Central limit theorems for biased randomly trapped random walks on Z

We prove CLTs for biased randomly trapped random walks in one dimension. In particular, we will establish an annealed invariance principal by considering a sequence of regeneration times under the assumption that the trapping times have finite second moment. In a quenched environment, an environment dependent centring is determined which is necessary to achieve a central limit theorem. As our main motivation, we apply these results to biased walks on subcritical Galton-Watson trees conditioned to survive and prove a tight bound on the bias required to obtain such limiting behaviour.

A technique is developed in [5] which can be used to extend an annealed invariance principal to a quenched result. This is applied in [25] to prove a quenched functional central limit theorem for the walk on the supercritical tree when the offspring distribution has exponential moments and no deaths. The condition of exponential moments is purely technical. However, because the offspring law has no deaths, the supercritical tree does not have traps which represents a significant simplification of the problem. Due to the similarity of the traps in the supercritical and subcritical GW-trees with leaves, a key motivation of this paper is to be able to extend the result of [25] to allow for deaths in the offspring law. This will be done in [7].
The argument of [5] relies on multiple copies of the walk eventually having separate escape paths; this creates a mixing of the environment which removes the dependency on the specific environment. In the one-dimensional model, the walk is forced to escape along a single route and therefore visits every trap in the positive segment of the environment. This results in the walk accumulating environment dependent fluctuations. In Section 3 we prove a quenched central limit theorem with environment dependent centring which demonstrates why this occurs in further detail.
We next introduce the models of interest and results in more detail. We consider the randomly trapped random walk model in which the underlying walk (Y k ) k≥0 is a simple, biased random walk on Z. That is, we write Y k := k j=1 χ j for a sequence of i.i.d. random variables (χ j ) j≥1 satisfying P (χ j = −1) = (β + 1) −1 = 1 − P (χ j = 1). For x ∈ Z write L(x, n) := n k=0 1 {Y k =x} for the local time of Y at site x by time n. A random environment ω is a sequence of (0, ∞)-valued probability measures (ω x ) x∈Z with environment law P := π ⊗Z for some fixed law π. For a fixed environment ω, let (η x,i ) x∈Z,i≥0 be independent with η x,i ∼ ω x . Writing and we then define the randomly trapped random walk by This process is then a continuous time random walk on Z with k th holding time η k := η Y k ,L(Y k ,k) and we write η := (η k ) k≥0 to be the sequence of holding times. For convenience we will define S t = S t where t := max{k ∈ Z : k ≤ t} for non-integer t ∈ R. Let P ω to denote the law over X for fixed environment ω and P(·) = P ω (·)P(dω) the annealed law. In Section 2 we prove a functional law of large numbers and central limit theorem for the randomly trapped random walk on Z. That is, in Corollary 2.2, we show that if the bias is positive (β > 1) and the expected holding time η 0 (under P) is finite, then X nt /n converges to the deterministic process νt where ν is a known constant (called the speed). In Corollary 2.3 we then show that this speed satisfies an Einstein relation; more specifically, the derivative of the speed with respect to the bias approaches half of the diffusion coefficient of the unbiased walk as β → 1 + . The main result of the section is Theorem 1 which states that, if β > 1 and E[η 2 0 ] < ∞ then for some ς ∈ (0, ∞) converges in distribution to a Brownian motion. We prove this by considering a renewal argument similar to that of [27].
In Section 3 we adapt the technique used in [16] (to prove a quenched CLT for a random walk in random environment) to derive a quenched central limit theorem with an environment dependent centring for the RTRW (Theorem 2). That is, for a fixed environment ω, we show that if β > 1, E[η 2 0 ] < ∞ and for some ε > 0 that E[E ω [η 0 ] 2+ε ] < ∞ then there exists an environment dependent centring G ω (n) such that for some ϑ ∈ (0, ∞) X n − G ω (n) ϑ √ n converges in distribution to a standard Gaussian. In particular, we show that the known function G ω (n) can be written as the annealed, deterministic centring with an environment dependent correction where this correction is a sum of centred i.i.d. random variables with non-zero variance under the environment law. This shows that the correction obeys a central limit theorem under P, thus has √ n fluctuations and is, therefore, necessary.
In Section 4 we then apply these results to the biased random walk on the subcritical GW-tree conditioned to survive. Let ξ denote the offspring distribution of a GW-process with mean µ ∈ (0, 1) and variance σ 2 < ∞ then let |X n | denote the graph distance between the walk at time n and the root of the tree. In Lemma 4.4 and Corollary 4.5 we show that if β ∈ (1, µ −1 ) then the expected trapping time is finite and the law of large numbers holds for |X n | with speed .
By Lemma 4.4 the expected trapping time is infinite when σ 2 = ∞ or β ≥ µ −1 . Moreover, when β < 1 the walk is recurrent. It therefore follows that the conditions σ 2 < ∞ and β ∈ (1, µ −1 ) are also necessary for the walk the be ballistic. We then develop the speed result to an Einstein relation in Lemma 4.6. In Corollary 4.11 we prove that if E[ξ 3 ] < ∞ and β ∈ (1, µ −1/2 ), then converges in distribution, under the annealed law P, to a Brownian motion. In Proposition 4.13 we prove the quenched analogue. That is, if we also have that E[ξ 3+ε ] < ∞ for some ε > 0 then for P-a.e. tree T , converges in distribution to a standard Gaussian under P T where G T is an environment dependent centring. Moreover, under these conditions, (ν β n − G T (n))/ √ n converges in distribution to a Gaussian random variable (with strictly positive variance) under P, which confirms the need for an environment dependent centring in the quenched CLT for the biased random walk on the subcritical GW-tree conditioned to survive. We also show, in Lemma 4.12, that the conditions β 2 µ < 1 and E[ξ 3 ] < ∞ are necessary for the previous results.
Let β c denote the smallest bias such that the walk is sub-ballistic whenever β > β c ; then, note that β c = µ −1 for the walk on the subcritical GW-tree. The necessity that β < µ −1/2 from Lemma 4.12 supports [3, Conjecture 3.1] which states that a quenched central limit should hold on the supercritical tree only when β < β 1/2 c which, as noted above, we will prove in an accompanying paper.

A functional law of large numbers and central limit theorem
The main aim of this section is to derive an annealed functional central limit theorem for the RTRW model with positive bias. That is, we show that (X nt − ntν(β))n − 1 2 converges in law to a Brownian motion under a second moment condition by considering a regeneration argument similar to that used in [27].
We first prove a straightforward speed result for the walk. By using ergodicity of the sequence of holding times we show that the sum of the first nt holding times converges almost surely, and then extend this to a result for the walk. Notice that we consider the unbiased case (β = 1) as well as the positive bias case. This will be used when we prove an Einstein relation for the walk. The key to Proposition 2.1 is [9, Lemma 2.1] which states that the left shift on sequences (θ(η 0 , η 1 , ...) = (η 1 , η 2 , ...)) acts ergodically on η under P. This holds for any non-degenerate random walk on a fixed environment with i.i.d. traps which is why we we can extend the following result to the unbiased case.
Proposition 2.1. Suppose β ≥ 1 and that E[η 0 ] < ∞ then S nt /n and S −1 nt /n converge P-a.s. on D([0, ∞), R) endowed with the Skorohod metric to the deterministic processes S t = E[η 0 ]t and S −1 Proof. We begin by showing convergence of S nt /n. Since θ acts ergodically on η under P and E[η 0 ] < ∞ we have that f (η) := η 0 is integrable. Therefore, by the ergodic theorem almost surely. The sequence of functions S nt /n are increasing in t and the limit S t = E[η 0 ]t is continuous therefore the convergence holds uniformly over t ∈ [0, T ] for T < ∞. Since S t is strictly increasing we have the desired convergence of S −1 nt /n by [30,Corollary 13.6.4].
Using Proposition 2.1, we are now able to show the speed result for the randomly trapped random walk.
Proof. Notice that By the law of large numbers n −1 Y n converges a.s. to (β − 1)/(β + 1) therefore by Proposition 2.1 and using that S −1 t is continuous we indeed have the desired result.
An additional result that can be deduced from Proposition 2.1 and Corollary 2.2 is that the following Einstein relation holds.
The unbiased (β = 1) walk X nt n −1/2 converges in P-distribution on D([0, ∞), R) endowed with the Skorohod metric to a scaled Brownian motion with variance Υ = where ν is the speed calculated in Corollary 2.2 for the β-biased walk.
Proof. For β = 1 we have that Y nt is the sum of i.i.d. copies of the random variable χ satisfying P (χ = 1) = 1/2 = P (χ = −1) thus by Donsker's invariance principle , and therefore we indeed have that lim We now move on to proving an annealed functional CLT which is the main result of this section. That is, we show that converges in P-distribution to a standard Brownian motion for some ς 2 ∈ (0, ∞). We want to write X nt as approximately the sum of i.i.d. centred random variables with finite second moments. Let κ 0 = 0 and, for j = 1, 2, ..., define κ j := inf{m > κ j−1 : {Y l } m−1 l=0 ∩ {Y l } ∞ l=m = φ} to be the regeneration times of the walk Y . We then have that S κj for j ≥ 1 are regeneration times for X and we write An important result of [11,Lemma 5.1] and the remark leading to it is that the time between regenerations of Y has exponential moments, that is for any j ≥ 1 and some constants C, c. Moreover, Y κj+1 − Y κj ≤ κ j+1 − κ j therefore the distance between regeneration points also has exponential moments. Proof. By [11] we have that the sections of the walk , for j ≥ 1, are i.i.d. therefore, since the traps (ω x ) x∈Z are i.i.d. and independent of the walk Y , we have that the sequences (η k ) It remains to show that Z j are centred. Since the distribution of a given holding time is independent of the regeneration times of Y and E[η 0 ] < ∞ we have that We want to show this is equal to ]. By (2.1) the time between regenerations and distance between regeneration points have exponential moments hence, by the law of large numbers, where Y κ1 /(κ m − κ 1 ), κ 1 /(κ m − κ 1 ) converge P-a.s. to 0. Furthermore, by the law of large numbers, Y κm /κ m converges P-a.s. to (β − 1)/(β + 1) therefore Therefore Z j are centred, as desired.
In Theorem 1 we show that B n t can be approximated by a sum of Z j which, by Lemma 2.4, are i.i.d. centred random variables. With the aim of proving a central limit theorem, we now show that they also have finite second moments.
Proof. Since {Z j } j≥2 are i.i.d. under P we have that Var P (Z j ) = Var P (Z 2 ) for all j ≥ 2. By properties of regenerations times Y κ2 ≥ Y κ1 and S κ2 ≥ S κ1 almost surely therefore we have that For the second term we have (2.6) By conditioning on Y we have that the holding times at separate vertices are independent therefore the second term in this expression can be written as By (2.1), the time between regenerations κ 2 − κ 1 has exponential moments therefore E[(κ 2 − κ 1 ) 2 ] < ∞. Furthermore, since Y moves in discrete time and has jumps of length 1 Combining (2.7) with (2.5) and (2.6), in order to show that Var P (Z 2 ) < ∞ it remains to show that Conditioning on Y this expectation is equal to which is finite by the assumptions of the theorem and equation (2.7).
We now conclude the proof of the annealed functional central limit theorem by showing that B n t can be suitably approximated by a sum of Z j .
converges in P-distribution on D(R + , R) endowed with the Skorohod metric to a standard Brownian motion.
Proof. By Lemmas 2.4 and 2.5 The random variables X Sκ 1 , S κ1 and | min k Y k | are all almost surely finite therefore the first fraction converges to 0 P-a.s. For ε > 0, by a union bound and Markov's inequality as n → ∞, and the supremum distance between (B n t ) t∈[0,T ] and (Σ mtn /ς √ n) t∈[0, T ] converges to 0 in P-probability. It therefore suffices to prove an invariance principle for Σ mtn .
For s ∈ R + let Σ s denote the linear interpolation of Σ m then by Donsker's invariance principle we T ] converges in distribution to a scaled Brownian motion. By the law of large numbers we have that κ n /n converges P-a.s. to E[κ 2 − κ 1 ] as n → ∞. Therefore, by [30,Corollary 13.6.4], we have that m tn /n converges P-a.s. on D([0, ∞), R) with the Skorohod metric to the deterministic process By [30,Theorem 13.2.1] it follows that the sequence (Σ mtn / √ n) t∈[0,T ] converges to the same limit as T ] which is a scaled Brownian motion. In particular, choosing we have that B n t converges to a standard Brownian motion.

Quenched central limit theorem
In this section we prove a quenched central limit theorem for the randomly trapped random walk under a 2 + ε moment condition. We do this by adapting the method used in [16] and first proving a quenched CLT for the first hitting time of n. Write τ n := inf{t ≥ 0 : X t = n} and, for ω fixed, Let ζ k := τ k+1 − τ k be the time taken between hitting k and k + 1 for the first time then the elements of (ζ k ) k≥1 are independent under P ω and Proof. By definition of τ n , H ω (n) and ζ k It therefore suffices to show that Lindeberg's conditions (see [13,Theorem 3.4.5]) hold: Recall, from the remark prior to Proposition 2.1, that θ is the shift map which is ergodic by [9, Lemma 2.1]. For the first condition we have that By Birkhoff's ergodic theorem, for P-a.e. ω, this converges to which converges to 0 as K → ∞ by dominated convergence.
Write τ Y n := inf{m ≥ 0 : Y m = n} to be the first hitting time of level n by the underlying walk. The following lemma describes the probability that the underlying walk moves back k levels before moving forward n. The result is the classical Gambler's ruin therefore we omit the proof.
By Lemma 3.1 we have a central limit theorem for the first hitting time of vertex n. The environment dependent centring H ω (n) can be written as the sum of n identically distributed random variables E ω [ζ k ]. These are not independent however; they are only locally dependent. Recall that η k,i is the i th holding We now show that H ω andH ω do not differ too much and therefore Lemma 3.1 also holds with H ω replaced byH ω . Under P, the functionH ω (n) is a sum of i.i.d. random variables with non-zero variance unless E ω [η 0 ] is constant. We thus have a central limit theorem forH ω (and therefore H ω ), which shows that the environment dependent centring is necessary. Notice that this is the first point at which we introduce the extra 2 + ε moment condition however we do require the condition later in Lemma 3.4 as well.
Proof. Recall that L(k, m) denotes the local time of Y at vertex k by time m and that the trapping times η k,j do not depend on the underlying walk, then We need to determine the expected local times at sites up to reaching level n. That is, (by the strong Markov property).
Let (τ Y n ) + := inf{m > 0 : Y m = n} be the first return time to level n by the underlying walk. By Lemma 3.2, the number of visits to k before reaching n for a walk started at k is geometrically distributed with escape probability Therefore, and is increasing in n and converges to β k (β + 1)/(β − 1). In particular, is increasing in n and converges to (β + 1)/(β − 1). In particular, The first term converges to 0 for P-a.e. ω by the strong law of large numbers. For the second term we have that, for δ, > 0, by Markov's inequality converges to 0 for P-a.e. ω.
We now prove a technical lemma that allows us to control the difference betweenH ω and its expected value under P which is important in proving the quenched CLT for the walk.
Proof. By Theorem 17 of Chapter IX.3 in [26], if Z n are i.i.d. centred random variables, a n is an increasing, diverging sequence and 3. a k /a n ≤ Ck/n for all k ≥ n then n k=1 Z k an converges to 0, P-a.s. Write Z n := E ω [η n,0 ] − E[η n,0 ] then Z n are i.i.d. and centred under P; moreover, the sequence a n = n 1+c 2 is increasing and diverges. By Chebyshev we have that which gives condition 1. Since n/a 2 n = n −c , an integral test gives condition 2. For k ≥ n we have that a k /a n = (k/n) 1+c 2 ≤ k/n so long as c ≤ 1 which gives 3. We therefore have that for any c > 0, J (n)n − 1+c 2 → 0 for P-a.e. ω hence the first statement holds. The process J (m) is a martingale therefore by the L p -maximal inequality we have that It therefore suffices to show that is bounded above. By the Marcinkiewicz-Zygmund inequality [24,Theorem 5] we have that using Jensen's inequality. Using that E E ω [η 0 ] 2+ε < ∞ for some ε > 0, it then follows that for δ > 0 sufficiently small and some constant C We now prove the main result of the section which is a quenched central limit theorem for the randomly trapped random walk. Recall from Corollary 2.
−1 is the P-a.s. limit of X n /n and from Lemma 3.1 that ς 2 = E[Var ω (τ 1 )] is the variance in the quenched CLT for the first hitting times τ n ; write ϑ := ςν 3/2 and Notice that, since νt is the centring of X t in the annealed CLT (Theorem 1) and G ω (t) − νt is a sum of i.i.d. random variables under the environment law, whenever these random variables have non-zero variance we obtain a central limit theorem for G ω (t)−νt with respect to P and therefore proving Theorem 2 also shows that there is no quenched CLT for X t with a deterministic centring.
uniformly in x as n → ∞.
Proof. Let t := sup{|X s | : s ≤ t} be the furthest point reached by X up to time t; then τ t ≤ t < τ t+1 and X t ≤ X τ t = t . We then have that |X t − t | = |X t − X τ t | ≤ sup s≥τ t X τ t − X s . Since X nt /n converges P-a.s. to νt uniformly over t we can choose a constant C T such that for n sufficiently large X t ≤ C T n for all t ≤ nT . Write to be the event that the walk never backtracks distance C log(n) up to reaching vertex C T n . By Lemma 3.2 we then have that Therefore, choosing C such that C log(β) > 2, by Borel-Cantelli we have that there exists only finitely many n such that the walk backtracks distance C log(n) up to time nT . In particular, on this event |X t − t |n −1/2 ≤ C log(n)n −1/2 which converges deterministically to 0 uniformly over t ≤ T . It therefore suffices to show that for P-a.e. ω The sequence I ω (t) is increasing in t and diverges; in particular, by the law of large numbers t/I ω (t) converges to ν −1 for P-a.e. ω. The result then follows from Lemma 3.

Subcritical Galton-Watson trees
A subcritical Galton-Watson tree conditioned to survive consists of a semi-infinite path with GW-trees as leaves. The walk on the tree does not deviate too far from this path and therefore behaves like a randomly trapped random walk on Z with holding times distributed as excursion times in GW-trees. Biased walks on subcritical GW-trees are, therefore, a natural example of the randomly trapped random walk. In this section we prove the required conditions for a biased random walk on a subcritical GW-tree conditioned to survive to satisfy the convergence results of the previous sections. That is, we prove a speed result, an Einstein relation, an annealed functional CLT and a quenched CLT with an environment dependent centring. To begin, we set up the model and show that it suffices to consider a randomly trapped random walk on Z with trapping times distributed as excursion times in trees. Let f (s) = ∞ k=0 p k s k denote the generating function of a GW-process with mean µ ∈ (0, 1) and variance σ 2 < ∞. To avoid the trivial case in which no traps form, we also assume that there exists k ≥ 2 such that p k > 0. We denote ξ as a random variable with the offspring law P(ξ = k) = p k and ξ * a random variable with the size-biased law given by the probabilities P(ξ * = k) = kp k µ −1 . Let Z n denote the n th generation size of the f -GW-process with this law. That is, started from a single progenitor, each individual independently has k offspring with probability p k . Such a process gives rise to a random rooted tree T where individuals in the process are represented by vertices (with the unique progenitor as the root ρ) and undirected edges connect individuals with their offspring. More generally, we will use Z T n to denote the n th generation size of a tree T . It has been shown in [19] that there is a well defined probability measure P over f -GW trees conditioned to survive which arises as the limit as n → ∞ of probability measures over GW-trees conditioned to survive up to generation n. It can be seen (e.g. [18]) that the tree can be constructed by attaching i.i.d. finite trees to a single infinite path Y := (ρ 0 = ρ, ρ 1 , ...). More specifically, starting with a single special vertex ρ 0 , at each generation let every normal vertex give birth to normal vertices according to independent copies of the original offspring distribution and every special vertex give birth to vertices according to independent copies of the size-biased distribution, one of which is chosen uniformly at random to be special (and denoted ρ k in the k th generation). We will use T to denote an f -GW-tree and T * an f -GW-tree conditioned to survive. We refer to Y as the backbone of the tree and the finite connected components T * \ {(ρ i , ρ i+1 ); i ≥ 0} appended to Y as branches. We denote by T * − x , the branch rooted at x ∈ Y; that is, the descendants of x which are not in the descendant tree of the unique child of x on the backbone. A β-biased random walk on a fixed, rooted tree T is a random walk (X n ) n≥0 on T which is β-times more likely to make a transition to a given child of the current vertex than the parent (which are the only options). More specifically, let ρ denote the root of the T , ← − x the parent of x ∈ T , c(x) the set of children of x and d x := |c(x)| the number of children, then the random walk X is the Markov chain started from X 0 = z defined by the transition probabilities We use P ρ (·) = P T * ρ (·)P(dT * ) for the annealed law obtained by averaging the quenched law P T * ρ over a law P on f -GW-trees conditioned to survive (with a fixed root ρ). For the remainder of the section we shall assume that β > 1 so that the walk is P-a.s. transient.
In [8] it is shown that either a strong bias or heavy tails of the offspring law can slow the walk into a sub-ballistic phase. More specifically, when β ≥ µ −1 the walk spends a large amount of time at the deep parts of the largest traps and when σ 2 = ∞ the walk takes a large number of excursions into the branches. These slowing effects cause the walk to become sub-ballistic; that is, |X n |n −1 converges P-a.s. to 0 in either case. Further to this, an exponent γ has been determined such that the walk escapes polynomially with this exponent.
Let |X n | denote the graph distance between the walk at time n and the root of the tree. We show that if β ∈ (1, µ −1 ) and E[ξ 2 ] < ∞ then |X n |n −1 converges P-a.s. to .
We then prove that if E[ξ 3 ] < ∞ and β ∈ (1, µ −1/2 ) then converges in distribution under P to a Brownian motion. Following this we show that if β ∈ (1, µ −1/2 ) and E[ξ 3+ε ] < ∞ for some ε > 0 then for P-a.e. tree T * , converges in distribution to a standard Gaussian under P T * where is an environment dependent centring. Moreover, under these conditions, G T * (n)/ √ n converges in distribution under P to a Gaussian random variable with strictly positive variance. This confirms the need for an environment dependent centring. We also show that the conditions β 2 µ < 1 and E[ξ 3 ] < ∞ are necessary for the previous results.
In Section 4.1 we couple the walk on the tree with a randomly trapped random walk in such a way that our main results of this section can be deduced from corresponding results for the RTRW. We then derive moment bounds on the excursion times in random trees that allow us to apply the results of the previous sections. Asymptotics for the generation size of a GW-tree will be important throughout the section; before discussing the walk in detail we give several useful properties. An important relation we shall use throughout is that the sequence P(Z n > 0)µ −n is decreasing and converges, which follows from [21, Theorem B] since µ ∈ (0, 1) and σ 2 < ∞, but can also be traced back further to [17] and [31]. In particular, we shall write by c µ , the limiting constant: Lemma 4.1 shows bounds on the expected moments of the generation sizes. A simple extension shows that these upper bounds are tight.
Lemma 4.1. Let Z n denote the n th generation size of an f -GW-process with offspring distribution ξ and mean µ ∈ (0, 1).
Proof. If P(ξ < 2) = 1 then Z n only takes values 0 and 1. Moreover, the number of generations until extinction is geometrically distributed with termination probability 1 − µ therefore the result follows. We therefore assume that P(ξ ≥ 2) > 0 which implies that f (1) > 0.
Let f n denote the generating function of Z n then statement 1 follows from For the second moment we have that E Applying this recursively we see that For m > n, by stationarity of GW-processes and (4.2) we have that Therefore, statement 2 follows by < ∞ therefore differentiating f n (s) and evaluating at s = 1 gives us that Iterating gives us that which proves that, for some c 1 > 0, n be independent copies of Z n then by convexity and (4.3) therefore, combining this with (4.6) we have that where the final inequality follows by (4.5), which gives statement 3.

Describing the walk on the subcritical tree as a randomly trapped random walk
We now wish to construct an almost equivalent model which allows us to consider the walk on the GWtree in our randomly trapped random walk framework. To begin, we show that the walk never deviates too far from the backbone. For a fixed tree T with root ρ we write H(T ) := max x∈T d(ρ, x) to be the height of the tree. LetX n be the projection of X n onto Y; that is,X n is the unique vertex on Y which satisfies d(X n ,X n ) = min y∈Y d(X n , y).
Proof. The distance d(X m ,X m ) between the walk and the backbone is at most the height of the largest branch seen up to time nT therefore, since the walk can have seen at most M branches by time M , by a union bound we have that for C > 0 By properties of the size-biased distribution we have that Therefore, by (4.1), We can therefore choose C sufficiently large so that thus the result follows by Borel-Cantelli.
For x ∈ T recall that |x| := d(ρ, x) then |X n | has the same distribution as a randomly trapped random walk on N with holding times distributed as excursion times in trees. The traps formed at different vertices are independent and identically distributed except at ρ since the root does not have an ancestor and therefore the transition probabilities from ρ differ from those at other vertices on the backbone. We now show that we can extend from N to Z with i.i.d. traps.
We begin by constructing the holding times of the randomly trapped random walk via a sequence of i.i.d. trees. Start with an initial vertex ρ and a unique ancestor ρ. Attach ξ * − 1 offspring to ρ where ξ * is size-biased as above. Note that this could result in zero offspring of ρ in which case the tree ends with only vertices ρ, ρ. Otherwise, attach independent f -GW trees to the offspring of ρ. This creates a tree T which has the distribution of a branch with an additional vertex connected to the root.

Recall that ← −
x denotes the parent of x ∈ T and c(x) the set of children of x. Consider a walk (W n ) n≥0 on T with transition probabilities otherwise.
An excursion in T started from ρ until absorption in ρ has the same distribution as the time taken to move between backbone vertices of T * (except at the root of T * ). Let ω = (T x ) x∈Z denote a sequence of independent trees with this law. For ω fixed let (η x,i ) x∈Z,i≥0 be independent with where ρ, ρ are the vertices in T x corresponding with the construction. Recall that for a discrete time process W we let The processŶ A −1 n is equal in distribution to |Ỹ n | therefore, without loss of generality, we may coupleỸ toŶ in the construction of X so that |Ỹ n | =Ŷ A −1 n without changing the distribution of X. LetŜ WriteŜ −1 n := inf{k ≥ 0 :Ŝ k > n} thenX n :=ŶŜ−1 n is a randomly trapped random walk coupled toX with trapping times equal in distribution to (η x,i ) x∈Z,i≥0 . The following lemma shows thatX andX never deviate too far. Proof. Using that |X|,X are discrete time processes with jump size 1, by the coupling of the two process we have that which is independent of n. That is, the supremum distance between the two processes is at most the total time spent by the two processes where the holding times differ.
Recall that τ + x := inf{m > 0 : X m = x} is the first return time to x for a β biased random walk. For a fixed tree T rooted at ρ with n th generation size Z n where Z 1 > 0 it is classical (e.g. [23,Chapter 2]) that The holding times η x,k are distributed as first hitting times of vertex ρ by the walk W n on T started from vertex ρ. The transition probabilities only differ from those of a β-biased walk at ρ and ρ but W n is more likely to move to ρ than the β-biased walk. It follows that the holding times are stochastically dominated by first return times to ρ of a β-biased random walk on a copy of T started from ρ. This means that where T is an f -GW-tree and we've used that for n ≥ 2 which follows from independence and stationarity of GW-trees. By statement 1 of Lemma 4.1 we have that E Z T n = µ n therefore which is finite since βµ < 1 and σ 2 < ∞. Similarly, the holding timesη 0,k are stochastically dominated by excursion times of a β-biased random walk on T started from ρ therefore we have that E[η ρ0,1 ] < ∞. For a β-biased walk on Z the last hitting time of 0 has exponential moments since it must occur before the first regeneration time and by (2.1) the time of the first regeneration has exponential moments. Therefore, using the above bound we can conclude that which is sufficient to prove the result by (4.8).
As a result of Lemmas 4.2 and 4.3 we can consider the randomly trapped random walk model in order to prove results for a biased random walk on a subcritical GW-tree conditioned to survive. In the remainder of the section we derive conditions for the tree which yield the moment bounds required in Theorems 1 and 2 and Corollary 2.2.

The speed of the walk
We now wish to prove a bound on the expected holding time for the randomly trapped random walk. Let η k := ηŶ k ,L(Ŷ k ,k) be the k th holding time of the randomly trapped random walk. Recall that η 0 is distributed as the first hitting time of ρ by the walk W n on the tree T described at the beginning of the section, and that T is a random tree rooted at ρ with a single ancestor ρ of the root, ξ * − 1 buds attached as children of ρ (where ξ * has the size biased law) and independent f -GW-trees attached to the buds.
The quantity η 0 is distributed as the first hitting time of ρ in the random tree T by the walk W started from ρ. Let be the number of return times to the root ρ before reaching its unique ancestor ρ. That is, N is the number of excursions to the trees attached to ρ before the walk reaches ρ. Let τ  Lemma 4.4. Suppose βµ < 1, σ 2 < ∞ and β ≥ 1, then Proof. By (4.11) we have that where Z 1 denotes the first generation size of an f -GW-tree T . The number of excursions N is geometrically distributed under P T 0 with termination probability 1 − p ex where It therefore follows that Using the formula (4.9) for the expected time spent in a fixed tree and statement 1 of Lemma 4.1 for the expected size of the k th generation we have that If β < µ −1 then this is equal to 2/(1 − βµ); otherwise, the sum does not converge. It follows that By Lemma 4.2 and 4.3 we have that sup m≤nT ||X m | −X m |n −1 converges to 0 P-a.s. therefore the following result follows from Corollary 2.2 and Lemma 4.4 sinceX has the distribution of a randomly trapped random walk with trapping times equal in distribution to (η x,i ) x∈Z,i≥0 .
We now extend the Einstein relation for the randomly trapped random walk to the walk on the GW-tree. This is a non-trivial extension because, in the tree model, the bias affects the trapping times and the unbiased walk is significantly influenced by the restriction to the half line. For this reason we observe convergence to a reflected Brownian motion and cannot simply apply Corollary 2.3. Lemma 4.6. Suppose µ < 1 and σ 2 < ∞. The unbiased (β = 1) walk X nt n −1/2 converges in Pdistribution on D([0, ∞), R) endowed with the Skorohod metric to |B t | where B t is a scaled Brownian where ν β is the speed calculated in Corollary 4.5 for the β-biased walk.
Proof. Recall thatX n is an randomly trapped random walk on Z which, by assumption and Lemma 4.4, is unbiased and has finite expected holding times. By [2, Theorem 2.9], for P-a.e. ω, the rescaled processX nt n −1/2 converges in P ω distribution to a scaled Brownian motion B with variance E[η 0 ] −1 .
The scaled local time at the origin n −1 LŶ (0, n − 1) converges P -a.s. to 0. Moreover, the holding times (η 0,i ) i≥1 are i.i.d. under P ω therefore, by the law of large numbers, n i=1 n −1 η 0,i converges P ω -a.s. to E ω [η 0,1 ] for P-a.e. ω. The same holds for (η 0,i ) i≥1 therefore the scaled sums LŶ (0,n−1) i=1 η 0,i n and LŶ (0,n−1) i=1η 0,i n converge to 0, P-a.s. It follows that the process X n : obeys the same central limit theorem asX. That is, we may replace the trap at the origin with the slightly different trap which corresponds to the branch at the root of the GW-tree, and still obtain a central limit theorem. Define the time spent above 0 by X and the associated limiting Brownian motion B as By substitution we then have that . Recall thatX is the projection of X onto the backbone. By definition of A X t , we have that Using Corollary 4.5 and taking the limit as β → 1 + we have that by Lemma 4.4, which completes the proof.

An annealed functional central limit theorem
We now prove an annealed functional central limit theorem for the biased walk on the subcritical GWtree conditioned to survive. By Lemmas 4.2 and 4.3 it will suffice to show the result holds for the corresponding randomly trapped random walk. We obtain the result by using the annealed invariance principle Theorem 1. That is, we show conditions on the tree and the bias which ensure that E[η 2 0 ] < ∞. In order to show this we will use a decomposition which counts the number of visits to each vertex. The return probability given in Lemma 3.2 will be important. For z 1 , z 2 , z 3 ∈ T write to be the probability that the walk started from z 1 hits z 2 before z 3 . We also require a similar relation for a walk on a tree. Let T x,y denote a tree with root ρ in which every vertex has a single offspring except the vertices w, x, y where w has two offspring and x, y have none. Denote these offspring w x , w y then let x, y be a descendants of w x , w y respectively (possibly w x , w y ). For vertices z 1 , z 2 , z 3 , z 4 write as the probability that the walk started at z 1 reaches z 2 before z 3 or z 4 by a β-biased walk on T x,y . Lemma 4.7 gives the probability that the walk started at w reaches ρ before x or y. Alternatively, this can be shown by comparing with an electrical network with conductances β k between vertices in generations k, k + 1 and then using network reduction (see, for example, [23,Chapter 2]).
Combining these gives us that For any x, y ∈ T there exists a unique vertex w x,y which is the closest ancestor of both x and y. We will often write w instead of w x,y when it is clear to which vertices we are referring. Moreover where, by comparison with a simple biased random walk on Z, we have that P T (τ + wx,y < τ + ρ ) ∈ [1−β −1 , 1]. We now prove a bound on E T w [v x v y ] following a similar method to that used in [19] for the unbiased case. Recall that c(x) is the set of children of x in T .
Lemma 4.8. For β > 1, there exists a constant C β such that for any finite tree T , Proof. When w = ρ at least one of x and y is never reached therefore v x v y = 0 and we may assume |w| ≥ 1. There are now three cases to consider; these are: 1. x = y = w x,y ; 2. x = w x,y = y; 3. x = w x,y = y.
In case 1 we have that v x is geometrically distributed with termination probability q x (ρ, x) therefore .
We therefore have that x ] ≤ C β (|c(x)|β + 1) 2 β 2|x| . In case 2, the number of visits to x from x is geometrically distributed as in case 1. For each visit to x (except the last) the walk reaches y before returning to x with probability q x (y, x)/q x (x, ρ) since, due to the tree structure, the walk cannot move from ρ to y without hitting x. From y, the walk returns to y a geometric number of times before returning to x. More specifically, where, conditional on the event {v x = j}, we have that v y is equal in distribution to the sum of B j x,y ∼ Bin(j − 1, q x (y, x)/q x (x, ρ)) independent geometric random variables G i x,y ∼ Geo(q y (x, y)). Under P T the number of excursions are independent therefore .
We therefore have that (4.13) Using Lemma 3.2 we then have that Combining these with (4.13) we have that In case 3, started from w x,y , the walk reaches either x or y before returning to ρ with probability q wx,y ({x, y}, ρ). From x the walk has a geometric number of returns to x before returning to w x,y . Moreover, from x, the walk must return to w x,y before reaching either ρ or y by definition of w x,y . The same also holds switching x and y. Letting since q x (w, x) −1 is the expected number of visits to x (started from x) before returning to w (and similarly for y) which are independent.
Substituting back into (4.14) it follows that The terms in the numerator can all be bounded below by half of the escape probability 1 − β −1 therefore we gain nothing using their exact expressions and bound them above by 1. Using Lemmas 3.2 and 4.7 for the other terms we have that Since |y| ≥ 1 we have that β |y| ≥ 1 therefore ≤ C β (|c(x)|β + 1)(|c(y)|β + 1)β |x|+|y| .
Recall from (4.10) that N is the number of return times to the root ρ before reaching its unique ancestor ρ and ζ k is the duration of the k th such excursion. Letting T = T 0 , by (4.11) we have that We want to show that E E ω η 2 0 < ∞. Lemma 4.9 shows that this can be reduced to showing that the expected value of the first sum in the quenched expectation is finite.
Proof. By (4.11) we have that which has finite expectation under P by Lemma 4.4 since β 2 µ < 1 implies that βµ < 1.
The variable N is geometrically distributed with termination probability 1 − p ex ; that is, and ξ * + 1 is the number of neighbours attached to ρ in T . We then have that by independence of excursion times under P T . Let T • denote the tree T without the ancestor of the root ρ and T be an f -GW-tree. Write Z n and Z T n to be the n th generation sizes of T • and T respectively then for j = 1, 2, ... let Z T ,j n be independent copies of Z T n . By the construction of T • using GW-trees we have that Using this with Lemma 4.1, for j ≥ k ≥ 1 we have that (4.17) Using (4.16) and the formula (4.9) for the expected time spent in a tree we have that By (4.17) we then have that Each of these terms is finite since β 2 µ < 1 and E[ξ 3 ] < ∞ where we recall from the definition of the size-biased distribution that E[(ξ * − 1) 2 ] ≤ CE[ξ 3 ].
In order to show that E[η 2 0 ] < ∞ it remains to prove Lemma 4.10 which follows similarly to Lemma 4.9 with the use of Lemma 4.8. Proof. Recall that T • denotes the tree T without the ancestor of the root ρ. Since the separate excursions are independent under P T and N is geometrically distributed we have that Labelling ρ 1 , ..., ρ ξ * −1 as the neighbours of ρ in T • , and T ρ k as the tree consisting of ρ, ρ k and the descendants of ρ k we have that when ξ * = 1 and 0 otherwise. Moreover, it then follows that since the subtraps are independent. Since E[ξ * − 1] ≤ CE[ξ 2 ] < ∞, it suffices to show that where T is a tree (equal in distribution to T ρ1 ) with root ρ, single first generation vertex − → ρ and, under P, the subtree rooted at − → ρ is a subcritical GW-tree with the original offspring distribution.
Recall that T z denotes the descendent tree of T at z. By (4.12) and Lemma 4.8 we have that By collecting terms in the k th generation we have that where Z T k is the size of the k th generation of T . For k ≥ 0 the tree T satisfies Z T k+1 = Z k for a GWprocess Z k with Z 0 = 1; therefore, using that β 2 µ < 1 and Lemma 4.1, we have that E Z T k Z T j ≤ Cµ j , for j ≥ k. In particular, By Lemmas 4.9 and 4.10 we have that E[η 2 0 ] < ∞ therefore, by Lemmas 4.2 and 4.3 and Theorem 1, we have the following annealed functional central limit theorem: Corollary 4.11. If β 2 µ < 1 and E[ξ 3 ] < ∞ then E[η 2 0 ] < ∞ and, in particular, there exists ς 2 < ∞ such that converges in P-distribution on D(R + , R) endowed with the Skorohod metric to a standard Brownian motion.