Limit theorems for vertex-reinforced jump processes on regular trees
aa r X i v : . [ m a t h . P R ] J u l LIMIT THEOREMS FOR VERTEX-REINFORCED JUMP PROCESSES ONREGULAR TREES
By Andrea Collevecchio (19 June, 2009) Abstract.
Consider a vertex-reinforced jump process defined on a regular tree, where eachvertex has exactly b children, with b ≥
3. We prove the strong law of large numbers andthe central limit theorem for the distance of the process from the root. Notice that it is stillunknown if vertex-reinforced jump process is transient on the binary tree. Dipartimento di Matematica applicata, Universit`a Ca’ Foscari – Venice, Italy. [email protected]
ANDREA COLLEVECCHIO
1. Introduction
Let D be any graph with the property that each vertex is the end point of only a finite numberof edges. Denote by Vert( D ) the set of vertices of D . The following, together with the vertexoccupied at time 0 and the set of positive numbers { a ν : ν ∈ Vert( D ) } , defines a right-continuousprocess X = { X s , s ≥ } . This process takes as values the vertices of D and jumps only tonearest neighbors, i.e. vertices one edge away from the occupied one. Given X s , ≤ s ≤ t ,and { X t = x } , the conditional probability that, in the interval ( t, t + d t ), the process jumps tothe nearest neighbor y of x is L ( y, t )d t , with L ( y, t ) := a y + Z t { X s = y } d s, a y > , where 1l A stands for the indicator function of the set A . The positive numbers { a ν : ν ∈ Vert( D ) } are called initial weights, and we suppose a ν ≡
1, unless specified otherwise. Such a process issaid to be a Vertex Reinforced Jump Process (VRJP) on D .Consider VRJP defined on the integers, which starts from 0. With probability 1 / −
1. The time of the first jump is an exponential random variable withmean 1 /
2, and is independent on the direction of the jump. Suppose the walk jumps towards1 at time z . Given this, it will wait at 1 an exponential amount of time with mean 1 / (2 + z ).Independently of this time, the jump will be towards 0 with probability (1 + z ) / (2 + z ).In this paper we define a process to be recurrent if it visits each vertex infinitely manytimes a.s., and to be transient otherwise. VRJP was introduced by Wendelin Werner, and itsproperties were first studied by Davis and Volkov (see [8] and [9]). This reinforced walk definedon the integer lattice is studied in [8] where recurrence is proved. For fixed b ∈ N := { , , . . . } ,the b -ary tree, which we denote by G b , is the infinite tree where each vertex has b + 1 neighborswith the exception of a single vertex, called the root and designated by ρ , that is connected to b vertices. In [9] is shown that VRJP on the b -ary tree is transient if b ≥
4. The case b = 3 wasdealt in [4], where it was proved that the process is still transient. The case b = 2 is still open.Another process which reinforces the vertices, the so called Vertex-Reinforced Random Walk(VRRW), shows a completely different behaviour. VRRW was introduced by Pemantle (see[17]). Pemantle and Volkov (see [19]) proved that this process, defined on the integers, getsstuck in at most five points. Tarr`es (see [23]) proved that it gets stuck in exactly 5 points.Volkov (in [24]) studied this process on arbitrary trees.The reader can find in [18] a survey on reinforced processes. In particular, we would liketo mention that little is known regarding the behaviour of these processes on infinite graphswith loops. Merkl and Rolles (see [13]) studied the recurrence of the original reinforced randomwalk, the so-called linearly bond-reinforced random walk, on two-dimensional graphs. Sellke(see [21]) proved than once-reinforced random walk is recurrent on the ladder.We define the distance between two vertices as the number of edges in the unique self-avoidingpath connecting them. For any vertex ν , denote by | ν | its distance from the root. Level i isthe set of vertices ν such that | ν | = i . The main result of this paper is the following. IMIT THEOREMS FOR VRJP ON TREES 3
Theorem 1.1.
Let X be VRJP on G b , with b ≥ . There exist constants K (1) b ∈ (0 , ∞ ) and K (2) b ∈ [0 , ∞ ) such that lim t →∞ | X t | t = K (1) b a.s. , (1.1) | X t | − K (1) b t √ t = ⇒ Normal (0 , K (2) b ) , (1.2) where we took the limit as t → ∞ , ⇒ stands for weak convergence and Normal (0 , stands forthe Dirac mass at . Durrett, Kesten and Limic have proved in [11] an analogous result for a bond-reinforced randomwalk, called one-time bond-reinforced random walk, on G b , b ≥
2. To prove this, they breakthe path into independent identically distributed blocks, using the classical method of cutpoints. We also use this approach. Our implementation of the cut point method is a strongimprovement of the one used in [3] to prove the strong law of large numbers for the originalreinforced random walk, the so-called linearly bond-reinforced random walk, on G b , with b ≥ G b , with b ≥
2. Preliminary definitions and properties
From now on, we consider VRJP X defined on the regular tree G b , with b ≥
3. For ν = ρ ,define par( ν ), called the parent of ν , to be the unique vertex at level | ν | − ν . Avertex ν is a child of ν if ν = par( ν ). We say that a vertex ν is a descendant of the vertex ν if the latter lies on the unique self-avoiding path connecting ν to ρ , and ν = ν . In this case, ν is said to be an ancestor of ν . For any vertex µ , let Λ µ be the subtree consisting of µ , itsdescendants and the edges connecting them, i.e. the subtree rooted at µ . Define T i := inf { t ≥ | X t | = i } . We give the so-called Poisson construction of VRJP on a graph D (see [20]). For eachordered pair of neighbors ( u, v ) assign a Poisson process P ( u, v ) of rate 1, the processes beingindependent. Call h i ( u, v ), with i ≥
1, the inter-arrival times of P ( u, v ) and let ξ := inf { t ≥ X t = u } . The first jump after ξ is at time c := ξ + min v h ( u, v ) (cid:0) L ( v, ξ ) (cid:1) − , where theminimum is taken over the set of neighbors of u . The jump is towards the neighbor v for whichthat minimum is attained. Suppose we defined { ( ξ j , c j ) , ≤ j ≤ i − } , and let ξ i := inf (cid:8) t > c i − : X t = u (cid:9) , and j v − j u,v − X jumped from u to v by time ξ i . The first jump after ξ i happens at time c i := ξ i + min v h j v ( u, v ) (cid:0) L ( v, ξ i ) (cid:1) − , and the jump istowards the neighbor v which attains that minimum. Definition 2.1.
A vertex µ , with | µ | ≥ , is good if it satisfies the following h ( µ , µ ) < h (cid:0) µ , par( µ ) (cid:1) h (cid:0) par( µ ) , µ (cid:1) where µ = par( µ ) . (2.1) ANDREA COLLEVECCHIO
By virtue of our construction of VRJP, (2.1) can be interpreted as follows. When the process X visits the vertex µ for the first time, if this ever happens, the weight at its parent is exactly1 + h (cid:0) par( µ ) , µ (cid:1) while the weight at µ is 1. Hence condition (2.1) implies that when theprocess visits µ (if this ever happens) then it will visit µ before it returns to par( µ ), if thisever happens.The next Lemma gives bounds for the probability that VRJP returns to the root after thefirst jump. Lemma 2.2.
Let α b := P (cid:0) X t = ρ for some t ≥ T (cid:1) , and let β b be the smallest among the positive solutions of the equation x = b X k =0 x k p k , (2.2) where, for k ∈ { , , . . . , b } , p k := k X j =0 (cid:18) bk (cid:19)(cid:18) kj (cid:19) ( − j Z ∞ zj + b − k + 1 + z e − z d z. (2.3) We have Z ∞ zb + 1 + z b e − bz d z ≤ α b ≤ β b . (2.4) Proof.
First we prove the lower bound in (2.4). The left-hand side of this inequality is theprobability that the process returns to the root with exactly two jumps. To see this, noticethat L ( ρ, T ) is equal 1 + min ν : | ν | =1 h ( ρ, ν ). Hence T = L ( ρ, T ) − /b . Given that T = z , the probability that the second jump is from X T to ρ is equal to (1 + z ) / ( b + 1 + z ). Hence the probability that the process returns to theroot with exactly two jumps is Z ∞ zb + 1 + z b e − bz d z. As for the upper bound in (2.4) we reason as follows. We give an upper bound for the probabilitythat there exists an infinite random tree which is composed only of good vertices and whichhas root at one of the children of X T . If this event holds, then the process does not return tothe root after time T (see the proof of Theorem 3 in [4]). We prove that a particular clusterof good vertices is stochastically larger than a branching process which is supercritical. Weintroduce the following color scheme. The only vertex at level 1 to be green is X T . A vertex ν , with | ν | ≥
2, is green if and only if it is good and its parent is green. All the other verticesare uncolored. Fix a vertex µ . Let C be any event in H µ := σ ( h i ( η , η ) : i ≥ , with η ∼ η and both η and η / ∈ Λ µ ) , (2.5)that is the σ -algebra that contains the information about X t observed outside Λ µ . Next weshow that given C ∩ { µ is green } , the distribution of h (par( µ ) , µ ) is stochastically dominatedby an exponential(1). To see this, first notice that h (par( µ ) , µ ) is independent of C . Let D := { par( µ ) is green } ∈ H µ and set W := h (cid:0) µ , par( µ ) (cid:1) h (cid:0) par( µ ) , µ (cid:1) where µ = par( µ ) . (2.6) IMIT THEOREMS FOR VRJP ON TREES 5
The random variable W is independent of h (par( µ ) , µ ) and is absolutely continuous withrespect the Lebesgue measure. By the definition of good vertices we have { µ is green } = { h (par( µ ) , µ ) < W } ∩ D. Denote by f W the conditional density of W given D ∩ C ∩ { h (par( µ ) , µ ) < W } . We have P (cid:16) h (par( µ ) , µ ) ≥ x (cid:12)(cid:12) { µ is green } ∩ C (cid:17) = P (cid:16) h (par( µ ) , µ ) ≥ x (cid:12)(cid:12) { h (par( µ ) , µ ) < W } ∩ C ∩ D (cid:17) = Z ∞ P (cid:16) h (par( µ ) , µ ) ≥ x (cid:12)(cid:12) { h (par( µ ) , µ ) < w } ∩ C ∩ D ∩ { W = w } (cid:17) f W ( w )d w (2.7)Using the facts that h (par( µ ) , µ ) is independent of W, C and D and P ( h (par( µ ) , µ ) ≥ x | h (par( µ ) , µ ) < w ) ≤ P ( h (par( µ ) , µ ) ≥ x ) , we get that the expression in (2.7) is less or equal to P (cid:16) h (par( µ ) , µ ) ≥ x (cid:17) . Summarising P (cid:16) h (par( µ ) , µ ) ≥ x | { µ is green } ∩ C (cid:17) ≥ P (cid:16) h (par( µ ) , µ ) ≥ x (cid:17) . (2.8)The inequality (2.7) implies that if µ is a child of µ and C ∈ H µ we have P (cid:16) µ is green | { µ is green } ∩ C (cid:17) ≥ P (cid:16) µ is green (cid:17) . (2.9)To see this, it is enough to integrate over the value of h (par( µ ) , µ ) and use the fact that,conditionally on h (par( µ ) , µ ), the events { µ is green } and { µ is green } ∩ C are independent.The probability that µ is good conditionally on { h (par( µ ) , µ ) = x } is a non-increasing func-tion of x , while the distribution of h (par( µ ) , µ ) is stochastically smaller than the conditionaldistribution of h (par( µ ) , µ ) given { µ is green } ∩ C , as shown in (2.8).Hence the cluster of green vertices is stochastically larger than a Galton–Watson tree whereeach vertex has k offspring, k ∈ { , , . . . , b } , with probability p k defined in (2.3). To seethis, fix a vertex µ and let µ i , with i ∈ { , , . . . , b } be its children. It is enough to realizethat p k is the probability that exactly k of the h ( µ, µ i ), with i ∈ { , , . . . , b } , are smallerthan (cid:0) h (par( µ ) , µ ) (cid:1) − h (cid:0) µ, par( µ ) (cid:1) . As the random variables h ( µ, µ i ) , h (cid:0) µ, par( µ ) (cid:1) and h (par( µ ) , µ ) are independent exponentials with parameter one, we have p k = (cid:18) bk (cid:19) Z ∞ Z ∞ P (cid:0) h ( µ , µ ) < y z (cid:1) k P (cid:0) h ( µ , µ ) ≥ y z (cid:1) b − k e − y e − z d y d z = (cid:18) bk (cid:19) Z ∞ Z ∞ (cid:0) − e − y z (cid:1) k e − y z ( b − k ) e − y e − z d y d z = k X j =0 Z ∞ Z ∞ (cid:18) bk (cid:19)(cid:18) kj (cid:19) ( − j e − y ( j + b − k +1+ z ) / (1+ z ) e − z d y d z = k X j =0 (cid:18) bk (cid:19)(cid:18) kj (cid:19) ( − j Z ∞ zj + b − k + 1 + z e − z d z. (2.10) ANDREA COLLEVECCHIO
From the basic theory of branching processes we know that the probability that this Galton–Watson tree is finite (i.e. extinction) equals the smallest positive solution of the equation x − b X k =0 x k p k = 0 . (2.11)The proof of (2.4) follows from the fact that 1 − β b ≤ − α b . This latter inequality is aconsequence of the fact that the cluster of green vertices is stochastically larger than the Galton-Watson tree, hence its probability of non-extinction is not smaller. As b ≥
3, the Galton-Watsontree is supercritical (see [4]),hence β b < (cid:3) For example, if we consider VRJP on G , Lemma 2.2 yields0 . ≤ α ≤ . . Definition 2.3.
Level j ≥ is a cut level if the first jump after T j is towards level j + 1 , andafter time T j +1 the process never goes back to X T j , and L ( X T j , ∞ ) < and L (par( X T j ) , ∞ ) < . Define l to be the cut level with minimum distance from the root, and for i > , l i := min { j > l i − : j is a cut level } . Define the i -th cut time to be τ i := T l i . Notice that l i = | X τ i | . l has an exponential tail For any vertex ν ∈ Vert( G b ), we define fc( ν ), which stands for first child of ν , to be the(a.s.) unique vertex connected to ν satisfying h ( ν, fc( ν )) = min (cid:8) h ( ν, µ ) : par( µ ) = ν (cid:9) . (3.1)For definiteness, the root ρ is not a first child. Notice that condition (3.1) does not imply thatthe vertex fc( ν ) is visited by the process. If X visits it, then it is the first among the childrenof ν to be visited.For any pair of distributions f and g , denote by f ∗ g the distribution of P Vk =1 M k , where • V has distribution f , and • { M k , k ∈ N } is a sequence of i.i.d random variables, independent of V , each withdistribution g .Recall the definition of p i , i ∈ { , . . . , b } , given in (2.3). Denote by p (1) the distribution whichassigns to i ∈ { , . . . , b } probability p i . Define, by recursion, p ( j ) := p ( j − ∗ p (1) , with j ≥ p ( j ) describes the number of elements, at time j , in a population which evolveslike a branching process generated by one ancestor and with offspring distribution p (1) . If welet m := b X j =1 jp j , IMIT THEOREMS FOR VRJP ON TREES 7 then the mean of p ( j ) is m j . The probability that a given vertex µ is good is, by definition, P (cid:16) h ( µ , µ ) < h (cid:0) µ , par( µ ) (cid:1) h (cid:0) par( µ ) , µ (cid:1) (cid:17) where µ = par( µ ) . As the h (cid:0) par( µ ) , µ (cid:1) is exponential with parameter 1, conditioning on its value and usingindependence between different Poisson processes, we have that the probability above equals P (cid:16) h ( µ , µ ) <
11 + z h (cid:0) µ , par( µ ) (cid:1)(cid:17) e − z d z = Z ∞
12 + z e − z d z = 0 . . . . . (3.2)Hence m = b · . > , because we assumed b ≥ q = p + p , and for k ∈ { , , . . . , b − } set q k = p k +1 . Set q to be the distributionwhich assigns to i ∈ { , . . . , b − } probability q i . For j ≥
2, let q ( j ) := p ( j − ∗ q . Denote by q ( j ) i the probability that the distribution q ( j ) assigns to i ∈ { , . . . , ( b − b j − } . The mean of q ( j ) is m j − ( m − ζ denotes the smallest positive integer in { , , . . . , } such that m ζ − ( m − > . (3.3)Next we want to define a sequence of events which are independent and which are closelyrelated to the event that a given level is a cut level. For any vertex ν of G b let Θ ν be the set ofvertices µ such that • µ is a descendant of ν , • the difference | µ | - | ν | is a multiple of ζ , • µ is a first child.By subtree rooted at ν we mean a subtree of Λ ν that contains ν . Set e ν = fc( ν ) and let A ( ν ) := (cid:8) ∃ an infinite subtree of G b root at a child of e ν , which is composed only bygood vertices and which contains none of the vertices in Θ ν } (3.4)For i ∈ N , let A i := A (cid:0) X T i (cid:1) . Notice that if the process reaches the first child of ν and if A ( ν )holds, then the process will never return to ν . Hence if A i holds, and if X T i +1 = X T i + 1, then i is a cut level, provided that the total weights at X T i and its parent are less than 2. Proposition 3.1.
The events A iζ , with i ∈ N , are independent. Proof.
We recall that ζ ≥
2. We proceed by backward recursion and show that the events A iζ depend on disjoint Poisson processes collections. Choose integers 0 < i < i < . . . < i k ,with i j ∈ ζ N := { ζ , ζ , ζ , . . . } for all j ∈ { , , . . . , k } . It is enough to prove that P (cid:16) k \ j =1 A i j (cid:17) = k Y j =1 P (cid:0) A i j (cid:1) . (3.5)Fix a vertex ν at level i k . The set A ( ν ) belongs to the sigma-algebra generated by (cid:8) P ( u, w ) : u, w ∈ Vert(Λ ν ) (cid:9) . On the other hand, the set T k − j =1 A i j ∩ { X T ik = ν } belongs ANDREA COLLEVECCHIO to (cid:8) P ( u, w ) : u / ∈ Vert(Λ ν ) (cid:9) . As the two events belong to disjoint collections of independentPoisson processes, they are independent. As P ( A ( ν )) = P ( A ( ρ )), we have P (cid:16) A i k ∩ k − \ j =1 A i j (cid:17) = X ν : | ν | = i k P (cid:16) A i k ∩ k − \ j =1 A i j ∩ { X T ik = ν } (cid:17) = X ν : | ν | = i k P (cid:16) A ( ν ) ∩ k − \ j =1 A i j ∩ { X T ik = ν } (cid:17) = X ν : | ν | = i k P (cid:0) A ( ν ) (cid:1) P (cid:16) k − \ j =1 A i j ∩ { X T ik = ν } (cid:17) = P (cid:0) A ( ρ ) (cid:1) X ν : | ν | = i k P (cid:16) k − \ j =1 A i j ∩ { X T ik = ν } (cid:17) = P (cid:0) A ( ρ ) (cid:1) P (cid:16) k − \ j =1 A i j (cid:17) . (3.6)The events A ( ν ) and { X T ik = ν } are independent, and by virtue of the self-similarity propertyof the regular tree we get P (cid:0) A ( ρ ) (cid:1) = P (cid:0) A i k (cid:1) . Hence P (cid:16) A i k ∩ k − \ j =1 A i j (cid:17) = P (cid:0) A i k (cid:1) P (cid:16) k − \ j =1 A i j (cid:17) . (3.7)Reiterating (3.7) we get (3.5). (cid:3) Lemma 3.2.
Define γ b to be the smallest positive solution of the equation x = b − X k =0 x k q ( ζ ) k , (3.8) where ζ and ( q ( n ) k ) have been defined at the beginning of this section. We have P ( A i ) ≥ − γ b > , ∀ i ∈ N . (3.9) Proof.
Fix i ∈ N and let ν ∗ = X T i . We adopt the following color scheme. The vertex fc (cid:0) X T i (cid:1) is colored blue . A descendant µ of ν ∗ is colored blue if it is good, its parent is blue , and either • | µ | − | ν ∗ | is not a multiple of ζ , or • ζ (cid:0) | µ | − | ν ∗ | (cid:1) ∈ N and µ is not a first child.Vertices which are not descendants of ν ∗ are not colored. Following the reasoning given in theproof of Lemma 2.2, we can conclude that the number of blue vertices at levels | ν ∗ | + jζ , with j ≥
1, is stochastically larger than the number of individuals in a population which evolves likea branching process with offspring distribution q ( ζ ) , introduced at the beginning of this section.Again, from the basic theory of branching processes we know that the probability that this treeis finite equals the smallest positive solution of the equation (3.8). By virtue of (3.3) we havethat γ b < (cid:3) The proof of the following Lemma can be found in [10] pages 26-27 and 35.
Lemma 3.3.
Suppose U n is Bin ( n, p ) . For x ∈ (0 , consider the entropy H ( x | p ) := x ln xp + (1 − x ) ln 1 − x − p . We have the following large deviations estimate, for s ∈ [0 , , P ( U n ≤ sn ) ≤ exp {− n inf x ∈ [0 ,s ] H ( x | p ) } . IMIT THEOREMS FOR VRJP ON TREES 9
Proposition 3.4. i) Let ν be a vertex with | ν | ≥ . The quantity P (cid:0) A ( ν ) | h ( ν, fc( ν )) = x (cid:1) is a decreasing function of x , with x ≥ . ii) P (cid:0) A ( ν ) | h ( ν, fc( ν )) ≤ x (cid:1) ≥ P (cid:0) A ( ν ) (cid:1) , for any x ≥ . Proof.
Suppose { fc( ν ) = ν } . Given (cid:8) h ( ν, ν ) = x (cid:9) , the set of good vertices in Λ ν is afunction of x . Denote this function by T : R + → { subset of vertices of Λ ν } . A child of ν , say ν , is good if and only if h ( ν, ν ) < h ( ν, ν )1 + x . Hence the smaller x is, the more likely ν is good. This is true for any child of ν . As fordescendants of ν at level strictly greater than | ν | + 2, their status of being good is independentof h ( ν, fc( ν )). Hence T ( x ) ⊃ T ( y ) for x < y . This implies that the connected component ofgood vertices contining ν is larger if { h ( ν, ν ) = x } rather than { h ( ν, ν ) = y } , for x < y . Hence P (cid:0) A ( ν ) | h ( ν, fc( ν )) = x, fc( ν ) = ν (cid:1) ≥ P (cid:0) A ( ν ) | h ( ν, fc( ν )) = y, fc( ν ) = ν (cid:1) , for x < y. Using symmetry we get i). In order to prove ii), use i) and the fact that the distributionof h ( ν, fc( ν )) is stochastically larger that the conditional distribution of h ( ν, fc( ν )) given { h ( ν, fc( ν )) ≤ x } . (cid:3) Denote by [ x ] the largest integer smaller than x . Theorem 3.5.
For VRJP defined on G b , with b ≥ , and s ∈ (0 , , we have P (cid:0) l [ sn ] ≥ n (cid:1) ≤ exp n − [ n/ζ ] inf x ∈ [0 ,s ] H (cid:16) x (cid:12)(cid:12) (1 − γ b ) ϕ b (cid:17)o , (3.10) where γ b was defined in Lemma 3.2, and ϕ b := (cid:0) − e − b (cid:1) (cid:0) − e − ( b +1) (cid:1) bb + 2 . (3.11) Proof.
By virtue of Proposition 3.1 the sequence 1l A kζ , with k ∈ N , consists of i.i.d. ran-dom variables. The random variable P [ n/ζ ] j =1 A jζ has binomial distribution with parameters (cid:0) P (cid:0) A ( ρ ) (cid:1) , [ n/ζ ] (cid:1) . We define the event B j := { the first jump after T j is towards level j + 1 and L (cid:0) X T j , T j +1 (cid:1) < L (cid:0) par( X T j ) , T j +1 (cid:1) < } . Let F t be the smallest sigma-algebra defined by the collection { X s , ≤ s ≤ t } . For anystopping time S define F S := (cid:8) A : A ∩ { S ≤ t } ∈ F t (cid:9) . Now we show P (cid:0) B j | F T i − (cid:1) ≥ (cid:0) − e − b (cid:1) (cid:0) − e − ( b +1) (cid:1) bb + 2 = ϕ b , (3.12)where the inequality holds a.s.. In fact, by time T i the total weight of the parent of X T i isstochastically smaller than 1+ an exponential of parameter b , independent of F T i − . Hence theprobability that this total weight is less than 2 is larger than 1 − e − b . Given this, the probabilitythat the first jump after T i is towards level i + 1 is larger than b/ ( b + 2). Finally, the conditional probability that T i +1 − T i < − e − ( b +1) . This implies, together with ζ ≥
2, thatthe random variable P [ n/ζ ] j =1 B j is stochastically larger than a binomial( n, ϕ b ). For any i ∈ N ,and any vertex ν with | ν | = iζ , set Z := min (cid:16) , h (cid:0) ν, par( ν ) (cid:1) h (cid:0) par( ν ) , ν (cid:1) (cid:17) E := { X T iζ = ν } ∩ { L (par( ν ) , T iζ ) < } . We have B iζ ∩ { X T iζ = ν } = { h ( ν, fc( ν )) < Z } ∩ E. Moreover, the random variable Z and the event E are both measurable with respect the sigma-algebra e H ν := σ n P (par( ν ) , ν ) , (cid:8) P ( u, w ) : u, w / ∈ Vert(Λ ν ) (cid:9)o . Let f Z be the density of Z given { h ( ν, fc( ν )) < Z } ∩ E. Using 3.4, ii), and the independencebetween h ( ν, fc( ν )) and e H ν , we get P (cid:0) A iζ (cid:12)(cid:12) B iζ ∩ { X T iζ = ν } (cid:1) = P (cid:0) A ( ν ) (cid:12)(cid:12) { h ( ν, fc( ν )) < Z } ∩ E (cid:1) = Z ∞ P (cid:0) A ( ν ) (cid:12)(cid:12) { h ( ν, fc( ν )) < z } (cid:1) f Z ( z )d z ≥ P (cid:0) A ( ν ) (cid:1) = X ν : | ν | = iζ P (cid:0) A ( ν ) ∩ { X T iζ = ν } (cid:1) = P ( A iζ ) . (3.13)The first equality in the last line of (3.13) is due to symmetry. Hence P ( A iζ (cid:12)(cid:12) B iζ ) ≥ P ( A iζ ) . (3.14)If A k ∩ B k holds then k is a cut level. In fact, on this event, when the walk visits level k forthe first time it jumps right away to level k + 1 and never visits level k again. This happensbecause X T k +1 = fc( X T k ) has a child which is the root of an infinite subtree of good vertices.Moreover the total weights at X T k and its parent are less than 2. Define e n := [ n/ζ ] X i =1 A iζ ∩ B iζ . By virtue of (3.9), (3.12), (3.14) and Proposition 3.1 we have that e n is stochastically largerthan a bin([ n/ζ ], (1 − γ b ) ϕ b ). Applying Lemma 3.3, we have P (cid:0) l [ sn ] ≥ n (cid:1) ≤ P (cid:0) e n ≤ [ sn ] (cid:1) ≤ exp n − [ n/ζ ] inf x ∈ [0 ,s ] H (cid:16) x (cid:12)(cid:12) (1 − γ b ) ϕ b (cid:17)o . (cid:3) The function H (cid:0) x (cid:12)(cid:12) (1 − γ b ) ϕ b (cid:1) is decreasing in the interval (0 , (1 − γ b ) ϕ b ). Hence for n > / (cid:0) (1 − γ b ) ϕ b (cid:1) , we have inf x ∈ [0 , /n ] H (cid:0) x (cid:12)(cid:12) (1 − γ b ) ϕ b (cid:1) = H (cid:0) /n (cid:12)(cid:12) (1 − γ b ) ϕ b (cid:1) . Corollary 3.6.
For n > / (cid:0) (1 − γ b ) φ b (cid:1) , by choosing s = 1 /n in Theorem 3.5, we have P (cid:0) l ≥ n (cid:1) ≤ exp n − [ n/ζ ] inf x ∈ [0 , /n ] H (cid:16) x (cid:12)(cid:12) (1 − γ b ) ϕ b (cid:17)o = exp n − [ n/ζ ] H (cid:16) n (cid:12)(cid:12) (1 − γ b ) ϕ b (cid:17)o , (3.15) IMIT THEOREMS FOR VRJP ON TREES 11 where, from the definition of H we have lim n →∞ H (cid:16) n (cid:12)(cid:12) (1 − γ b ) ϕ b (cid:17) = ln 11 − (1 − γ b ) ϕ b > . τ has finite (2 + δ ) -moment The goal of this section is to prove the finiteness of the 11 / • first we prove the finiteness of all moments for the number of vertices visited by time τ , then • we prove that the total time spent at each of these sites has finite 12 / n ∈ N and letΠ n := number of distinct vertices that X visits by time T n , Π n,k := number of distinct vertices that X visits at level k by time T n . Let T ( ν ) := inf { t ≥ X t = ν } . For any subtree E of G b , b ≥
1, define δ ( a, E ) := sup (cid:26) t : Z t { X s ∈ E } d s ≤ a (cid:27) . The process X δ ( t,E ) is called the restriction of X to E . Proposition 4.1 ( Restriction principle (see [8] )).
Consider VRJP X defined on a tree J rooted at ρ . Assume this process is recurrent, i.e. visits each vertex infinitely often, a.s..Consider a subtree e J rooted at ν . Then the process X δ ( t, e J ) is VRJP defined on e J . Moreover,for any subtree J ∗ disjoint from e J , we have that X δ ( t, e J ) and X δ ( t, J ∗ ) are independent. Proof.
This principle follows directly from the Poisson construction and the memorylessproperty of the exponential distribution. (cid:3)
Definition 4.2.
Recall that P ( x, y ) , with x, y ∈ Vert (cid:0) G b (cid:1) are the Poisson processes used togenerate X on G b . Let J be a subtree of G b . Consider VRJP V on J which is generatedby using (cid:8) P ( u, v ) : u, v ∈ Vert ( J ) (cid:9) , which is the same collection of Poisson processes used togenerate the jumps of X from the vertices of J . We say that V is the extension of X in J .The processes V t and X δ ( t, J ) coincide up to a random time, that is the total time spent by X in J . We construct an upper bound for Π n,k , with 2 ≤ k ≤ n −
1. . Let G ( k ) be the finite subtreeof G b composed by all the vertices at level i with i ≤ k −
1, and the edges connecting them.Let V be the extension of X to G ( k ). This process is recurrent, because is defined on a finitegraph. The total number of first children at level k − b k − , and we order them according towhen they are visited by V , as follows. Let η be the first vertex at level k − V . Suppose we have defined η , . . . , η m − . Let η m be the first child at level k − { η , η , . . . , η m − } , to be visited. The vertices η i , with 1 ≤ i ≤ b k − are determined by V . All the other quantities and events such as T ( ν ) and A ( ν ), with ν runningover the vertices of G b , refer to the process X . Define f n ( k ) := 1 + b inf { m ≥ A (par( η m )) = 1 } . Let J := inf { n : T ( η n ) = ∞} , if the infimum is over an empty set, let J = ∞ . Suppose that A ( η m ) holds, then X , after time T ( η m ), is forced to remain inside Λ η m , and never visits fc( η m )again. This implies that T ( η m +1 ) = ∞ . Hence, if J = m then T m − i =1 ( A (par( η i ))) c holds, and f n ( k ) ≥ b ( m − J = ∞ then f n ( k ) = 1 + b b k − = 1 + b k , which is anobvious upper bound for the number of vertices at level k which are visited by X . On theother hand, if J = m then the number of vertices at level k which are visited by X is at most1 + ( m − b . In fact, the processes X and V coincide up to the random time when the formerprocess leaves G ( k ) and never returns to it. Hence if T ( η i ) < ∞ then X visited exactly i − k − T ( η i ). On the event { J = m } we have that { T ( η m − < ∞} ∩ { T ( η m ) = ∞} , hence exactly m − k − m − b vertices at level k are visited.We conclude that f n ( k ) overcounts the number of vertices at level k which are visited, i.e.Π n,k ≤ f n ( k ).Recall that h ( ν, fc( ν )), being the minimum over a set of b independent exponentials withrate 1, is distributed as an exponential with mean 1 /b . Lemma 4.3.
For any m ∈ N , we have P (cid:0) f n ( k ) > mb (cid:1) ≤ ( γ b ) m . Proof.
Given T m − i =1 ( A (par( η i )) c the distribution of h (par( η m ) , η m ) is stochastically smallerthan an exponential with mean 1 /b . Fix a set of vertices ν i with 1 ≤ i ≤ m − k − η i = ν i for i ≤ m −
1, consider the restriction of V tothe finite subgraph obtained from G ( k ) by removing each of the ν i and par( ν i ), with i ≤ m − V to this subgraph is VRJP, independent of T m − i =1 ( A (par( η i )) c , and the totaltime spent by this process in level k − /b . This total time is anupper bound for h (par( η m ) , η m ). This conclusion is independent of our choice of the vertices ν i with 1 ≤ i ≤ m −
1. Finally, using Proposition 3.4 i), we have P (cid:0) f n ( k ) > mb | f n ( k ) > m − b (cid:1) = P (cid:0) ( A (par( η m ))) c | m − \ i =1 ( A (par( η i )) c (cid:1) ≤ P (cid:0) ( A (par( η m ))) c (cid:1) ≤ γ b . (4.1) (cid:3) Let a n , c n be numerical sequences. We say that c n = O ( a n ) if c n /a n is bounded. Lemma 4.4.
For p ≥ , we have E [Π pn ] = O ( n p ) . Proof.
Consider first the case p >
1. Notice that Π n, = Π n,n = 1. By virtue of Lemma 4.3,we have that sup n E [ f pn ] < ∞ . By Jensen’s inequality E [Π pn ] = E " n − X k =1 Π n,k ) ! p ≤ n p E " n − X k =1 Π pn,k n + 2 p n ≤ n p E " n − X k =1 f pn ( k ) n + 2 p n = O ( n p ) . (4.2) IMIT THEOREMS FOR VRJP ON TREES 13
As for the case p = 1, E [Π n ] ≤ n − X k =1 E [ f n ( k )] = O ( n ) . (cid:3) Let Π := X ν { ν is visited before time τ } . where the sum is over the vertices of G b . In words, Π is the number of vertices visited before τ . Lemma 4.5.
For any p > we have E [Π p ] < ∞ . Proof.
By virtue of Lemma 4.4, q E (cid:2) Π pn (cid:3) ≤ C (1) b,p n p , for some positive constant C (1) b,p . Henceusing Cauchy-Schwartz, E [Π p ] = ∞ X n =1 E (cid:2) Π pn { l = n } (cid:3) ≤ ∞ X n =1 q E (cid:2) Π pn (cid:3) P ( l ≥ n ) ≤ C (1) b,p ∞ X n =1 n p exp n −
12 [ n/ζ ] H (cid:16) n (cid:12)(cid:12) (1 − γ b ) ϕ b (cid:17)o < ∞ . In the last inequality we used Corollary 3.6. (cid:3)
Next, we want to prove that the 12 / L ( ρ, ∞ ) is finite. We start with threeintermediate results. The first two can be found in [9]. We include the proofs here for the sakeof completeness. Lemma 4.6.
Consider VRJP on { , } , which starts at , and with initial weights a = c and a = 1 . Define ξ ( t ) := inf n s : L (1 , s ) = t o . We have sup t ≥ E "(cid:18) L (0 , ξ ( t )) t (cid:19) = c + 3 c + 3 c. (4.3) Proof.
We have L (0 , ξ ( t + d t )) = L (0 , ξ ( t )) + χη , where χ is a Bernoulli which takes value 1with probability L (0 , ξ ( t ))d t , and η is exponential with mean 1 /t . Given L (cid:0) , ξ ( t ) (cid:1) , the randomvariables χ and η are independent. Hence E h L (0 , ξ ( t + d t )) i − E h L (0 , ξ ( t )) i = E [ L (0 , ξ ( t ))] t d t, i.e. E [ L (0 , ξ ( t ))] is solution of the equation y ′ ( t ) = y ( t ) /t , with initial condition y (1) = c (see[8]). Hence E [ L (0 , ξ ( t ))] = ct. Similarly E h L (0 , ξ ( t + d t )) i = E h L (0 , ξ ( t )) i + 2 E h L (0 , ξ ( t )) E h χ | L (0 , ξ ( t )) ii E [ η ] + E h χ | L (0 , ξ ( t )) i E [ η ]= E h L (0 , ξ ( t )) i + (2 /t ) E h L (0 , ξ ( t )) i d t + (2 /t ) E h L (0 , ξ ( t )) i d t = E h L (0 , ξ ( t )) i + (2 /t ) E h L (0 , ξ ( t )) i d t + (2 c/t )d t. Thus E h L (0 , ξ ( t )) i satisfies the equation y ′ = (2 /t ) y + (2 c/t ), with y (1) = c . Then, E h L (0 , ξ ( t )) i = − c + (cid:0) c + c (cid:1) t . Finally, reasoning in a similar way, we get that E h L (0 , ξ ( t )) i satisfies the equation y ′ = (3 /t ) y +6( c + c ), with y (1) = c . Hence, E h L (0 , ξ ( t )) i = − c + c ) t + (cid:0) c + 3 c + 3 c (cid:1) t . Divide both sides by t , and use the fact that c > (cid:3) A ray σ is a subtree of G b containing exactly one vertex of each level of G b . Label the verticesof this ray using { σ i , i ≥ } , where σ i is the unique vertex at level i which belongs to σ . Denoteby S the collection of all rays of G b . Lemma 4.7.
For any ray σ , consider VRJP X ( σ ) := { X ( σ ) t , t ≥ } , which is the extension of X to σ . Define T ( σ ) n := inf { t > X ( σ ) t = σ n } ,L ( σ ) ( σ i , t ) := 1 + Z t { X ( σ ) s = σ i } d s. We have that E (cid:2) L ( σ ) ( σ , T ( σ ) n ) (cid:3) ≤ (37) n . (4.4) Proof.
By the tower property of conditional expectation, E h(cid:0) L ( σ ) ( σ , T ( σ ) n ) (cid:1) i = E (cid:0) L ( σ ) ( σ , T ( σ ) n ) (cid:1) E L ( σ ) ( σ , T ( σ ) n ) L ( σ ) (cid:0) σ , T ( σ ) n (cid:1) ! (cid:12)(cid:12)(cid:12) L ( σ ) ( σ , T ( σ ) n ) . (4.5)At this point we focus on the process restricted to { , } . This restricted process is VRJPwhich starts at 1, with initial weights a = 1, and a = 1 + h ( σ , σ ) and σ = ρ . By applyingLemma 4.6, and using the fact that h ( σ , σ ) is exponential with mean 1, we have E L ( σ ) ( σ , T ( σ ) n ) L ( σ ) (cid:0) σ , T ( σ ) n (cid:1) ! (cid:12)(cid:12)(cid:12) L ( σ ) ( σ , T ( σ ) n ) ≤ E (cid:2) h ( σ , σ )) + (1 + h ( σ , σ )) + (1 + h ( σ , σ )) (cid:3) = 37 . (4.6) IMIT THEOREMS FOR VRJP ON TREES 15
Then E h(cid:0) L ( σ , T n ) (cid:1) i = E E L ( σ ) ( σ , T ( σ ) n ) L ( σ ) (cid:0) σ , T ( σ ) n (cid:1) ! (cid:12)(cid:12)(cid:12) L ( σ ) ( σ , T ( σ ) n ) (cid:0) L ( σ ) ( σ , T ( σ ) n ) (cid:1) ≤ E h(cid:0) L ( σ ) ( σ , T ( σ ) n ) (cid:1) i . (4.7)The Lemma follows by recursion and restriction principle. (cid:3) Next, we prove that L ( ρ, T ( σ n )) ≤ L ( σ ) ( σ , T ( σ ) n ) . (4.8)In fact, we have equality if T ( σ n ) < ∞ , because the restriction and the extension of X to σ coincide during the time interval [0 , T ( σ n )]. If T ( σ n ) = ∞ , it means that X left the ray σ at atime s < T ( σ ) n . Hence L ( ρ, T ( σ n )) = L ( σ ) ( σ , s ) ≤ L ( σ ) ( σ , T ( σ ) n ) . Hence, for any ν , with | ν | = n , we have E (cid:2) L ( ρ, T ( ν )) (cid:3) ≤ (37) n . (4.9) Lemma 4.8. E h ( L ( ρ, ∞ )) / i < ∞ . Proof.
Recall the definition of A ( ν ) from (3.4) and set D k := [ ν : | ν | = k − A ( ν ) . If A ( ν ) holds, after the first time the process hits the first child of ν , if this ever happens, itwill never visit ν again, and will not increase the local time spent at the root. Roughly, ourstrategy is to use the extensions on paths to give an upper bound of the total time spent atthe root by time T k and show that the probability that T ki =1 D ci decreases quite fast in k .Using the independence between disjoint collections of Poisson processes, we infer that A ( ν ),with | ν | = k − A ( ν ) is determined by the Poisson processesattached to pairs of vertices in Λ ν . Hence P ( D ck ) ≤ ( γ b ) b k − (4.10)Define d = inf { n ≥ D n = 1 } . Fix k ∈ N . On the set { d = k } , define µ to be one of the firstchildren at level k − A (par( µ )) holds. On { T ( µ ) < ∞} ∩ { d = k } , we clearly have L ( ρ, ∞ ) = L ( ρ, T ( µ )). On the other hand, on { T ( µ ) < ∞} ∩ { d = k } , we have that, after theprocess reaches µ it will never return to the root. Hence L ( ρ, ∞ ) = 1 + Z T ( µ )0 { X u = ρ } d u + Z ∞ T ( µ ) { X u = ρ } d u = 1 + Z T ( µ )0 { X u = ρ } d u = L ( ρ, T ( µ )) . Using this fact, combined with L ( ρ, T ( µ )) ≤ X ν : | ν | = k − L ( ρ, T (fc( ν )) , and 1l { d = k } ≤ { d>k − } ≤ D ck − , we have L ( ρ, ∞ )1l { d = k } = L ( ρ, T ( µ ))1l { d = k } ≤ (cid:16) X ν : | ν | = k − L (cid:0) ρ, T (fc( ν )) (cid:1)(cid:17) { d = k } ≤ (cid:16) X ν : | ν | = k − L (cid:0) ρ, T (fc( ν )) (cid:1)(cid:17) D ck − . (4.11)Using (4.11), Holder’s inequality (with p = 5 /
4) and (4.10) we have E h ( L ( ρ, ∞ )) / i = ∞ X k =1 E h(cid:0) L ( ρ, ∞ ) (cid:1) / { d = k } i = ∞ X k =1 E h(cid:0) L ( ρ, ∞ )1l { d = k } (cid:1) / i ≤ ∞ X k =1 E X ν : | ν | = k − L (cid:0) ρ, T (fc( ν )) (cid:1) D ck − / ≤ ∞ X k =1 E X ν : | ν | = k − L (cid:0) ρ, T (fc( ν )) (cid:1) / ( γ b ) b k − / ≤ ∞ X k =1 E X ν : | ν | = k − L (cid:0) ρ, T (fc( ν )) (cid:1) ( γ b ) b k − / (using L ( ρ, t ) ≥ ≤ ∞ X k =1 b k X ν : | ν | = k − E (cid:2) L (cid:0) ρ, T (fc( ν )) (cid:1) (cid:3) ( γ b ) b k − / (by Jensen) ≤ ∞ X k =1 b k (37) k ( γ b ) b k − / < ∞ . (cid:3) Lemma 4.9.
For ν = ρ , there exists a random variable ∆ ν which is σ (cid:8) P ( u, v ) : u, v ∈ Vert (Λ ν ) (cid:9) -measurable, such that i) L ( ν, ∞ ) ≤ ∆ ν , and ii) ∆ ν and L ( ρ, ∞ ) are identically distributed. Proof.
Let e X := { e X t , t ≥ } be the extension of X on Λ ν . Define∆ ν := 1 + Z ∞ { e X t = ν } d t. By construction, this random variable satisfies i) and ii) and is σ (cid:8) P ( u, v ) : u, v ∈ Vert(Λ ν ) (cid:9) -measurable. (cid:3) Theorem 4.10. E (cid:2) ( τ ) / (cid:3) < ∞ . Proof.
Suppose we relabel the vertices that have been visited by time τ , using θ , θ , . . . , θ Π ,where vertex ν is labeled θ k if there are exactly k − ν . Notice that ∆ ν and { θ k = ν } are independent, because they are determined bydisjoint non-random sets of Poisson processes (∆ ν is σ (cid:8) P ( u, v ) : u, v ∈ Vert(Λ ν ) (cid:9) -measurable).As the variables ∆ ν , with ν ∈ Vert( G b ), share the same distribution, for any p >
0, we have E [∆ pθ k ] = E [∆ pν ] = E [ L ( ρ, ∞ ) p ] . IMIT THEOREMS FOR VRJP ON TREES 17
By Jensen’s and Holder’s (with p = 12 /
11) inequalities, Lemma 4.9 i) and ii), and Lemma 4.8,we have E (cid:2) ( τ ) / (cid:3) ≤ E Π X k =1 ∆ θ k ! / ≤ E " Π (11 / − X k =1 (∆ θ k ) / = E " ∞ X k =1 ∆ / θ k Π / { Π ≥ k } ≤ ∞ X k =1 E h ∆ / θ k i / E (cid:2) Π / (Π ≥ k ) (cid:3) / ≤ C (3) b ∞ X k =1 E (cid:2) Π / (cid:3) / P (Π ≥ k ) / (by Cauchy-Schwartz and Lemma 4.8) ≤ C (4) b ∞ X k =1 P (Π ≥ k ) / , (by Lemma 4.5) , for some positive constants C (3) b and C (4) b . It remains to prove the finiteness of the last sum.We use the fact lim k →∞ k P (Π ≥ k ) = 0 . (4.12)The previous limit is a consequence of the well-known formula ∞ X k =1 k P (Π ≥ k ) = E [Π ] , (4.13)and the finiteness of E [Π ] by virtue of Lemma 4.5. ∞ X k =1 P (Π ≥ k ) / = ∞ X k =1 k (cid:16) k P (Π ≥ k ) (cid:17) / < ∞ . (cid:3) Lemma 4.11. sup x ∈ [1 , E [( L ( ρ, ∞ )) / | L ( ρ, T ) = x ] < ∞ . Proof.
Using 4.9, and the fact that ∆ X T is independent of L ( ρ, T ), we havesup x ∈ [1 , E [( L ( X T , ∞ )) / | L ( ρ, T ) = x ] ≤ sup x ∈ [1 , E [(∆ X T ) / | L ( ρ, T ) = x ]= E [(∆ X T ) / ] = E [( L ( ρ, ∞ )) / ] < ∞ . (4.14)Given L ( ρ, T ) = x , the process X restricted to { ρ, X T } is VRJP which starts from X T ,with initial weights a ρ = x and 1 on X T . This process runs up to the last visit of X to one ofthese two vertices. Using Lyapunov inequality, i.e. E [ Z q ] /q ≤ E [ Z p ] /p whenever 0 < q ≤ p ,Lemma 4.7, and the fact x ≥
1, we have E h(cid:16) L ( ρ, T n ) L ( X T , T n ) (cid:17) / | L ( X T , T n ) , { L ( ρ, T ) = x } i ≤ E h(cid:16) L ( ρ, T n ) L ( X T , T n ) (cid:17) | L ( X T , T n ) , { L ( ρ, T ) = x } i / ≤ ( x + 3 x + 3 x ) / ≤ x + 3 x + 3 x. (4.15) Finally E [( L ( ρ, T n )) / | L ( ρ, T ) = x ] = E h(cid:16) L ( ρ, T n ) L ( X T , T n ) (cid:17) / ( L ( X T , T n )) / | L ( ρ, T ) = x i ≤ ( x + 3 x + 3 x ) E h ( L ( X T , T n )) / | L ( ρ, T ) = x i ≤ ( x + 3 x + 3 x ) E h ( L ( X T , ∞ )) / | L ( ρ, T ) = x i ≤ ( x + 3 x + 3 x ) E [( L ( ρ, ∞ )) / ] . (4.16)By sending n → ∞ and taking the suprema over x ∈ [1 ,
2] we getsup x ∈ [1 , E [( L ( ρ, T n )) / | L ( ρ, T ) = x ] ≤ E [( L ( ρ, ∞ )) / ] < ∞ . (cid:3) Theorem 4.12. sup x ∈ [1 , E (cid:2) ( τ ) / | L (cid:0) ρ, T (cid:1) = x (cid:3) < ∞ . Proof.
Label the vertices at level 1 by µ , µ , . . . , µ b . Let τ ( µ i ) be the first cut time of theextension of X on Λ µ i . This extension is VRJP on Λ µ i with initial weights 1, hence we canapply Theorem 4.10 to get E [( τ ( µ i )) / ] < ∞ . (4.17)Hence, it remains to prove that for x ∈ [1 , E h(cid:0) τ (cid:1) / | L (cid:0) ρ, T (cid:1) = x i ≤ E h(cid:16) L ( ρ, ∞ ) + max i τ ( µ i ) (cid:17) / (cid:12)(cid:12)(cid:12) L (cid:0) ρ, T (cid:1) = x i ≤ E h(cid:16) L ( ρ, ∞ ) + b X i =1 τ ( µ i ) (cid:17) / (cid:12)(cid:12)(cid:12) L (cid:0) ρ, T (cid:1) = x i ≤ ( b + 1) / − E h(cid:16) L ( ρ, ∞ ) (cid:17) / | L (cid:0) ρ, T (cid:1) = x i + ( b + 1) / E h(cid:0) ( τ ( µ ) (cid:1) / i < ∞ , where we used Jensen’s inequality, the independence of τ ( µ i ) and T and Lemma 4.11. In fact,as L (cid:0) ρ, ∞ (cid:1) ≥ E [( L ( ρ, ∞ )) / | L ( ρ, T ) = x ] ≤ E [( L ( ρ, ∞ )) / | L ( ρ, T ) = x ] < ∞ . (cid:3)
5. Splitting the path into one-dependent pieces
Define Z i = L ( X τ i , ∞ ), with i ≥ Lemma 5.1.
The process Z i , with i ≥ is a homogenous Markov chain with state space [1 , . Proof.
Fix n ≥
1. On { Z n = x } ∩ { X τ n = ν } the random variable Z n +1 is determined bythe variables { P ( u, v ) , u, v ∈ Λ ν , u = ν } . In fact these Poisson processes, on the set { Z n = x } ∩ { X τ n = ν } , are the only ones used to generate the jumps of the process { X T (fc( ν )+t } t ≥ . Let E , E , . . . , E n − , E n +1 be Borel subsets of [0 , { Z n = x } ∩ { X τ n = ν } , the IMIT THEOREMS FOR VRJP ON TREES 19 two events { Z n +1 ∈ E n +1 } and { Z ∈ E , Z ∈ E , . . . Z n − ∈ E n − } are independent becauseare determined by disjoint collections of Poisson processes. By symmetry P ( Z n +1 ∈ E n +1 | { Z n = x } ∩ { X τ n = ν } )does not depend on ν . Hence P ( Z n +1 ∈ E n +1 | Z ∈ E , Z ∈ E , . . . Z n − ∈ E n − , Z n = x )= X ν P ( Z n +1 ∈ E n +1 | Z ∈ E , . . . , Z n − ∈ E n − , Z n = x, X τ n = ν ) P ( X τ n = ν | Z ∈ E , . . . , Z n = x )= P ( Z n +1 ∈ E n +1 | Z n = x, X τ n = ν ) = P ( Z n +1 ∈ E n +1 | Z n = x ) . This implies that Z is a Markov chain. The self-similarity property of G b and X yields thehomegeneity. (cid:3) From the previous proof, we can infer that given Z i = x , the random vectors ( τ i +1 − τ i , l i +1 − l i )and ( τ i − τ i − , l i − l i − ), are independent. Proposition 5.2. sup i ∈ N sup x ∈ [1 , E h(cid:0) τ i +1 − τ i (cid:1) / (cid:12)(cid:12) Z i = x i < ∞ (5.1)sup i ∈ N sup x ∈ [1 , E h(cid:0) l i +1 − l i (cid:1) / | Z i = x i < ∞ . (5.2) Proof.
We only prove (5.1), the proof of (5.2) being similar. Define C := (cid:8) X t = ρ, ∀ t > T (cid:9) and fix a vertex ν . Notice that by the self-similarity property of G b , we have E h ( τ i +1 − τ i ) / | { Z i = x } ∩ { X τ i = ν } i = E h ( τ ) / |{ L ( ρ, T ) = x } ∩ C i . By the proof of Lemma 2.2, we have thatinf ≤ x ≤ P (cid:0) C (cid:12)(cid:12) L ( ρ, T ) = x (cid:1) ≥ bb + x P ( A ) ≥ (1 − γ b ) bb + 2 > . (5.3)Hence sup x : x ∈ [1 , E h ( τ ) / (cid:12)(cid:12) L ( ρ, T ) = x i ≥ sup x : x ∈ [1 , E h ( τ ) / (cid:12)(cid:12) { L ( ρ, T ) = x } ∩ C i P ( C | L ( ρ, T ) = x ) ≥ (1 − γ b ) bb + 2 sup x : x ∈ [1 , E h ( τ ) / (cid:12)(cid:12) { L ( ρ, T ) = x } ∩ C i ≥ (1 − γ b ) bb + 2 sup x : x ∈ [1 , E h ( τ i +1 − τ i ) / | { Z i = x } ∩ { X τ i = ν } i Hence E h ( τ i +1 − τ i ) / | { Z i = x } ∩ { X τ i = ν } i ≤ b + 2 b (1 − γ b ) sup ≤ x ≤ E h ( τ ) / | { L ( ρ, T = x } i . (cid:3) Next we prove that Z satisfies the Doeblin condition. Lemma 5.3.
There exists a probability measure φ ( · ) and < λ ≤ , such that for every Borelsubset B of [1 , , we have P (cid:0) Z i +1 ∈ B | Z i = z (cid:1) ≥ λ φ ( B ) ∀ z ∈ [1 , . (5.4) Proof. As Z i is homogeneous, it is enough to prove (5.4) for i = 1. In this proof we show thatthe distribution of Z is absolutely continuous and we compare it to 1+ an exponential withparameter 1 conditionated on being less than 1. The analysis is technical because Z i dependon the behaviour of the whole process X . Our goal is to find a lower bound for P (cid:16) Z ∈ ( x, y ) (cid:12)(cid:12) Z = z (cid:17) , with z ∈ [1 , . (5.5)Moreover, we require that this lower bound is independent of z ∈ [1 , ε ∈ (0 , { Z ∈ ( x, y ) , Z ∈ I ε ( z ) } , where I ε ( z ) := ( z − ε, z + ε ). Fix z ∈ [1 ,
2] and consider the functione − ( b + u )( t − − ( b + 1)e − ( b +2) e − ( t − . (5.6)Its derivative with respect t is( b + 1)e − ( b +2) − ( t − − ( b + u )e − ( b + u )( t − , which is non-positive for t ∈ [1 ,
2] and u ∈ [1 , b + 1)e − ( b +2) − ( t − − ( b + u )e − ( b + u )( t − ≤ ( b + 1)e − ( b +2) − (1 − − ( b + u )e − ( b + u )(2 − = ( b + 1)e − ( b +2) − ( b + u )e − ( b + u ) ≤ . Hence for fixed u ∈ [1 , t ∈ [1 , ≤ x < y ≤ − ( b + u )( x − − e − ( b + u )( y − ≥ ( b + 1)e − ( b +2) (cid:0) e − ( x − − e − ( y − (cid:1) . (5.7)We use this inequality to get a lower bound for the probability of the event { Z ∈ ( x, y ) , Z ∈ I ε ( z ) } . Our strategy is to calculate the probability of a suitable subset of the latter set. Considerthe following event. Suppose thata) T <
1, thenb) the process spends at X T an amount of time enclosed in ( z − − ε, z − ε ), thenc) it jumps to a vertex at level 2, spends there an amount of time t where t + 1 ∈ ( x, y ),andd) it jumps to level 3 and never returns to X T .In the event just described, levels 1 and 2 are the first two cut levels, and { Z ∈ ( x, y ) , Z ∈ I ε ( z ) } holds. The probability that a) holds is exactly e − b . Given T = s −
1, the time spentin X T before the first jump is exponential with parameter ( b + s ). Hence b) occurs withprobability larger than inf s ∈ [1 , (cid:16) e − ( b + s )( z − ε ) − e − ( b + s )( z + ε ) (cid:17) . Given a) and b), the process jumps to level 2 and then to level 3 with probability larger than (cid:0) b/ ( b + 2) (cid:1)(cid:0) b/ ( b + z + ε )). The conditional probability, given a) and b), that the time gapbetween these two jumps lies in ( x − , y −
1) is larger thaninf u ∈ I ε ( z ) (cid:16) e − ( b + u )( x − − e − ( b + u )( y − (cid:17) . IMIT THEOREMS FOR VRJP ON TREES 21
At this point, a lower bound for the conditional probability that the process never returns to X T is bb + y (1 − α b ) ≥ bb + 2 (1 − α b ) . We have P (cid:16) Z ∈ ( x, y ) , Z ∈ I ε ( z ) (cid:17) ≥ e − b b ( b + 2) ( b + z + ε ) inf s ∈ [1 , (cid:16) e − ( b + s )( z − ε ) − e − ( b + s )( z + ε ) (cid:17) inf u ∈ I ε ( z ) (cid:16) e − ( b + u )( x − − e − ( b + u )( y − (cid:17) (1 − α b ) ≥ (1 − α b )e − b b ( b + 1)( b + 2) ( b + z + ε ) e − ( b +2) (cid:0) e − ( x − − e − ( y − (cid:1) inf s ∈ [1 , (cid:16) e − ( b + s )( z − ε ) − e − ( b + s )( z + ε ) (cid:17) , (5.8)where in the last inequality we used (5.7). Notice that there exists a constant C (4) b > ε ∈ (0 , inf z,s ∈ [1 , ε (cid:16) e − ( b + s )( z − ε ) − e − ( b + s )( z + ε ) (cid:17) ≥ C (4) b . (5.9)Summarizing, we have P (cid:16) Z ∈ ( x, y ) , Z ∈ I ε ( z ) (cid:17) ≥ C (5) b (cid:0) e − ( x − − e − ( y − (cid:1) ε, (5.10)where C (5) b depends only on b .In order to find a lower bound for (5.5) we need to prove thatsup ε ∈ (0 , ε P (cid:16) Z ∈ I ε ( z ) (cid:17) ≤ C (6) b , (5.11)for some positive constant C (6) b . To see this, recall the definition of B j from the proof ofTheorem 3.5, and ζ from (3.3). The event that level i is not a cut level is subset of ( B i ∩ A i ) c (see the proof of Theorem 3.5). Denote by m i = h (cid:0) X T i , fc( X T i ) (cid:1) , which is exponential withmean 1 /b . Then P ( Z ∈ I ε ( z )) ≤ ∞ X i P (cid:16) m i ∈ I ε ( z ) (cid:17) P (cid:16) i − \ k =1 ( B k ∩ A k ) c | m i ∈ I ε ( z ) (cid:17) ≤ Cε ∞ X i P (cid:16) i − \ k =1 ( B k ∩ A k ) c | m i ∈ I ε ( z ) (cid:17) , where the constant C is independent of ε and z . It remains to prove that the sum in theright-hand side is bounded by a constant independent of ε . Notice that, for i > ζ , A i − ζ and B i − ζ are independent of m i . Moreover the events A i − ζ ∩ B i − ζ , A i − ζ ∩ B i − ζ , A i − ζ ∩ B i − ζ , . . . are independent by the proof of Proposition 3.1. Hence P ( Z ∈ I ε ( z )) ≤ Cε ∞ X i P (cid:16) [( i − /ζ ] \ k =1 ( B i − kζ ∩ A i − kζ ) c | m i ∈ I ε ( z ) (cid:17) = Cε ∞ X i P (cid:16) [( i − /ζ ] \ k =1 ( B i − kζ ∩ A i − kζ ) c (cid:17) (by independence)= Cε ∞ X i P (cid:16) ( B i − kζ ∩ A i − kζ ) c (cid:17) [( i − /ζ ] < ∞ . Combining (5.8), (5.9) and (5.11), we get P (cid:16) Z ∈ ( x, y ) (cid:12)(cid:12) Z = z (cid:17) = lim ε ↓ P (cid:16) Z ∈ I ε ( z ) (cid:17) P (cid:16) Z ∈ ( x, y ) , Z ∈ I ε ( z ) (cid:17) ≥ λ (cid:0) e − ( x − − e − ( y − (cid:1) (1 − e − ) , (5.12)for some λ >
0. . A finite measure defined on field A can be extended uniquely to the sigma-field generated by A , and this extension coincides with the outer measure. We apply this resultto prove that (5.12) holds for any Borel set C ⊂ [1 , E , the right-hand side of (5.12) can be written inan integral form as λ Z E e − x +1 (1 − e − ) d x. Fix a Borel set C ⊂ [1 ,
2] and ε > E i ⊂ [1 , i ≥
1, with C ⊂ S ∞ i =1 E i , such that P ( Z ∈ C (cid:12)(cid:12) Z = z ) ≥ ∞ X i =1 P ( Z ∈ E i (cid:12)(cid:12) Z = z ) − ε ≥ λ ∞ X i =1 Z E i e − x +1 (1 − e − ) d x − ε ≥ λ Z C e − x +1 / (cid:0) − e − (cid:1) d x − ε. The first inequality is true because of the extension theorem, and the fact that the right-handside is a lower bound for the outer measure, for a suitable choice of the E i s. The inequality(5.4), with φ ( C ) = R C e − x +1 / (1 − e − ) d x , follows by sending ε to 0. (cid:3) The proof of the following Proposition can be found in [2].
Proposition 5.4.
There exists a constant ̺ ∈ (0 , and a sequence of random times { N k , k ≥ } , with N = 0 , such that • the sequence { Z N k , k ≥ } consists of independent and identically distributed randomvariables with distribution φ ( · ) • N i − N i − , i ≥ , are i.i.d. with a geometric distribution( ρ ), i.e. P ( N − N = j ) = (1 − ̺ ) j − ̺, with j ≥ . IMIT THEOREMS FOR VRJP ON TREES 23
Lemma 5.5. sup i ∈ N E [( τ N i +1 − τ N i ) ] < ∞ . Proof.
It is enough to prove E [( τ N − τ N ) ] < ∞ . By virtue of Jensen’s inequality, we havethat E h ( τ k − τ m ) / i = E h ( k − m X j =1 τ m + j − τ m + j − ) / i ≤ ( k − m ) / E [( τ − τ ) / ] . (5.13)Using Holder with p = 11 /
10, we have E [( τ N − τ N ) ] = ∞ X k =2 k − X m =1 E (cid:2) ( τ k − τ m ) { N = m, N = k } (cid:3) ≤ ∞ X k =2 k − X m =1 E h(cid:0) τ k − τ m (cid:1) / i / P ( N = m, N − N = k − m ) / = ∞ X k =2 k − X m =1 E h(cid:0) τ k − τ m (cid:1) / i / P ( N = m ) / P ( N − N = k − m ) / ≤ ∞ X k =2 k − X m =1 ( k − m ) E [( τ − τ ) / ] / ̺ / (cid:0) − ̺ (cid:1) ( k − / ≤ ̺ / E [( τ − τ ) / ] / ∞ X k =2 k (cid:0) − ̺ (cid:1) ( k − / < ∞ , where we used the fact that 0 < ̺ < (cid:3) With a similar proof we get the following result.
Lemma 5.6. sup i ∈ N E (cid:2)(cid:0) l N i +1 − l N i (cid:1) (cid:3) < ∞ . Definition 5.7.
A process { Y k , k ≥ } , is said to be one-dependent if Y i +2 is independentof { Y j , with ≤ j ≤ i } . Lemma 5.8.
Let Υ i := (cid:0) τ N i +1 − τ N i , l N i +1 − l N i (cid:1) , for i ≥ . The process Υ := (cid:8) Υ i , i ≥ (cid:9) isone-dependent. Moreover Υ i , i ≥ , are identically distributed. Proof.
Given Z N i − , Υ i is independent of { Υ j , j ≤ i − } . Thus, it is sufficient to prove thatΥ i is independent of Z N i − . To see this, it is enough to realize that given Z N i , Υ i is independentof Z N i − , and combine this with the fact that Z N i and Z N i − are independent. The variables Z N i are i.i.d., hence { Υ i , i ≥ } , are identically distributed. (cid:3) The Strong Law of Large Numbers holds for one-dependent sequences of identically dis-tributed variables bounded in L . To see this, just consider separately the sequence of randomvariables with even and odd indices and apply the usual Strong Law of Large Numbers to eachof them.Hence, for some constants 0 < C (7) b , C (8) b < ∞ , we havelim i →∞ τ N i i → C (7) b , and lim i →∞ l N i i → C (8) b , a.s.. (5.14) Proof of Theorem 1. If τ N i ≤ t < τ N i +1 , then by the definition of cut level, we have l N i ≤ | X t | < l N i +1 . Hence l N i τ N i +1 ≤ | X t | t < l N i +1 τ N i . Let K (1) b = E [ l N − l N ] E [ τ N − τ N ] , (5.15)which are the constants in (5.14). Thenlim sup t →∞ | X t | t ≤ lim i →∞ l N i +1 τ N i = lim i →∞ l N i +1 i + 1 iτ N i = K (1) b , a.s..Similarly, we can prove that lim inf t →∞ | X t | t ≥ K (1) b , a.s..Now we turn to the proof of the central limit theorem. First we prove that there exists aconstant C ≥ l N m − K (1) b τ N m √ m = ⇒ Normal(0 , C ) , (5.16)where Normal(0 ,
0) stands for the Dirac mass at 0. To prove (5.16) we use a theorem from[15]. The reader can find the statement of this theorem in the Appendix, Theorem 6.1, (seealso [22]). In order to apply this result we first need to prove that the quantity1 m E h(cid:0) l N m − K (1) b τ N m (cid:1) i = E h(cid:16) l N m − K (1) b τ N m √ m (cid:17) i (5.17)converges. Call Y = l N − K (1) b ( τ N and let Y i = l N i − l N i − − K (1) b ( τ N i − τ N i − ), with i ≥
2. Thequantity in (5.17) can be written as 1 m E h(cid:0) m X i =1 Y i (cid:1) i . The random variables Y i are identically distributed with the exception of Y . From the definitionof K (1) b given in (5.15), we have E [ Y i ] = E [ l N − l N ] − E [ l N − l N ] = 0 . Hence Y i , with i ≥
1, is a zero-mean one-dependent process, and we get E h(cid:0) l N m − K (1) b τ N m (cid:1) i = E h(cid:16) m X i =1 Y i (cid:17) i = ( m − E [ Y ] + 2( m − E [ Y Y ] + E [ Y ] + 2 E [ Y Y ] . (5.18)This proves that the limit in (5.17) exists and is equal to E [ Y ] + 2 E [ Y Y ]. Now we face twooptions. If the limit is equal to zero, then using Chebishev we get thatlim m →∞ P (cid:16)(cid:12)(cid:12)(cid:12) l N m − Cτ N m √ m (cid:12)(cid:12)(cid:12) > ε (cid:17) = lim m →∞ P (cid:16)(cid:12)(cid:12)(cid:12) √ m m X i =1 Y i (cid:12)(cid:12)(cid:12) > ε (cid:17) ≤ lim m →∞ ε E h(cid:16) P mi =1 Y i √ m (cid:17) i = 0 . If the limit of the quantity in (5.17) is positive, then we can apply Theorem 6.1 and deducecentral limit theorem for Y i , i ≥
1, yielding (5.16).
IMIT THEOREMS FOR VRJP ON TREES 25
Now we use (5.16) to prove the central limit theorem for | X t | . If τ N m ≤ t < τ N m +1 , then | X t | − K (1) b tK (2) b √ t ≥ l N m − K (1) b τ N m +1 K (2) b √ τ N m +1 = r mτ N m +1 (cid:16) l N m − K (1) b τ N m √ m + K (1) b √ m ( τ N m − τ N m +1 ) (cid:17) = r mτ N m +1 (cid:16) P mi =1 Y i √ m − Y m K b √ m (cid:17) . (5.19)The last expression converges, by virtue of the Slutzky’s lemma, either to a Normal distributionor to a Dirac mass at 0, depending on whether the limit in (5.17) is positive or is zero. To seethis, notice that lim m →∞ r mτ N m +1 = s E [ τ N − τ N ] , a.s. P mi =1 Y i √ m = ⇒ Normal(0 , C )lim m →∞ Y m K b √ m = 0 , a.s. . Similarly | X t | − K (1) b tK (2) b √ t ≤ s m + 1 τ N m (cid:16) P m +1 i =1 Y i √ m + 1 + Y m +1 K b √ m (cid:17) , and the right-hand side converges to the same limit of the right-hand side of (5.19). (cid:3)
6. Appendix
We include a corollary to a result of Hoeffding and Robbins (see [15] or [22]).
Theorem 6.1 (Hoeffding-Robbins) . Suppose Y := { Y i , i ≥ } is a one-dependent processwhose components are identically distributed with mean 0. If • E [ Y δi ] < ∞ , for some δ > , • lim n →∞ n Var( P ni =1 Y i ) converges to a positive finite constant K , then P ni =1 Y i − n E [ Y ] K √ n = ⇒ Normal(0 , . Aknowledgement.
The author was supported by the DFG-Forschergruppe 718 ”Analysis andstochastics in complex physical systems”, and by the Italian PRIN 2007 grant 2007TKLTSR”Computational markets design and agent-based models of trading behavior”. The authorwould like to thank Burgess Davis and an anonymous referee for helpful suggestions.
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