Almost sure convergence of vertex degree densities in the vertex-splitting model
AALMOST SURE CONVERGENCE OF VERTEX DEGREE DENSITIES IN THEVERTEX–SPLITTING MODEL
SIGURDUR ¨O. STEF ´ANSSON AND ERIK TH ¨ORNBLADA
BSTRACT . We study the limiting degree distribution of the vertex splitting model introducedin [3]. This is a model of randomly growing ordered trees, where in each time step the tree isseparated into two components by splitting a vertex into two, and then inserting an edge betweenthe two new vertices. Under some assumptions on the parameters, related to the growth of themaximal degree of the tree, we prove that the vertex degree densities converge almost surelyto constants which satisfy a system of equations. Using this we are also able to strengthen andprove some previously non–rigorous results mentioned in the literature.
1. I
NTRODUCTION
The vertex splitting model is a recent model of randomly growing ordered trees introducedin [3]. It is a modification of a model of randomly growing trees encountered in the theory ofRNA-folding [4]. The parameters of the model are non-negative weights ( w i,j ) i,j ≥ , symmetricin the indices i and j , and the trees are grown randomly in discrete time steps according to thefollowing rules: Let T be an ordered tree, V T its set of vertices and denote the degree of a vertex v by deg( v ) .(1) Start with some finite tree T at time t := | V T | .(2) Given a tree T at time t ≥ t , select a vertex v in T with probability w deg( v ) (cid:80) v (cid:48) ∈ V T w deg( v (cid:48) ) where w i := i i +1 (cid:88) j =1 w j,i +2 − j . (1.1)(3) Partition the edges which contain v into disjoint sets of adjacent edges: E (cid:48) of size k − and E (cid:48)(cid:48) of size deg( v ) − k + 1 , with probability w k, deg( v )+2 − k w deg( v ) .(4) Remove the vertex v and the edges containing it and insert two new vertices v (cid:48) and v (cid:48)(cid:48) .Connect v (cid:48) by an edge to all vertices u such that uv is an edge in E (cid:48) and connect v (cid:48)(cid:48) toall vertices w such that wv is an edge in E (cid:48)(cid:48) . Add the edge v (cid:48) v (cid:48)(cid:48) (see Fig 1).F IG . 1. The vertex v of degree i := deg( v ) is split into two new vertices v (cid:48) and v (cid:48)(cid:48) of degrees k := deg( v (cid:48) ) and (cid:96) := deg( v (cid:48)(cid:48) ) = i + 2 − k respectively withprobability w k,(cid:96) /w i . Here i = 5 , k = 3 and (cid:96) = 4 . Date : October 10, 2018.
Key words and phrases.
Vertex splitting, almost sure convergence, degree densities, random trees. a r X i v : . [ m a t h . P R ] N ov S. ¨O. STEF ´ANSSON AND E. TH ¨ORNBLAD
Remark 1.1.
The numbers ( w i ) i ≥ defined by (1.1) are called splitting weights and the numbers ( w i,j ) i,j ≥ are referred to as partitioning weights . We say that the vertex v which is selected instep (2) is split into the vertices v (cid:48) and v (cid:48)(cid:48) in step (4).In step (3) of the growth rules, the adjacency of edges is well-defined since the trees areordered. There are in general many choices of the sets E (cid:48) and E (cid:48)(cid:48) . When deg( v ) is even and k − v ) / there are exactly deg( v ) / different choices but otherwise there are deg( v ) different choices.We are interested in studying the distribution of vertex degrees in large random trees T grownaccording to the above rules. More precisely, let n t,k be the number of vertices of degree k inthe tree at time t . In [3] the asymptotics of the expected values E ( n t,k ) were studied under thefollowing assumptions.(A1) The splitting weights are linear, i.e. satisfy w i = ai + b for some real numbers a and b such that w i ≥ for all i ≥ .(A2) There is a finite integer d max such that w j,k = 0 if either j or k exceeds d max (no verticesof degree greater than d max are created in the growth process) and w ,k = w k, > for all ≤ k ≤ d max (it is possible to create vertices of degree d max starting from anyinitial tree). (Corresponds to (1) in Lemma 2.3 in [3].)(A3) w i,d max +2 − i > for some i satisfying ≤ i ≤ d max − (it is possible to split verticesof degree d max ). (Corresponds to (2) in Lemma 2.3 in [3].)(A4) The d max × d max matrix B given by the matrix elements B ij = jw i,j +2 − i − δ ij w i , ≤ i, j ≤ d max is diagonalizable. (Appears in Theorem 2.5 in [3].)It was shown that under these assumptions the limits ρ k := lim t →∞ E ( n t,k ) /t exist for ≤ k ≤ d max and are the unique positive solutions to ρ k = − w k w ρ k + d max (cid:88) i = k − i w k,i +2 − k w ρ i (1.2)such that (cid:80) ∞ k =1 ρ k = 1 and (cid:80) ∞ k =1 kρ k = 2 . The last two should be compared to the equations d max (cid:88) i =1 n t,i = t and d max (cid:88) i =1 in t,i = 2 t − , (1.3)which is (2.7) in [3].The condition (A1) is a very convenient technical condition and we will assume that it holdsthroughout the paper. The reason is that for linear splitting weights, the total weight of selectinga vertex in T only depends on the number of vertices in T , namely (cid:88) v ∈ V T w deg( v ) = d max (cid:88) i =1 w i n t,i = (2 a + b ) t − a = w t − a =: W t (1.4)by (1.3).In this paper we prove stronger results concerning convergence of the random variables n t,k .First of all, we prove almost sure convergence of n t,k /t towards ρ k satisfying (1.2) which im-mediately implies, by the boundedness of n t,k /t and the dominated convergence theorem, thatthe expected value converges. Furthermore, we relax some of the conditions (A2)–(A4) statedabove as will be mentioned in the statement of the results in Theorem 1.2. In particular we donot require the matrix B to be diagonalizable when d max < ∞ and we obtain partial resultswhen there is no bound on the maximum degree.Throughout this paper we do not take into account the structural properties of the trees, butonly analyze the asymptotic vertex degree densities. This means that the analysis fits into the LMOST SURE CONVERGENCE OF VERTEX DEGREE DENSITIES IN THE VERTEX–SPLITTING MODEL 3 framework of generalized P´olya urn models. Consider d max urns, labelled , , . . . , d max , ini-tially containing some number of balls. At each time step, draw a ball from urn i with probabilityproportional to w i n i , where n i is the number of balls in the i :th urn. Then put two balls backin, one in the k :th urn and one in the ( i + 2 − k ) :th urn, where the pair ( k, i + 2 − k ) is chosenwith probability w k,i +2 − k /w i . In this framework each vertex of degree i in the vertex splittingtree corresponds to a ball in the i :th urn. The body of literature regarding P´olya urns is large. Inparticular, whenever d max < ∞ , our results follow from [5]. In fact stronger results regardingasymptotic joint normality are attainable in this regime, but we do not pursue this matter further.Even though the results of [5] holds only for a finite number of urns, sometimes a reduction froma situation from infinitely many urns to a situation with finitely many urns is possible. For thevertex splitting model, such a reduction unfortunately only works for the subclass of splittingtrees for which, for all large enough i , the only positive partitioning weights are w ,i +1 , w ,i .By choosing the splitting and partitioning weights appropriately, the vertex splitting modelcontains several other known models. For instance, by letting w ,i +1 be the only positive parti-tioning weight, we may retrieve the random recursive trees by choosing w i = 1 and the randomplane recursive trees by choosing w i = i . These were analyzed in [6] by using the aforemen-tioned connection to generalized P´olya urns.Without giving complete details, we also extend our results to the setting considered in themotivating paper [4], in which a correspondence between coloured splitting trees and arch depo-sition models is exploited to analyze the secondary structure of RNA folding. In this case eachvertex is coloured either black or white. If a black vertex is chosen, it is recoloured white. If awhite vertex is chosen, it splits (similar to the aforementioned 1–coloured case) into two blackvertices. By and large the same methods apply, and moreover it turns out that there is a clearcorrespondence between the 1–coloured version and the 2–coloured version.1.1. Main results.
In the following we let d max be a positive integer or infinite. If d max < ∞ we will assume that initially no vertex has degree > d max and that Condition (A2) above issatisfied. Let s := inf { iw ,i +1 : 1 ≤ i < d max } . For each k such that ≤ k < d max + 1 ,define the sequence ( a ( j ) k ) j ≥ as follows. For k = 1 let a (0)1 = 0 ,a ( j +1)1 = 1 w + s (cid:32) s + ∞ (cid:88) i =2 ( iw ,i +1 − s ) a ( j ) i (cid:33) , and when ≤ k < d max + 1 let a (0) k = 0 ,a ( j +1) k = 1 w + w k ∞ (cid:88) i = k − iw k,i − k +2 a ( j ) i . The main result of the paper is the following.
Theorem 1.2.
Suppose that inf { iw ,i +1 : 1 ≤ i < d max } > . Then for each k such that ≤ k < d max + 1 the following limits exist and it holds almost surely that lim t →∞ n t,k t = lim j →∞ a ( j ) k =: a k and ( a k ) ≤ k We note that Theorem 1.2 requires fewer assumptions than those in (A1)–(A4).However, we have not provided conditions which guarantee that ( a k ) ∞ k =1 is a unique positivesolution to (1.5) satisfying (1.6).Whenever d max < ∞ however, uniqueness is guaranteed sincethe condition that inf { iw ,i +1 : 1 ≤ i < d max } > is equivalent to Condition (A2) abovewhich guarantees that (1.2) has rank at least d max − . Equations (1.6) fix the remaining constant.In the case d max = ∞ we do not know how to prove that the system has a unique solution, butwe comment on some special examples in the next section.The case d max < ∞ is implicitly covered by the case d max = ∞ by assuming that there issome i for which w ,i +1 = 0 . This allows for the identification of three regimes as follows.I. There is some i ≥ such that w ,i +1 = 0 .II. There is no i ≥ such that w ,i +1 = 0 , and inf { iw ,i +1 : 1 ≤ i < d max } = 0 .III. It holds that inf { iw ,i +1 : 1 ≤ i < d max } > .Theorem 1.2 deals with Case I and III, but we do not know how to extend these results to caseII. The assumption inf { iw ,i +1 : 1 ≤ i < d max } > in some sense ensures that the probabilityof performing the split i (cid:55)→ (1 , i + 1) does not become too small as i grows large.The rest of the paper is outlined as follows. In Section 2 we give some examples of thecase d max = ∞ to which Theorem 1.2 is applicable and where uniqueness of the solution to(1.5)–(1.6) is proved. The proof of Theorem 1.2 along with some technical results is presentedin Section 3. Finally, in Section 4 we sketch similar results for a two–coloured vertex splittingmodel that was originally considered in [4].2. E XAMPLES Before we turn to the proof of Theorem 1.2, we consider some explicit examples that can beanalysed using Theorem 1.2. Throughout we assume that w i = ai + b where ai + b > for all ≤ i ≤ d max − . If a (cid:54) = 0 , it is easy to see that the growth rules defined in the introductionare not changed if we take w i = i + x with x = b/a , so for notational convenience we willsometimes use this definition instead, unless we are interested in the case of constant splittingweights.We will focus on the case where there is no bound on the vertex degrees since the case when d max < ∞ is already covered. In general we cannot show that the system of equations inTheorem 1.2 has a unique solution. However, in some specific cases this is possible, and wepresent some of these here.2.1. Preferential attachment. Preferential attachment–type models are obtained by attachingnew edges to existing vertices, where the vertex is chosen with probability proportional to thevertex degree. In the vertex–splitting model with d max = ∞ , this is obtained by setting allpartitioning weights to zero, except for w i +1 , = w ,i +1 = w i i for all i ≥ . The conditions ofTheorem 1.2 are satisfied, so the limiting degree distribution ( a i ) ∞ i =1 satisfies the equations ( w + w ) a = ∞ (cid:88) i =1 w i a i ( w k + w ) a k = w k − a k − , k ≥ . LMOST SURE CONVERGENCE OF VERTEX DEGREE DENSITIES IN THE VERTEX–SPLITTING MODEL 5 The relations (cid:80) ∞ i =1 a i = 1 and (cid:80) ∞ i =1 ia i = 2 imply that (cid:80) ∞ i =1 w i a i = w , so a = w w + w .Then, iterating (2.1) we find that a k = w k − w k + w a k − = · · · = a k (cid:89) i =2 w i − w i + w = w w k k (cid:89) i =1 w i w i + w , for all k ≥ . It is not difficult to analyse the asymptotics of the sequence ( a k ) ∞ k =1 . One canshow that when a (cid:54) = 0 a k = (2 + x )Γ (2 x + 3) Γ ( k + x + 1)( k + x )Γ ( x + 1) Γ ( k + 2 x + 3) ∼ (2 + x )Γ (2 x + 3)Γ ( x + 1) k − − x as k → ∞ (2.1)by using standard asymptotics for the Gamma function.If w i = i , then the preferential attachment model is equivalent to the model of random planerecursive trees. In this case the exact solution is given by a k = 4 k ( k + 1)( k + 2) , k ≥ . (2.2)This was originally proved by M´ori [7] using martingale methods. He also achieved results onjoint normality of the degree densities. The case w i = 1 is equivalent to the case of randomrecursive trees; the exact solution being a k = 2 − k , k ≥ , (2.3)in this case. Using the connection to generalized P´olya urns mentioned in the introduction,Janson [6] proved stronger results that imply the almost sure convergence to the densities in(2.2) and (2.3).2.2. Uniform partitioning weights. Let w i = i + x , where x > − . In [3], the expecteddegree densities were studied in the case of uniform partitioning weights, i.e. w i,k +2 − i = w k / (cid:18) k + 12 (cid:19) = 2 w k k ( k + 1) . The methods in [3] were non-rigorous in this case but the results were correct as we confirmhere. By Theorem 1.2 the asymptotic vertex degree densities ( a i ) ∞ i =1 satisfy ( w + w k ) a k = ∞ (cid:88) i = k − i + 1 w i a i , k ≥ . Subtracting the k :th equation from the ( k + 1) :st yields the recursion ( w + w k ) a k − ( w + w k +1 ) a k +1 = 2 k w k − a k − , k ≥ (2.4)where we have defined a = 0 . The solution is given by a k = 1 C ( x ) 2 k − Γ( k + x )Γ( k )Γ( k + 3 + 2 x ) ( k + 1 + 2 x ) , (2.5)where C ( x ) = e √ π − − x I + x (1)2 + x and I + x is the modified Bessel function of the first kind. In particular, for x = 0 , i.e. splittingweights w k = k , we have a k = 2 k +2 ( k + 1)( e − k + 2)! . S. ¨O. STEF ´ANSSON AND E. TH ¨ORNBLAD It is easy to see that the obtained solutions are in fact unique. Namely, from (2.4) one caninductively determine constants C , C . . . such that a k = C k a for all k ≥ . The condition (cid:80) ∞ i =1 a i = 1 determines a uniquely as a function of these constants.If the splitting weights are constant, i.e. w i = b for some b > , then the solution to (2.4) isgiven by a k = 1 e k − . However, in this case iw ,i +1 = O ( i − ) , so the conditions of Theorem 1.2 are not satisfied.This falls into Case II identified in Remark 1.3, so it does not follow by our results that these arealso the almost sure limiting vertex densities.2.3. An infinite class. Our next example provides an infinite class of splitting trees for which aunique solution to the system of equations is attainable. It includes the preferential attachmentmodel and more generally a model of trees which grow by attachment and grafting , studied in[8] (see below).Let ( α i ) ∞ i =1 be a sequence in (0 , , such that inf i ≥ w i α i > . Suppose there exists some M ≥ such that iw ,i +1 = α i w i , iw ,i = (1 − α i ) w i for all i ≥ M with the exception that w , = (1 − α ) w when M = 2 .The conditions of Theorem 1.2 are satisfied. In particular, for k > M the asymptotic degreesequence satisfies ( w k + w ) a k = α k − w k − a k − + (1 − α k ) w k a k which is similar to the expressions in Section 2.1. Iterating we find a k = a M k (cid:89) i = M +1 α i − w i − w + α i w i =: C k a M for all k > M .For any ≤ k ≤ M ( w k + w ) a k = ∞ (cid:88) i = k − iw k,i − k +2 a i = M (cid:88) i = k − iw k,i − k +2 a i + a M ∞ (cid:88) i = M +1 iw k,i − k +2 C i where the final sum is zero unless k = 1 or k = 2 . For instance, choosing k = M yields ( w M + w ) a M = ( M − w M, a M − + M w M, a M so we can determine C M − so that a M − = C M − a M . Continue inductively to find a sequence ( C i ) M − i =1 such that a k = C k a M for ≤ k < M . In the case k = 1 and k = 2 one gets asystem of two equations involving a and a which may easily be seen to have a unique solutionwhich is a multiple of a M . Now, the condition (cid:80) ∞ i =1 a i = 1 means that a M (cid:80) ∞ i =1 C i = 1 , so a M = ( (cid:80) ∞ i =1 C i ) − , where we have put C M = 1 for consistency. But this allows us to uniquelydetermine the entire sequence ( a k ) ∞ k =1 .An example of a family of weights which belongs to the above class appears in [8] (with aminor modification in the dynamics which does not affect the limiting densities). The weightsin [8] are defined in terms of two parameters α, γ ∈ [0 , by w i = (cid:16) α − γ (cid:17) i + 2 γ − α − , for i ≥ LMOST SURE CONVERGENCE OF VERTEX DEGREE DENSITIES IN THE VERTEX–SPLITTING MODEL 7 and by choosing M = 2 and α i = 1 − αi w i for all i ≥ M . Note that when α = 0 this is thepreferential attachment model. In [8] the solution to (1.5) when < γ < was found to be a = 1 − α γ − α and a k = γ Γ (cid:16) − α − γ − γ (cid:17) Γ (cid:16) k − − α − γ (cid:17) (1 + γ − α )(2 − α )Γ (cid:16) − α − γ (cid:17) Γ (cid:16) k − − α − γ (cid:17) for k ≥ and for γ = 1 a = 1 − α − α and a k = 1(2 − α ) (cid:18) − α − α (cid:19) k − for k ≥ . The results agree with (2.1) and (2.3) when α = 0 (preferential attachment). There was no proofin [8] that these solutions are the limiting degree densities but our Theorem 1.2 along with theuniqueness of the solution confirms that they are the almost sure limit. In the case < γ < ,standard asymptotics of the Gamma function yield the power law a k ∼ γ Γ (cid:16) − α − γ − γ (cid:17) (1 + γ − α )(2 − α )Γ (cid:16) − α − γ (cid:17) k − − γ − γ as k → ∞ and when γ = 1 the densities decay exponentially with rate (1 − α ) / (2 − α ) .3. P ROOF OF T HEOREM d max = ∞ , the discussion being even simpler when d max is finite. First we state a key lemma,that appears in slightly more general form in [1]. Lemma 3.1 (Backhausz, M´ori [1]) . Let ( F t ) ∞ t =0 be a filtration. Let ( ξ t ) ∞ t =0 be a non-negativeprocess adapted to ( F t ) ∞ t =0 , and let ( u t ) ∞ t =1 , ( v t ) ∞ t =1 be non–negative predictable processes suchthat u t < t for all t ≥ and lim t →∞ u t = u > exists almost surely. Let w be a positiveconstant. Suppose that there exists δ > such that E [( ξ t − ξ t − ) |F t ] = O ( t − δ ) . If lim inf t →∞ v t w ≥ v for some constant v ≥ and E [ ξ t |F t − ] ≥ (cid:16) − u t t (cid:17) ξ t − + v t , then lim inf t →∞ ξ t tw ≥ vu + 1 a.s. We use Lemma 3.1 to prove the following lemma. This essentially follows the approach takenin the papers [2, 9]. We note here that the s –term present below does not appear in Theorem 1.2since we shall later prove that (cid:80) ∞ i =1 a i = 1 , but this is a priori not known. S. ¨O. STEF ´ANSSON AND E. TH ¨ORNBLAD Lemma 3.2. Suppose that s := inf { iw ,i +1 : i ≥ } > . Then lim j →∞ a ( j ) k =: a k exist foreach k , ( a k ) k ≥ is a positive bounded sequence satisfying ( w + w ) a = ∞ (cid:88) i =1 iw ,i +1 a i + s (cid:32) − ∞ (cid:88) i =1 a i (cid:33) , ( w + w k ) a k = ∞ (cid:88) i = k − iw k,i +2 − k a i ( k ≥ . and lim inf t →∞ n t,k t ≥ a k holds almost surely, Let us first state the idea behind the proof of Lemma 3.2. To ease notation we define A k =lim inf t →∞ n t,k t for each k ≥ .(1) For each k ≥ , show by induction that a ( j ) k ≤ A k for all j ≥ .(2) Prove that ( a ( j ) k ) ∞ j =1 is monotonically increasing (in j ) for each k . Since each suchsequence lies in the bounded set [0 , , the limit lim j →∞ a ( j ) k = a k exists. Then wehave that a k ≤ A k . Proof. Recall the definition of the total weight W t in (1.4). The following expressions for theexpected number of vertices of degree k , conditional on the tree at the previous time step, easilyfollow from the growth rules. For k = 1 we have that E [ n t, | F t − ] = n t − , + 1 W t − ∞ (cid:88) i =2 iw ,i +1 n t − ,i = n t − , (cid:18) − sW t − (cid:19) + sW t − n t − , + 1 W t − ∞ (cid:88) i =2 iw ,i +1 n t − ,i = n t − , (cid:18) − sW t − (cid:19) + sW t − ( t − 1) + 1 W t − ∞ (cid:88) i =2 ( iw ,i +1 − s ) n t − ,i . In the last line we used the fact that n t − , = ( t − − (cid:80) ∞ i =2 n t − ,i .For k ≥ we have that E [ n t,k | F t − ] = n t − ,k (cid:18) − w k W t − (cid:19) + 1 W t − ∞ (cid:88) i = k − iw k,i − k +2 n t − ,i We shall use the above analysis along with Lemma 3.1. By induction we prove that a ( j ) k ≤ A k for all j ≥ . For j = 0 we clearly have a (0) k = 0 ≤ A k . Suppose that a ( j ) k ≤ A k for some j and for all k ≥ . We prove first that a ( j +1)1 ≤ A . For this, define the following variables: ξ t = n t, ,w = 1 ,u t = sW t − t,v t = sW t − ( t − 1) + W t − (cid:80) ∞ i =2 ( iw ,i +1 − s ) n t − ,i . We note that ( u t ) ∞ t =1 and ( v t ) ∞ t =1 are positive predictable sequences and that ξ t is non–negativeand adapted. Furthermore, we have that u t = sW t − t → sw =: u . The condition u t < t inLemma 3.1 is satisfied for large enough t , which is enough. In fact, the initial starting tree isirrelevant, so if u t ≥ t , one can grow the tree and wait until u t < t occurs, at which point LMOST SURE CONVERGENCE OF VERTEX DEGREE DENSITIES IN THE VERTEX–SPLITTING MODEL 9 Lemma 3.1 can be applied. Recall that s = inf { iw ,i +1 : i ≥ } . Using Fatou’s lemma andthe induction hypothesis we find that lim inf t →∞ v t ≥ sw + ∞ (cid:88) i =2 ( iw ,i +1 − s ) lim inf t →∞ n t − ,i W t − ≥ sw + ∞ (cid:88) i =2 ( iw ,i +1 − s ) a ( j ) i w = sw + 1 w ∞ (cid:88) i =2 ( iw ,i +1 − s ) a ( j ) i := v. Applying Lemma 3.1 we find that A = lim inf t →∞ n t, t ≥ vu + 1= sw + w (cid:80) ∞ i =2 ( iw ,i +1 − s ) a ( j ) isw + 1= 1 w + s (cid:32) s + ∞ (cid:88) i =2 ( iw ,i +1 − s ) a ( j ) i (cid:33) = a ( j +1)1 . For k ≥ define the following variables: ξ t = n t,k ,w = 1 ,u t = w k W t − t,v t = W t − (cid:80) ∞ i = k − iw k,i − k +2 n t − ,i . We note that ( u t ) ∞ t =1 and ( v t ) ∞ t =1 are positive predictable sequences and that ξ t is non–negativeand adapted. Furthermore, we have that u t = w k W t − t → w k w =: u . We now apply Fatou’s lemmaand use the induction hypothesis and find that lim inf t →∞ v t ≥ ∞ (cid:88) i = k − iw k,i − k +2 lim inf t →∞ n t − ,i W t − ≥ ∞ (cid:88) i = k − iw k,i − k +2 a ( j ) i w = 1 w ∞ (cid:88) i = k − iw k,i − k +2 a ( j ) i := v. Applying Lemma 3.1 we find that A k = lim inf t →∞ n t,k t ≥ vu + 1= w (cid:80) ∞ i = k − iw k,i − k +2 a ( j ) iw k w + 1= 1 w + w k ∞ (cid:88) i = k − iw k,i − k +2 a ( j ) i = a ( j +1)1 . We note finally that the technical condition E [( ξ t − ξ t − ) |F t ] = O ( t − δ ) is satisfied for all k ≥ . Indeed, the conditional expectation E [( ξ t − ξ t − ) |F t ] is bounded above by . Thiscompletes the Step 1, i.e. we have showed that a ( j ) k ≤ A k for all j ≥ and all k ≥ .We now prove that for each k ≥ , the sequence ( a ( j ) k ) ∞ j =0 is increasing. We use an inductiveargument. By construction we have that a (0)1 = 0 ≤ ss + w = a (1)1 . Suppose that the statement istrue for some j . Recall that s := inf { iw ,i +1 : i ≥ } > , so in particular iw ,i +1 − s ≥ for all i ≥ . Then a ( j +1)1 = 1 w + s (cid:32) s + ∞ (cid:88) i =2 ( iw ,i +1 − s ) a ( j ) i (cid:33) ≥ w + s (cid:32) s + ∞ (cid:88) i =2 ( iw ,i +1 − s ) a ( j − i (cid:33) ≥ a ( j )1 . This proves that the sequence ( a ( j ) k ) ∞ j =0 is increasing for k = 1 . The proof for k ≥ is similarand we omit this.For each k ≥ we thus have that the sequence ( a ( j ) k ) ∞ j =0 is an increasing sequence. It isbounded above by A k ≤ , so each such sequence must be convergent and we have the existenceof a limit lim j →∞ a ( j ) k = a k . By taking limits in (1.1–1.1) we find that a = 1 w + s (cid:32) s + ∞ (cid:88) i =2 ( iw ,i +1 − s ) a i (cid:33) and for each k ≥ a k = 1 w + w k ∞ (cid:88) i = k − iw k,i − k +2 a i . Interchanging limits and summation is justified since all terms are positive. Using w = w , , itis now easy to see that (3) is equivalent to ( w + w ) a = ∞ (cid:88) i =1 iw ,i a i + s (cid:32) − ∞ (cid:88) i =1 a i (cid:33) . This completes the proof. (cid:3) LMOST SURE CONVERGENCE OF VERTEX DEGREE DENSITIES IN THE VERTEX–SPLITTING MODEL 11 Next we show that the sequence ( a k ) ∞ k =1 , constructed in Lemma 3.2, defines a probabilitydistribution. After this we prove Theorem 1.2. First we prove the following lemma, which is inthe spirit of Lemma 2.3 in [3] Lemma 3.3. The sequence ( a k ) ∞ k =1 satisfies (cid:80) ∞ i =1 a i = 1 and (cid:80) ∞ i =1 ia i = 2 .Proof. Recall from Lemma 3.2 that for each j and k , a ( j ) k ≤ lim inf n →∞ n t,k t . Thus, for each j , ∞ (cid:88) k =1 a ( j ) k ≤ ∞ (cid:88) k =1 lim inf t →∞ n t,k t ≤ lim inf t →∞ t ∞ (cid:88) k =1 n t,k = 1 by Fatou’s lemma and (1.3). Similarly, for each j , (cid:80) ∞ k =1 ka ( j ) k ≤ . Letting j → ∞ it followsfrom monotonicity of a ( j ) k that the following series are convergent and satisfy ∞ (cid:88) k =1 a k ≤ and ∞ (cid:88) k =1 ka k ≤ . (3.1)Summing (3.2) and (3.2) over k = 1 , , . . . we find that w ∞ (cid:88) k =1 a k = − ∞ (cid:88) k =1 w k a k + ∞ (cid:88) k =1 ∞ (cid:88) i = k − iw k,i − k +2 a i + s (cid:32) − ∞ (cid:88) i =1 a i (cid:33) = − ∞ (cid:88) k =1 w k a k + 2 ∞ (cid:88) i =1 w i a i + s (cid:32) − ∞ (cid:88) i =1 a i (cid:33) = ∞ (cid:88) i =1 w i a i + s (cid:32) − ∞ (cid:88) i =1 a i (cid:33) (3.2)Similarly, multiplying the k :th equation by k and summing over k = 1 , , . . . , N we find byswapping sums and using i +1 (cid:88) k =1 kw k,i − k +2 = i + 22 i +1 (cid:88) k =1 w k,i − k +2 that w N (cid:88) k =1 ka k = − N (cid:88) k =1 kw k a k + N (cid:88) k =1 k ∞ (cid:88) i = k − iw k,i − k +2 a i + s (cid:32) − N (cid:88) i =1 a i (cid:33) = − N (cid:88) k =1 kw k a k + N − (cid:88) i =1 i (cid:32) i +1 (cid:88) k =1 kw k,i − k +2 (cid:33) a i + ∞ (cid:88) i = N i N (cid:88) k =1 kw k,i − k +2 a i + s (cid:32) − N (cid:88) i =1 a i (cid:33) = − N (cid:88) k =1 kw k a k + N − (cid:88) i =1 i i + 22 i +1 (cid:88) k =1 w k,i − k +2 a i + ∞ (cid:88) i = N i N (cid:88) k =1 kw k,i − k +2 a i + s (cid:32) − N (cid:88) i =1 a i (cid:33) = − N (cid:88) k =1 kw k a k + N − (cid:88) i =1 ( i + 2) w i a i + ∞ (cid:88) i = N i N (cid:88) k =1 kw k,i − k +2 a i + s (cid:32) − N (cid:88) i =1 a i (cid:33) which yields, with some simple rewriting w N (cid:88) k =1 ka k − N − (cid:88) k =1 w k a k − s (cid:32) − N (cid:88) i =1 a i (cid:33) = − N w N a N + ∞ (cid:88) i = N i N (cid:88) k =1 kw k,i − k +2 a i . The limit, as N → ∞ , of the left hand side exists by (3.1) and thus the limit of the right handside exists, denote it by x := lim N →∞ (cid:32) − N w N a N + ∞ (cid:88) i = N i N (cid:88) k =1 kw k,i − k +2 a i (cid:33) . Then w ∞ (cid:88) k =1 a k − ∞ (cid:88) k =1 w k a k − s (cid:32) − ∞ (cid:88) i =1 a i (cid:33) = x. (3.3)Finally, let A = (cid:80) ∞ i =1 a i and B = (cid:80) ∞ i =1 ia i . Putting w k = ak + b in (3.2) and (3.3) weobtain the linear system of equations (cid:40) (2 a + b ) A = aB + bA + s − sA, (2 a + b ) B = 2( aB + bA ) + s − sA + x having solutions A = 1 + axs ( a + b ) B = 2 + (2 a + s ) xs ( a + b ) . Since d max = ∞ , a ≥ and thus by (3.1) it necessarily holds that x ≤ . If x < then there isan M such that for all N ≥ M , N w N a N > − x/ . Therefore ∞ (cid:88) N =1 w N a N > ∞ (cid:88) N = M − x N = ∞ which contradicts (3.1). Thus x = 0 which gives A = 1 and B = 2 , as desired. (cid:3) Proof of Theorem 1.2. Already knowing that lim inf t →∞ n t,k t ≥ a k , the idea is to prove that lim sup t →∞ n t,k t ≤ a k for all k ≥ . By Lemma 3.3 we have that (cid:80) ∞ k =1 a k = 1 . By definitionit holds that (cid:80) ∞ k =1 n t,k t = 1 . The following calculation is routine and only uses Fatou’s lemmaand well–known facts about the limit inferior and limit superior. For any k ≥ we have that lim sup t →∞ n t,k t = lim sup t →∞ − ∞ (cid:88) j =1 j (cid:54) = k n t,j t ≤ − ∞ (cid:88) j =1 j (cid:54) = k lim inf t →∞ n t,j t ( Fatou’s Lemma ) ≤ − ∞ (cid:88) j =1 j (cid:54) = k a j ( Lemma. 3.2 )= a k . Thus, by the above along with Lemma 3.2 lim inf t →∞ n t,k t ≥ a k ≥ lim sup t →∞ n t,k t almost surely for all k ≥ . This implies that for all k ≥ t →∞ n t,k t = a k almost surely. Finally, Equations (1.5) and (1.6) follow from Lemma 3.2 and Lemma 3.3. (cid:3) LMOST SURE CONVERGENCE OF VERTEX DEGREE DENSITIES IN THE VERTEX–SPLITTING MODEL 13 4. A N EXTENSION TO TWO COLOURS As mentioned in the introduction, the motivation of this study originally comes from [4],wherein a two–coloured version of the vertex splitting tree was considered. In this model, eachvertex in the tree is coloured either white or black, the number of black and white vertices ofdegree i at time t denoted by n • t,i and n ◦ t,i respectively. The parameters of the model are the splitting weights ( w • i ) ∞ i =1 , ( w ◦ i ) ∞ i =1 and symmetric partitioning weights ( w ◦ i,j ) i,j ≥ , satisfying w ◦ i = i (cid:80) j +1 i =1 w ◦ j,i +2 − j . Start at time t = 2 with a single edge with both its endpoints black. Ateach time step, the tree evolves as follows.(1) Select a vertex v in the tree with probability proportional to w ◦ deg( v ) if v is white, and w • deg( v ) if v is black.(2) If v is black, make it white. If v is white, partition its edges into two disjoint setsof adjacent edges E (cid:48) of size k − and E (cid:48)(cid:48) of size deg( v ) − k + 1 with probability w ◦ k, deg( v )+2 − k w ◦ deg( v ) . Remove the vertex v and its incident edges. Insert two new black vertices v (cid:48) and v (cid:48)(cid:48) , such that v (cid:48) is connected to all vertices u such that uv is an edge in E (cid:48) , and v (cid:48)(cid:48) is connected to all vertices w such that wv is an edge in E (cid:48)(cid:48) . Add the edge v (cid:48) v (cid:48)(cid:48) .The paper [4] considered only the case corresponding to splitting weights w ◦ k = k + 1 and w • k = k (no bound on vertex degrees) and uniform partitioning weights w ◦ i,k +2 − i = w ◦ k / (cid:18) k + 12 (cid:19) for i = 1 , . . . , k + 1 . In this case the model is equivalent to a model of random RNA folding, see [4] where thecorrespondence is explained in detail. Under the assumption that the limit exists, the authors in[4] found that lim t →∞ E [ n ◦ t,k ] t = 2 k ke ( k + 2)! , lim t →∞ E [ n • t,k ] t = 2 k e ( k + 1)! , for all k ≥ . We consider a wider class of splitting weights and partitioning weights andare able to prove an analogue of Theorem 1.2 replacing E [ n • t,k ] t and E [ n ◦ t,k ] t with n • t,k t and n ◦ t,k t respectively and proving almost sure convergence. Thus we confirm, strengthen and generalizethe above results from [4].Let w ◦ k = (cid:0) a − b (cid:1) k + a and w • k = (cid:0) a − b (cid:1) k + b . This choice ensures that the total weightgrows linearly, i.e. ∞ (cid:88) i =1 (cid:0) w ◦ i n ◦ t,i + w • i n • t,i (cid:1) = ( a − b ) t + b (4.1)which can be showed by induction. Moreover, up to a multiplicative constant this is the uniquechoice of splitting weights resulting in linear growth for the total weight. Note also that a − b = w • / w ◦ / . It is also possible to show that ∞ (cid:88) i =1 (cid:0) n ◦ t,i + 2 n • t,i (cid:1) = t + 2 . (4.2)Equations (4.1) and (4.2) correspond to (1.3).In the following theorem, the sequences ( e ◦ k ) ∞ k =1 and ( e • k ) ∞ k =1 are constructed like the se-quence ( a k ) ∞ k =1 in Theorem 1.2, but we leave the exact details to the reader. Theorem 4.1. Suppose that inf { iw ◦ ,i +1 : 1 ≤ i < d max + 1 } > . Then there exist twonon–negative sequences ( e ◦ k ) ∞ k =1 and ( e • k ) ∞ k =1 such that lim t →∞ n ◦ t,k t a.s. = e ◦ k lim t →∞ n • t,k t a.s. = e • k satisfying ( w • k + w • / e • k = ∞ (cid:88) i = k − iw k,i − k +2 e ◦ k , ( w ◦ k + w ◦ / e ◦ k = w • k e • k , for all ≤ k < d max + 1 . Moreover (cid:80) ∞ i =1 (3 e ◦ i + 2 e • i ) = 1 and (cid:80) ∞ i =1 ( w ◦ i e ◦ i + w • i e • i ) = w • / . Note that the quantities n ◦ t,k t and n • t,k t are not degree densities any more, since the tree doesnot grow whenever a black vertex is selected. The actual asymptotic degree densities are givenby ρ • i = lim t →∞ n • t,i (cid:80) ∞ j =1 ( n • t,j + n ◦ t,j ) = lim t →∞ n • t,i t · t (cid:80) ∞ j =1 ( n • t,j + n ◦ t,j )= e • i (cid:80) ∞ j =1 ( e • j + e ◦ j ) and ρ ◦ i = e ◦ i (cid:80) ∞ j =1 ( e • j + e ◦ j ) . The proof of Theorem 4.1 is similar to that of Theorem 1.2, so we omit this. Instead wemention another result that relates the densities of any two–coloured process to a one–colouredprocess. Note however that this depends crucially on knowing that the solutions to the equationsin Theorems 1.2 and 4.1 are unique , something we do not know in general. If this is known, therest of the proof is straightforward and is omitted – it amounts to showing that the appropriateconditions and equations in Theorem 1.2 and Theorem 4.1 are satisfied. Proposition 4.2. Let w ◦ i and w • i be the splitting weights for the 2–colour model. Let w ◦ j,i +2 − j be the partitioning weights. Define a –colour process with splitting weights w i = w • i andpartitioning weights w j,i +2 − j = w • i w ◦ i w ◦ j,i +2 − j . Let ρ ◦ i and ρ • i be the degree densities of the 2–colour model, and let a i be the degree densities of the 1–colour model. If these are unique assolutions to the systems in Theorem 1.2 and Theorem 4.1 respectively, then ρ ◦ i + ρ • i = a i for all ≤ i < d max + 1 . Let us illustrate Proposition 4.2 by considering uniform partitioning weights in the the one–coloured and two–coloured cases, respectively. Indeed, this was the case considered in [4] andmentioned at the beginning of the section. In any case, it can be shown that ∞ (cid:88) k =1 (cid:18) k ke ( k + 2)! + 2 k e ( k + 1)! (cid:19) = e − e , LMOST SURE CONVERGENCE OF VERTEX DEGREE DENSITIES IN THE VERTEX–SPLITTING MODEL 15 so the asymptotic vertex densities are given by ρ ◦ k = 2 e e − k ke ( k + 2)! = 2 k +1 k ( e − k + 2)! ρ • k = 2 e e − k e ( k + 1)! = 2 k +1 ( e − k + 1)! . Now, if we follow the notation in Proposition 4.2 we obtain a one–coloured process, whichis precisely the uniform splitting model considered in Section 2.2, with weights w k = k andpartitioning weights w i,k +2 − i = w • k w ◦ k w ◦ i,k +2 − i = kk + 1 · k + 1 (cid:0) k +12 (cid:1) = k (cid:0) k +12 (cid:1) = w k (cid:0) k +12 (cid:1) . Recall (2.5), i.e. that the limiting vertex densities in this case were a k = 2 k +2 ( k + 1)( e − k + 2)! . In this particular case one verifies that a k = 2 k +2 ( k + 1)( e − k + 2)! = 2 k +1 k ( e − k + 2)! + 2 k +1 ( e − k + 1)! = ρ ◦ k + ρ • k , as predicted by Proposition 4.2. R EFERENCES [1] ´A. Backhausz and T. M´ori. A random model of publication activity. Discrete Appl. Math. , 162:78–89, 2014.[2] ´A. Backhausz and T. M´ori. Asymptotic properties of a random graph with duplications. Journal of AppliedProbability , 52(2):375–390, 2015.[3] F. David, M. Dukes, T. Jonsson, and S. ¨O. Stefansson. Random tree growth by vertex splitting. 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S TEF ´ ANSSON , D IVISION OF M ATHEMATICS , T HE S CIENCE I NSTITUTE , U NIVERSITY OF I CELAND ,D UNHAGA EYKJAVIK , I CELAND E-mail address : [email protected] E. T H ¨ ORNBLAD , D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF U PPSALA , B OX PP - SALA , S WEDEN E-mail address ::