y − ζ Y X k =1 η k (cid:12)(cid:12)(cid:12) ζ Y = 1 ≤ E ( ζ Y − ζ Y ) ζ Y − ζ Y X k =1 η k (cid:12)(cid:12)(cid:12) ζ Y = 1
4y convexity. Using convexity again with the decomposition (2.2) we can write this as E ( ζ Y − ζ Y ) ζ Y − ζ Y X k =1 N k X j =1 γ k,j (cid:12)(cid:12)(cid:12) ζ Y = 1 ≤ E ( ζ Y − ζ Y ) ζ Y − ζ Y X k =1 ( N k + 1) N k X j =1 γ k,j (cid:12)(cid:12)(cid:12) ζ Y = 1 . By conditioning on the backbone, buds and the walk on the backbone and buds we have that theindividual excursion times are independent of the regeneration times of Y and the number of excursions.The excursion times are also distributed as the first return time to ρ for a walk started from ρ on T h .We therefore have that the above expectation can be bounded above by E h E T h ρ h ( τ + ρ ) ii E ( ζ Y − ζ Y ) ζ Y − ζ Y X k =1 ( N k + 1) (cid:12)(cid:12)(cid:12) ζ Y = 1 . Where, by Lemma 2.1, we have that E h E T h ρ h ( τ + ρ ) ii < ∞ .Let ( z j ) ∞ j =0 denote the ordered distinct vertices visited by Y and L ( z, j ) := j X i =0 { Y j = z } , L ( z ) := L ( z, ∞ )the local times of the vertex z . Write W z,l := ∞ X j =0 { X j = z, X j +1 / ∈T g , L ( z,j )= l } to be the number of excursions from z (by X ) on the l th visit to z (by Y ) for l = 1 , ..., L ( z ) and M := |{ Y k } ζ Y − k =1 | the number of distinct vertices visited by Y between time 1 and time ζ Y − E ( ζ Y − ζ Y ) ζ Y − ζ Y X k =1 ( N k + 1) (cid:12)(cid:12)(cid:12) ζ Y = 1 = E ( ζ Y − ζ Y ) M X k =1 L ( z k ) X l =1 ( W z k ,l + 1) (cid:12)(cid:12)(cid:12) ζ Y = 1 = ∞ X k =1 ∞ X l =1 E (cid:2) ( ζ Y − ζ Y ) { k ≤ M, l ≤ L ( z k ) } ( W z k ,l + 1) | ζ Y = 1 (cid:3) ≤ ∞ X k =1 ∞ X l =1 (cid:16) E (cid:2) ( ζ Y − ζ Y ) { k ≤ M, l ≤ L ( z k ) } | ζ Y = 1 (cid:3) E (cid:2) ( W z k ,l + 1) | ζ Y = 1 (cid:3) (cid:17) / (2.3)by Cauchy-Schwarz. Conditional on ζ Y = 1, for all 1 ≤ k ≤ M we have that L ( z k ) ≤ ζ Y − ζ Y ; moreover, M ≤ ζ Y − ζ Y therefore { k ≤ M, l ≤ L ( z k ) } ≤ { k,l ≤ ζ Y − ζ Y } . Since the root does not have a parent, without any further information concerning the numberof children from a given vertex, we have that the walk is more likely to take an excursion into oneof the neighbouring traps when at the root than from this vertex. We can, therefore, stochasticallydominate the number of excursions from a vertex by the number of excursions from the root to see that E (cid:2) ( W z k ,l + 1) (cid:3) ≤ E (cid:2) ( W z , + 1) (cid:3) . Using this, Cauchy-Schwarz and that P ( ζ Y = 1) >
0, the expression(2.3) can be bounded above by P ( ζ Y = 1) − ∞ X k =1 ∞ X l =1 (cid:16) E h ( ζ Y − ζ Y ) { k,l ≤ ζ Y − ζ Y } i E (cid:2) ( W z k ,l + 1) (cid:3)(cid:17) / C E (cid:2) ( ζ Y − ζ Y ) (cid:3) / E (cid:2) ( W z , + 1) (cid:3) / ∞ X k =1 ∞ X l =1 P (cid:0) k, l ≤ ζ Y − ζ Y (cid:1) / . Since the offspring distribution ξ g has exponential moments we have that the time between regenerationshas finite fourth moments by [8, Proposition 3]. That is, E (cid:2) ( ζ Y − ζ Y ) (cid:3) < ∞ .Write Z n and Z gn to be the GW-processes associated with T and T g . The number of excursions fromthe root is geometrically distributed with termination probability 1 − p ex where p ex := Z − Z g Z . Using properties of geometric random variables we therefore have that E (cid:2) ( W z , + 1) (cid:3) ≤ C E [(1 − p ex ) − ] ≤ C E [ Z ] < ∞ since Z d = ξ which has exponential moments.It remains to show that ∞ X k =1 ∞ X l =1 P (cid:0) k, l ≤ ζ Y − ζ Y (cid:1) / (2.4)is finite. Note that P (cid:0) k, l ≤ ζ Y − ζ Y (cid:1) = P (cid:0) ζ Y − ζ Y ≥ l (cid:1) whenever l ≥ k . Using Chebyshev’s inequalitywe can then bound (2.4) above by2 ∞ X k =1 ∞ X l = k P (cid:0) ζ Y − ζ Y ≥ l (cid:1) / ≤ ∞ X k =1 ∞ X l = k E h(cid:0) ζ Y − ζ Y (cid:1) j i l j / for any integer j . In particular, by [8, Proposition 3] we have that E h(cid:0) ζ Y − ζ Y (cid:1) j i is finite for any integer j . Choosing j > x ∈ T let T x denote the subtree consisting of all descendants of x in T . Then, for y ∈ T g , let T − y be the branch at y ; that is, the subtree rooted at y consisting only of y , the children of y not on T g andtheir descendants. The tree T − y then has the law of a tree rooted at y with some random number M − y of h -GW-trees attached to y . Since M − y is dominated by ξ , by (1.1) we have that M − y has exponentialmoments. It therefore follows from [7, Theorem B] that there exists some constant C such that P ( H ( T − y ) ≥ n ) ≤ Cf ′ ( q ) n (2.5)where, for a fixed rooted tree T , H ( T ) := sup { d ( ρ, x ) : x ∈ T } is the height of T . Let H n :=max {H ( T − y ) : y ∈ { Y k } nk =0 } denote the largest branch seen by Y by time n . It follows thatsup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) B nt − Σ m tn σ √ n (cid:12)(cid:12)(cid:12)(cid:12) ≤ | X ζ X | + νζ X + H nT σ √ n + sup j =1 ,...,m nT | Y ζ Yj +1 | − | Y ζ Yj | + ν ( ζ Xj +1 − ζ Xj ) σ √ n . (2.6)Up to time nT , the walk Y can have visited at most nT vertices on T g therefore the probability that X has visited a branch of height at least C log( n ) by time nT is at most C T nf ′ ( q ) C log( n ) . In particular,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 C log( n ) by time nT . Since X ζ X and ζ X do not depend on n and have finite mean, we have that the first term in (2.6) converges P -a.s. to 0.By [8, Proposition 3], for any k ∈ Z + we have that E [( | Y ζ Y | − | Y ζ Y | ) k ] < ∞ therefore the distancebetween regeneration points is small. In particular, bounding m nT above by nT , using a union boundand Markov’s inequality we have that for any ε > P sup j =1 ,...,m nT | Y ζ Yj +1 | − | Y ζ Yj | σ √ n > ε ! ≤ C T,ε E (cid:20)(cid:16) | Y ζ Y | − | Y ζ Y | (cid:17) n | Y ζY |−| Y ζY | >ε √ n o (cid:21) n → ∞ by dominated convergence. Similarly, using Proposition 2.2, we havethat the same holds for the supremum of ζ Xj +1 − ζ Xj ; therefore, we have that P sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) B nt − Σ m tn σ √ n (cid:12)(cid:12)(cid:12)(cid:12) > ε ! converges to 0 as n → ∞ .By the law of large numbers and that ζ X /n converges P -a.s. to 0 we have that ζ Xn = ζ X + n X k =2 ( ζ Xk − ζ Xk − )converges P -a.s. It therefore follows by continuity of the inverse at strictly increasing functions, [10,Corollary 13.6.4], that m nt /n converges P -a.s. to a deterministic linear process.By Proposition 2.2 and the remark leading to it we have that Σ m is the sum of i.i.d. centred randomvariables with finite second moment. By Donsker’s theorem we therefore have that (Σ nt / √ n ) t ≥ con-verges to a scaled Brownian motion. By continuity of composition at continuous limits, [10, Theorem13.2.1], 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 σ > such that theprocess B nt := | X ⌊ nt ⌋ | − nνtσ √ n converges in P -distribution on D ([0 , ∞ ) , R ) endowed with the Skorohod J topology to a standard Brow-nian motion. Remark 2.4.
The branch of a subcritical GW-tree conditioned to survive can be constructed by attachinga random number of subcritical GW-trees to a root vertex. In [4, Lemma 4.12] it is shown that, conditionalon having a single vertex in the first generation of the branch, the second moment of the first return timeto the root is infinite whenever β ˜ µ ≥ where ˜ µ is the mean of the subcritical GW-law. It thereforefollows from this that E h E T h ρ h ( τ + ρ ) ii = ∞ whenever β f ′ ( q ) ≥ and µ > . In particular, if we have that β f ′ ( q ) ≥ then χ j have infinitesecond moments since Proposition 2.2 fails. In this case, we do not have a central limit theorem for Σ m from which it follows that B nt does not converge in distribution. This shows that the condition β f ′ ( q ) < is necessary for the annealed central limit theorem. We note here that when p = 0 we havethat q = 0 = f ′ ( q ) and, therefore, this condition is necessarily satisfied. We now extend Corollary 2.3 to a quenched functional central limit theorem. For each n ∈ N write B nt ( X ) to be the linear interpolation satisfying B nk/n ( X ) = | X k | − kνσ √ n for k ∈ N . We then have that B nt = B nt for t > nt ∈ N and | B nt − B nt | ≤ n − / ( ν + 1) /σ therefore it suffices to consider the interpolation. To begin, we prove the following lemma which is theanalogue 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 unction F : C ([0 , T ] , R ) → R and b ∈ (1 , we have that X k ≥ Var P (cid:16) E T h F (cid:16) B ⌊ b k ⌋ (cid:17)i(cid:17) < ∞ . (3.1) Then, for P -a.e. T , the process ( B nt ) t ≥ converges in P T -distribution on D ([0 , ∞ ) , R ) endowed with theSkorohod J topology to a standard Brownian motion.Proof. Suppose that for any bounded Lipschitz function F : C ([0 , T ] , R ) → R and b ∈ (1 ,
2) we have that P -a.s. E T [ F ( B ⌊ b k ⌋ )] → E [ F ( B )] (3.2)where B is a standard Brownian motion. For any δ, T >
0, the function F T,δ ( ω ) := sup {| ω ( s ) − ω ( t ) | ∧ s, t ≤ T, | t − s | ≤ δ } is bounded and Lipschitz; furthermore, for P -a.e. T lim δ → lim sup k →∞ E T [ F T,δ ( B ⌊ b k ⌋ )] = 0 (3.3)since, by properties of Brownian motion, E [ F T,δ ( B )] → δ →
0. In particular, by Markov’s inequalitywe then have that for any ε > δ → lim sup k →∞ P T sup s,t ≤ T : | s − t |≤ δ | B ⌊ b k ⌋ s − B ⌊ b k ⌋ t | > ε ≤ lim δ → lim sup k →∞ ε − E T h F T,δ ( B ⌊ b k ⌋ ) i = 0which gives tightness of ( B ⌊ b k ⌋· ) ∞ k =1 under P T .For n ∈ N let k n denote the unique integer such that ⌊ b k n ⌋ ≤ n < ⌊ b k n +1 ⌋ then, by Markov’sinequality and the definition of the interpolation, we have that for any ε ∈ (0 , δ → lim sup k →∞ P T sup s,t ≤ T : | s − t |≤ δ | B ns − B nt | > ε ≤ lim δ → lim sup n →∞ E T sup s,t ≤ T : | s − t |≤ δ | B ns − B nt | ∧ ε − ≤ lim δ → lim sup n →∞ E T sup s,t ≤ T : | s − t |≤ δ (cid:12)(cid:12)(cid:12)(cid:12) B ⌊ b kn ⌋ s n ⌊ bkn ⌋ − B ⌊ b kn ⌋ t n ⌊ bkn ⌋ (cid:12)(cid:12)(cid:12)(cid:12) ∧ ε − ≤ lim δ → lim sup k →∞ E T sup u,v ≤ T : | u − v |≤ δ | B ⌊ b k ⌋ u − B ⌊ b k ⌋ v | ∧ ε − since b < | s n ⌊ b kn ⌋ − t n ⌊ b kn ⌋ | < δ whenever | s − t | < δ . In particular, the above expressionis equal to 0 by (3.3) therefore the laws of B n · are tight under P T .Let F be bounded and Lipschitz; without loss of generality we may assume || F || ∞ , || F || Lip ≤
1. For n ∈ N we have thatlim sup n →∞ | E T [ F ( B n )] − E T [ F ( B ⌊ b kn ⌋ )] |≤ || F || Lip lim sup n →∞ E T (cid:20) sup s ≤ T | B ns − B ⌊ b kn ⌋ s | ∧ (cid:21) ≤ lim sup n →∞ E T " sup s ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r ⌊ b k n ⌋ n B ⌊ b kn ⌋ s n ⌊ bkn ⌋ − B ⌊ b kn ⌋ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∧ lim sup n →∞ E T " sup s ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ⌊ b k n ⌋ n − ! B ⌊ b kn ⌋ s n ⌊ bkn ⌋ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∧ + E T sup s,t ≤ T : | s − t |≤ T ( b − (cid:12)(cid:12)(cid:12) B ⌊ b kn ⌋ s − B ⌊ b kn ⌋ t (cid:12)(cid:12)(cid:12) ∧ ≤ | − b − / | E (cid:20) sup s ≤ bT | B s | (cid:21) + lim sup k →∞ E T h F T,T ( b − ( B ⌊ b k ⌋ ) 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 || Lip ≤ b ∈ (1 , P -a.s. we have that E T [ F ( B n )] → E [ F ( B )] . (3.4)Bounded Lipschitz functions are separable and dense in the space of continuous bounded functionstherefore we have that (3.4) holds P -a.s. ∀ F ∈ C b ( C ([0 , T ] , R )) which completes the quenched functionalCLT. It therefore remains to show that (3.1) implies (3.2).By Corollary 2.3 we have that for any bounded Lipschitz function, E h E T h F ( B ⌊ b k ⌋ ) ii = E h F ( B ⌊ b k ⌋ ) i → E [ F ( B )]as k → ∞ therefore, it suffices to show that for P -a.e. tree T we have that | E [ E T [ F ( B ⌊ b k ⌋ )]] − E T [ F ( B ⌊ b k ⌋ )] | converges to 0. By Chebyshev’s inequality, for ε > P (cid:16) | E [ E T [ F ( B ⌊ b k ⌋ )]] − E T [ F ( B ⌊ b k ⌋ )] | > ε (cid:17) ≤ ε − Var P (cid:16) E T h F (cid:16) B ⌊ b k ⌋ (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] toshow that condition (3.1) holds for any bounded Lipschitz function F : C ([0 , T ] , R ) → R and b ∈ (1 , Proof of Theorem 1.
For a fixed tree T , let X , X be independent β -biased walks on T and Y , Y thecorresponding backbone walks. For i = 1 , , k ∈ N and t, s ≥ B k,it,s = B ⌊ b k ⌋ t ( X i · + s ) − B ⌊ b k ⌋ t ( X is )be a random variable with law of the interpolation B ⌊ b k ⌋ started from the vertex X is . Define ϑ Y i k := min { m > ⌊ b k/ ⌋ : m ∈ { ζ Y i j } j ≥ } and ϑ X i k = min n m ≥ X im = Y iϑ Y ik o to be the first regeneration time of Y i after time ⌊ b k/ ⌋ and the corresponding time for X i .Let A k := n { Y s : s ≤ ϑ Y k } ∩ { Y ϑ Y k } = φ o = n { X s : s ≤ ϑ X k } ∩ { X ϑ X k } = φ o , A k := n { Y s : s ≤ ϑ Y k } ∩ { Y ϑ Y k } = φ o = n { X s : s ≤ ϑ X k } ∩ { X ϑ X k } = φ o and A k := A k ∩ A k be the event that, after the first regeneration times after time ⌊ b k/ ⌋ , the paths of Y , Y do not intersect. Write B k,i := { ϑ Y i k ≤ b k/ } to be the event that the first regeneration after time b k/ happens before time b k/ .Recall that for x ∈ T g we denote by H ( T − x ) the height of the branch attached to the vertex x . UsingLipschitz properties of B k,i we have thatsup t ≤ T (cid:12)(cid:12)(cid:12)(cid:12) B k,it, − B k,it,ϑ Xik (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup m ≤ T b k b − k/ (cid:12)(cid:12)(cid:12) | X im | − mν − | X im + ϑ Xik | + ( m + ϑ X i k ) ν + | X iϑ Xik | − ϑ X i k ν (cid:12)(cid:12)(cid:12) = sup m ≤ T b k b − k/ (cid:12)(cid:12)(cid:12) | X im | − | X im + ϑ Xik | + | X iϑ Xik | (cid:12)(cid:12)(cid:12) b − k/ max m ≤ T b k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | Y im | − | Y im + ϑ Y ik | (cid:12)(cid:12)(cid:12) + | Y iϑ Y ik | (cid:12)(cid:12)(cid:12) + b − k/ H iT b k where H iT b k is the height of the tallest branch seen by time T b k by Y i . By time T b k the walk Y i can visitat most T b k + 1 unique vertices. At the first hitting time of a vertex, the bud and backbone distributionfrom this vertex are independent of the past; therefore, by (2.5) P (cid:0) H iT b k ≥ C log( b k ) (cid:1) ≤ C T b k P ( H ( T − ρ ) ≥ C log( b k )) ≤ C T b k f ′ ( q ) C log( b k ) ≤ C T b − k (3.5)for C sufficiently large. Furthermore, by the Lipschitz property of Y i we have that b − k/ max m ≤ T b k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | Y im | − | Y im + ϑ Y ik | (cid:12)(cid:12)(cid:12) + | Y iϑ Y ik | (cid:12)(cid:12)(cid:12) ≤ ϑ Y i k b − k/ which is bounded above by 2 b − k/ on the event B k,i . Letting C k,i := {H iT b k < C log( b k ) } , we then havethat, on the event B k,i ∩ C k,i , (cid:12)(cid:12)(cid:12)(cid:12) F (cid:16) B k,i · , (cid:17) − F (cid:18) B k,i · ,ϑ Xik (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cb − k/ for any bounded Lipschitz function F : C ([0 , T ] , R ) → R .Using the Lipschitz and boundedness properties of F , we then have thatVar P (cid:16) E T h F (cid:16) B ⌊ b k ⌋ (cid:17)i(cid:17) = E (cid:20) E T h F ( B ⌊ b k ⌋ ) i (cid:21) − E h E T h F ( B ⌊ b k ⌋ ) ii = E (cid:2) F ( B k, ) F ( B k, ) (cid:3) − E (cid:2) F ( B k, ) (cid:3) E (cid:2) F ( B k, ) (cid:3) ≤ C (cid:16) P (cid:0) ( B k, ) c (cid:1) + P (cid:0) ( C k, ) c (cid:1) + b − k/ (cid:17) + E (cid:20) F (cid:18) B k, · ,ϑ X k (cid:19) F (cid:18) B k, · ,ϑ X k (cid:19)(cid:21) − E (cid:20) F (cid:18) B k, · ,ϑ X k (cid:19)(cid:21) E (cid:20) F (cid:18) B k, · ,ϑ X k (cid:19)(cid:21) . On the event A k we have that B k, · ,ϑ X k , B k, · ,ϑ X k are independent therefore E (cid:20) F (cid:18) B k, · ,ϑ X k (cid:19) F (cid:18) B k, · ,ϑ X k (cid:19) |A k (cid:21) − E (cid:20) F (cid:18) B k, · ,ϑ X k (cid:19) |A k (cid:21) E (cid:20) F (cid:18) B k, · ,ϑ X k (cid:19) |A k (cid:21) = 0 . Using the Lipschitz property of F we then have thatVar P (cid:16) E T h F (cid:16) B ⌊ b k ⌋ (cid:17)i(cid:17) ≤ C (cid:16) P (cid:0) ( A k, ) c (cid:1) + P (cid:0) ( B k, ) c (cid:1) + P (cid:0) ( C k, ) c (cid:1) + b − k/ (cid:17) . For i = 1 , Y i are biased random walks on a supercritical GW-tree without leaves T g ,whose offspring law has exponential moments. It follows that the estimates P (( A k, ) c ) , P (( B k, ) c ) ≤ b − ˜ ck given in the proof of [8, Theorem 3] still hold. Combining these with (3.5) we have that there exists c > k sufficiently largeVar P (cid:16) E T h F (cid:16) B ⌊ b k ⌋ (cid:17)i(cid:17) ≤ Cb − ck which shows (3.1) and therefore the result follows from Lemma 3.1. Acknowledgements
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