The k-cut model in deterministic and random trees

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Gabriel Berzunza

Department of Mathematical Sciences University of Liverpool

Liverpool, U.K.

gabriel.berzunza-ojeda@liverpool.ac.uk

Xing Shi Cai Cecilia Holmgren^∗

Mathematics Department Uppsala University

Uppsala, Sweden

{xingshi.cai,cecilia.holmgren}@math.uu.se

Submitted: Apr 1, 2020; Accepted: Nov 27, 2020; Published: Jan 29, 2021 c

The authors. Released under the CC BY-ND license (International 4.0).

Abstract

The k-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the k-cut number of conditioned Galton-Watson trees converge after proper rescaling, which implies convergence in distribution to the same limit law re- gardless of the offspring distribution of the trees. This extends the result of Janson.

Using the same method, we also show that the k-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees.

Mathematics Subject Classifications: 60C05, 60F05, 05C05

1 Introduction and main result

In order to measure the difficulty for the destruction of a resilient network Cai et al. [12]

introduced a generalization of the cut model of Meir and Moon [30] where each vertex (or edge) needs to be cut k ∈ N times (instead of only once) before it is destroyed. More precisely, consider that the resilient network is a rooted tree Tn, with n ∈ N vertices. We

∗This work is supported by the Knut and Alice Wallenberg Foundation, the Swedish Research Council and The Swedish Foundations’ starting grant from Ragnar S¨oderbergs Foundation.

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assume that sibling vertices in Tn are ordered. (Such trees sometimes are referred to as plane trees.) We destroy it by removing its vertices as follows: Step 1: Choose a vertex uniformly at random from the component that contains the root and cut the selected vertex once. Step 2: If this vertex has been cut k times, remove the vertex together with the edges attached to it from the tree. Step 3: If the root has been removed, then stop. Otherwise, go to step Step 1. We let K_k(Tn) denote the (random) total number of cuts needed to end this procedure the k-cut number, i.e., K_k(Tn) models how much effort it takes to destroy the network. (For simplicity, we will omit the subscript k and write K(Tn).) It should be clear that one can define analogously an edge deletion version of the previous algorithm, where one needs to cut an edge k times before removing it from the root component. Then, one would be interested in the number K_e(Tn) of edge cuts needed to isolate the root of Tn.

The case k = 1 (i.e., the traditional cutting model of Meir and Moon [30]) has been well-studied by several authors. More precisely, Meir and Moon estimated the first and second moment of the 1-cut number in the cases when Tn is a Cayley tree [30] and a recursive tree [31]. Subsequently, several weak limit theorems for the 1-cut number have been obtained for Cayley trees (Panholzer [33, 34]), complete binary trees (Janson [24]), conditioned Galton-Watson trees (Janson [25] and Addario-Berry et al. [1]), recursive trees (Drmota et al. [16], Iksanov and M¨ohle [23]), binary search trees (Holmgren [19]) and split trees (Holmgren [20]). In the general case k > 1, the authors in [12] established first moment estimates of K(Tn) for families of deterministic and random trees, such as paths, complete binary trees, split trees, random recursive trees and conditioned Galton- Watson trees. In particular, the authors in [12] have proven a weak limit theorem for K(Tn) when Tn is a path consisting of n vertices. More recently, Cai and Holmgren [11]

also obtained a weak limit theorem in the case when Tⁿ is a complete binary tree.

In this work, we continue the investigation of this general cutting-down procedure in conditioned Galton-Watson trees and show that K(Tn), after a proper rescaling, converges in distribution to a non-degenerate random variable. More precisely, let ξ be a non- negative integer-valued random variable such that

E[ξ] = 1 and 0 < σ² := V ar(ξ) < ∞. (1) We further assume that the distribution of ξ is aperiodic. This last condition is to avoid unnecessary complications, but our results can be extended to the periodic case. We then consider a Galton-Watson process with (critical) offspring distribution ξ. Let Tⁿ be the family tree conditioned on its number of vertices being n ∈ N, providing that this conditioning makes sense. The main result of this paper is the following. We write → to^d denote convergence in distribution. (In the rest of the paper CRT stands for Continuum Random Tree.)

Theorem 1. Let k ∈ N. Let Tn be a Galton-Watson tree conditioned on its number of vertices being n ∈ N with offspring distribution ξ satisfying (1). Then,

σ^−1/kn^−1+1/2kK(Tn)→ Z^d _CRT, as n → ∞, (2)

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where Z_CRT is a non-degenerate random variable whose law is determined entirely by its moments: E[ZCRT⁰ ] = 1, and for q ∈ N, E[ZCRT^q ] = η_k,q with

ηk,q := q!

Z ∞ 0

· · · Z ∞

0

y1(y1+ y2) · · · (y1+ · · · + yq)e⁻(y1+···+yq )2

2 Fq(yq) dyq· · · dy1, (3) where y_q = (y₁, . . . , y_q) ∈ R^q+ and

F_q(y_q) :=

Z ∞ 0

Z x1

0

. . . Z xq−1

0

exp −y₁x^k₁ + y₂x^k₂ + · · · + y_qx^k_q k!

!

dx_q· · · dx₂ dx₁.

Furthermore, if E[ξ^p] < ∞ for every p ∈ Z>0, then for every q ∈ Z>0, σ^−q/kn^−q+q/2kE[K(Tⁿ)^q] → E[ZCRT^q ], as n → ∞.

In the case k = 1, Theorem 1 reduces to a Z_CRT having a Rayleigh distribution with density xe^−x²^/2, for x ∈ R+. More precisely, one can verify that η_1,q = 2^q/2Γ(1 + q/2), for q ∈ Z>0, which are the moments of a random variable with the Rayleigh distribution;

in this paper Γ(·) denotes the well-known gamma function. As we mentioned earlier, the case k = 1 has been shown in [25, Theorem 1.6] (or Addario-Berry et al. [1]). We henceforth assume throughout this paper that k > 2.

It is also important to mention that we could not find a simpler expression (in general) for the moments η_k,q except for some particular instances. For q = 1, we have

η_k,1= 2⁻^2k¹ (k!)^k¹ k Γ 1

k

Γ

1 − 1

2k

.

Then Theorem 1 provides a proof of [12, Lemma 4.10], where an estimation of the first moment of K(Tn) was first announced but whose proof was left to the reader. One can also compute with the help of Mathematica the second moment of Z_CRT or other particular examples. However, the expressions are too involved and we decided not to include them.

On the other hand, let (U₁, . . . , U_q) be q i.i.d. leaves of a Brownian CRT and define the vector (L^CRT₀ , L^CRT₁ , . . . , L^CRT_q ) where L^CRT₀ = 0 and L^CRT_i is the total length of the minimal subtree of a Brownian CRT which connects its root and the leaves of U₁, . . . , U_i; see [3, Lemma 21] from where one can deduce explicitly the distribution of (L^CRT₀ , L^CRT₁ , . . . L^CRT_q ). From the proof of Theorem 1, we obtain, for q ∈ N, that

η_k,q = q!

Z ∞ 0

Z x1

0

. . . Z xq−1

0

E

exp

− Pq

i=1(L^CRT_i − L^CRT_i−1 )x^k_i k!

d ~x_q,

where _q~x = (x_q, . . . , x₁) ∈ R^q+. This suggests that it ought to be possible to build the random variable Z_CRT by some construction that can be interpreted as the k-cut model on the Brownian CRT defined by Aldous [2, 3]. The appearance of the Brownian CRT in this framework should not come as a surprise since it is well-known that if we assign length n^−1/2 to each edge of the Galton-Watson tree Tn, then the latter converges weakly

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to a Brownian CRT as n → ∞. We believe that this connection can be exploited even more than the one used in this work in order to obtain the precise distribution of Z_CRT. For example, ideas from [6] and [1] could be useful to answer this question.

The approach used in this work consists of implementing an extension of the idea of Janson [25], which was used in [12], in order to study the k-cut model on deterministic and random trees. The authors in [12] introduced an equivalent model that allows them to define K(Tn) in terms of the number of records in Tn when vertices are assigned random labels. More precisely, let (E_i,v)_i>1,v∈T_n be a sequence of independent exponential random variables with parameter 1; Exp(1) for short. Let G_r,v := P

16i6rE_i,v, for r ∈ N and v ∈ Tⁿ. Clearly, Gr,v has a gamma distribution with parameters (r, 1), which we denote by Gamma(r). Imagine that each vertex v ∈ Tn has an alarm clock and v’s clock fires at times (G_r,v)_r>1. If we cut a vertex when its alarm clock fires, then due to the memoryless property of exponential random variables, we are actually choosing a vertex uniformly at random to cut. However, this also means that we are cutting vertices that have already been removed from the tree. Thus, for a cut on vertex v at time G_r,v (for some r ∈ {1, . . . , k}) to be counted in K(Tⁿ), none of its strict ancestors can already have been cut k times, i.e.,

G_r,v < min{G_k,u: u ∈ Tn and u is a strict ancestor of v}.

When the previous event happens, we say that G_r,v, or simply v, is an r-record and let I_r,v :=JGr,v < min{G_k,u: u ∈ Tn and u is a strict ancestor of v}K, (4) whereJ·K denotes the Iverson bracket, i.e., JS K = 1 if the statement S is true and JS K = 0 otherwise. Let K_r(Tn) be the number of r-records, i.e., K_r(Tn) := P

v∈TnI_r,v. Then, it should be clear that

K(Tn)=^d X

16r6k

K_r(Tn), (5)

where = denotes equal in distribution.^d

Loosely speaking, we then consider the well-known depth-first search walk or contour function Vn = (Vn(t), t ∈ [0, 2(n − 1)]) of the (ordered) tree Tⁿ as depicted in Figure 1, that is, V_n(t) is “the depth of the t-th vertex” visited in this walk; this will be made precise in the next section. As it is well-known (see Aldous [3, Theorem 23 with Remark 2] or [29, Theorem 1]), when Tⁿ is a conditioned Galton-Watson with offspring distribution satisfying (1), we have that

(n^−1/2V_n(2(n − 1)t), t ∈ [0, 1])→ 2σ^d ⁻¹B^ex, as n → ∞,

in C([0, 1], R+), with its usual topology, and where B^ex = (B^ex(t), t ∈ [0, 1]) is a stan- dard normalized Brownian excursion. It has been shown in [12, Lemma 2.1] that¹

1For two sequences of non-negative real numbers (A_n)_n>1 and (B_n)_n>1 such that B_n > 0, we write A_n∼ B_n if A_n/B_n→ 1 as n → ∞

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1 2 3 4 5 6 7 8 9 10 1

2

Figure 1: An example of a depth-first search walk in a tree and the corresponding V_n.

E[Ir,v] ∼ C_r,kd_n(v)^−r/k, for some (explicit) constant C_r,k > 0, where d_n(v) is the depth of the vertex v ∈ Tn. Let ◦ denote the root of Tn. Thus, heuristically

E [K^r(Tn) | Tn] ≈ X

v∈Tn\{◦}

C_r,k

d_n(v)^r/k ≈ C_r,k 2

Z 2(n−1) 0

dt V_n(t)^r/k

≈ C_r,k n⁻¹⁺^2k^r

Z 1 0

V_n(2(n − 1)t)

√n

−^r

k

dt

≈ C_r,k n⁻¹⁺^2k^r

σ 2

_k^r Z 1 0

dt B^ex(t)^r/k,

in distribution, as n → ∞. (The reader should bear in mind that the above calculation is not rigorous at all and that the purpose is only to illustrate the idea of proof.) One then expects that

σ^−r/kn⁻¹⁺^2k^r E [Kr(Tn)] ∼ C_r,kE

Z 1 0

(2B^ex(t))^−r/kdt

, as n → ∞,

which coincides with the right-hand side of (3) when r = q = 1. Note that this informal computation suggests that² E [K^r(Tⁿ)] = O(n¹⁻^2k^r ), for r ∈ {1, . . . , k}. As a consequence, Markov’s inequality implies that n⁻¹⁺^2k¹ K_r(Tn) → 0 in probability, as n → ∞, for r ∈ {2, . . . , k}. As shown later, by the identity in (5), it would be enough to prove Theorem 1 for K₁(Tn) instead of K(Tn).

In the rest of the paper, Section 2 and Section 3 make the above argument precise and extend it to higher moments. This will allow us to use the method of moments for proving Theorem 1. In Section 4, we also apply the same idea to get all moments of the number of records in paths and several types of trees of logarithmic height, e.g., complete binary trees, split trees, uniform random recursive trees and scale-free trees.

2 Preliminary results

The purpose of this section is to establish a general convergence result for the number of 1-records K₁(Tn) of a deterministic rooted ordered tree Tn. The results of this section

2For two sequences of non-negative real numbers (A_n)_n>1 and (B_n)_n>1 such that B_n > 0, we write A_n= O(B_n) if lim sup_n→∞A_n/B_n< ∞.

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can also be viewed as a generalization of those in Janson [25] and in Cai, et al. [12].

Furthermore, these results will allow us to study the convergence of K(Tn) not only for conditioned Galton-Watson trees, but also for other classes of random trees in Section 4.

We start by defining a probability measure through a continuous function in the same spirit as in [25, Theorem 1.9]. Let I ⊆ R+ be an interval. For a function f : I → R+ and t₁, . . . , t_q ∈ I with q ∈ N, we define

L_f(t₁, . . . , t_q) :=

q

X

i=1

f (t_(i)) −

q−1

X

i=1

inf

t∈[t_(i),t_(i+1)]f (t), (6)

where t₍₁₎, . . . , t_(q)are t₁, . . . , t_qarranged in nondecreasing order. Notice that L_f(t₁, . . . , t_q) is symmetric in t₁, . . . , t_q and that L_f(t) = f (t) for t ∈ I. Define

D_f(t₁) := L_f(t₁) and D_f(t₁, . . . , t_q) := L_f(t₁, . . . , t_q) − L_f(t₁, . . . , t_q−1), for q > 2. (7) We also consider the functional

G_f(t_q, x_q) := exp −D_f(t₁)x^k₁+ · · · + D_f(t₁, . . . , t_q)x^k_q k!

!

, (8)

for x_q = (x₁, . . . , x_q) ∈ R^q+ and t_q = (t₁, . . . , t_q) ∈ I^q. If I = [0, 1], we further define, for q ∈ N, m0(f ) := 1 and

m_q(f ) := q!

Z 1 0

· · · Z 1

0

Z ∞ 0

Z x1

0

. . . Z xq−1

0

G_f(t_q, x_q) d ~x_qd ~t_q, for q > 2, (9) where _q~x = (x_q, . . . , x₁) and q~t = (t_q, . . . , t₁).

Theorem 2. Let k ∈ N. Suppose that f ∈ C([0, 1], R+) is such that R1

0 f (t)^−1/kdt < ∞.

Then there exists a unique probability measure ν_f on [0, ∞) with finite moments given by Z

[0,∞)

x^qν_f(dx) = m_q(f ), for q ∈ Z>0.

Proof. We only prove uniqueness here. The proof for existence follows along the lines of [25, Proof of Theorem 1.9, Pages 18-19] and details are left to the interested reader.

Informally speaking, the idea in [25] for the proof of existence is to build a sequence of functions that satisfy the conditions of Theorem 3 below. Define the function

H_f,q(t_q) :=

Z ∞ 0

Z x1

0

. . . Z xq−1

0

G_f(t_q, x_q) d ~x_q. (10) By changing the order of integration, we obtain that

H_f,q(t_q) = Z ∞

0

Z ∞ xq

. . . Z ∞

x2

G_f(t_q, x_q) dx_q,

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for x_q = (x₁, . . . , x_q) ∈ R^q+ and t_q = (t₁, . . . , t_q) ∈ [0, 1]^q. By making the change of variables x_q = w_q, x_q−1 = w_q+ w_q−1, . . . , x₁ = w_q+ · · · + w₁, we see that

Hf,q(tq) =

Z

[0,∞)^q

exp



−1 k!

q

X

i=1

Df(t1, . . . , ti)

q

X

j=i

wj

!k

dwq,

where w_q = (w₁, . . . , w_q) ∈ R^q+. From the inequality (x₁ + · · · + x_q)^k > x^k1+ · · · + x^k_q, we observe that

H_f,q(t_q) 6

q

Y

j=1

Z ∞ 0

exp −w^k_j k!

j

X

i=1

D_f(t₁, . . . , t_i)

! dw_j

= Γ (1 + 1/k)^qΓ(1 + k)^q/k

q

Y

j=1 j

X

i=1

D_f(t₁, . . . , t_i)

!^−1/k

= Γ (1 + 1/k)^qΓ(1 + k)^q/k

q

Y

i=1

L_f(t₁, . . . , t_i)^−1/k

6 Γ (1 + 1/k)^qΓ(1 + k)^q/k

q

Y

i=1

f (t_i)^−1/k, (11)

where for the last inequality we have used the fact that L_f(t₁, . . . , t_i) > max16j6if (t_j), for 1 6 i 6 q. The later follows from the symmetry of Lf; see [25, Lemma 4.1] for a proof.

Then, the previous inequality allows us to conclude that 0 6 m^q(f ) 6 q! Γ (1 + 1/k)^qΓ(1 + k)^q/k

Z 1 0

f (t)^−1/kdt

^q . We conclude that there exists a > 0 such that P∞

q=0m_q(f )^x_q!^q < ∞, for 0 6 x < a.

Then a probability measure with moments m_q(f ) has a finite generating function in a neighbourhood of 0. Thus, it is well-known that this implies that the probability measure is unique; see, e.g., [18, Section 4.10].

Consider a rooted ordered tree Tnwith root ◦ and n ∈ N vertices. We now explain how Tn can be encoded by a continuous function. We define the so-called depth-first search function [2, page 260], ψ_n : {0, 1, . . . , 2(n − 1)} → { vertices of Tn} such that ψ_n(i) is the (i + 1)-th vertex visited in a depth-first walk on the tree starting from the root ◦. Note that ψ_n(i) and ψ_n(i + 1) always are neighbours, and thus, we extend ψ to [0, 2(n − 1)] by letting, for 1 6 i < t < i + 1 6 2(n − 1), ψn(t) to be the one of ψ_n(i) and ψ_n(i + 1) that has largest depth (recall that the depth of a vertex v ∈ Tn is the distance, i.e., number of edges, between ◦ to v). Let d_n(v) be the depth of a vertex v ∈ Tn. We further define the depth-first walk V_n of Tn by

V_n(i) := d_n(ψ(i)), i ∈ {0, . . . , 2(n − 1)},

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and extend V_n to [0, 2(n − 1)] by linear interpolation. Thus V_n ∈ C([0, 2(n − 1)], R+). See Figure 1 for an example of V_n. Furthermore, we normalize the domain of V_n to [0, 1] by defining

Ve_n(t) := V_n(2(n − 1)t) and Vb_n(t) := dV_n(2(n − 1)t)e, (12) for t ∈ [0, 1]. Thus eVn∈ C([0, 1], R⁺). Note that dn(ψ(t)) = dVn(t)e, for t ∈ [0, 2(n − 1)].

Moreover,

maxv∈Tn

dn(v) = sup

t∈[0,2(n−1)]

Vn(t) = sup

t∈[0,1]

Ven(t). (13)

We now state the central result of this section, that is, a general limit theorem in distribution for the number of 1-records K₁(Tn) of a deterministic rooted tree Tn with n vertices. It is important to notice that K₁(Tn) is a random variable since the 1-records are random. From now on, we always assume that k > 2.

Lemma 3. Suppose that (Tn)_n>1 is a sequence of ordered (deterministic) rooted trees, and denote the corresponding normalized depth-first walks by eV_n and bV_n. Suppose that there exists a sequence (an)_n>1 of non-negative real numbers with limn→∞an= 0, limn→∞na^1/kn =

∞ and a function f ∈ C([0, 1], R+) such that (a) a_nVe_n(t) → f (t), in C([0, 1], R+), as n → ∞.

(b) Z 1

0

(a_nVb_n(t))^−1/k dt → Z 1

0

f (t)^−1/k dt < ∞, as n → ∞.

Then, for each q ∈ Z>0,

n^−qa^−q/k_n E[K1(Tn)^q] → m_q(f ),

as n → ∞, where m_q(f ) is defined in (9). Moreover, n⁻¹a^−1/kn K₁(Tn)→ Z^d _f, as n → ∞, where Zf is a random variable with distribution νf defined by Theorem 2.

Before proving Theorem 3, we need to establish some preliminary results and to in- troduce some further notation. For q ∈ N and vertices v1, . . . , v_q ∈ Tn, let L_n(v₁, . . . , v_q) be the number of edges in the subtree of Tn spanned by v₁, . . . , v_q and its root ◦ (i.e., the minimal number of edges that are needed to connect v₁, . . . , v_q and ◦). We write D_n(v₁) := L_n(v₁) and D_n(v₁, . . . , v_q) := L_n(v₁, . . . , v_q) − L_n(v₁, . . . , v_q−1) for q > 2. We also consider the functional

G_n(v_q, x_q) := exp −Dn(v1)x^k₁ + · · · + Dn(v1, . . . , vq)x^k_q k!

!

, (14)

for x_q = (x₁, . . . , x_q) ∈ R^q+ and v_q = (v₁, . . . , v_q) ∈ T^qn. We denote by Γ(k, ·) the upper incomplete gamma function of parameter k ∈ N, i.e.,

Γ(k, x) = Z ∞

x

t^k−1e^−tdt, for x > 0.

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Remark 4. Let Tn be an ordered (deterministic) rooted tree with depth-first search walk ψ_n and the corresponding function V_n. It is not difficult to see that L_n and LdV_ne are connected, in the sense that L_n(ψ_n(t₁), . . . , ψ_n(t_q)) = LdVne(t₁, . . . , t_q) for t₁, . . . , t_q ∈ [0, 2(n − 1)]; see [25, Lemma 4.4] for a proof of this fact.

Lemma 5. Let Tⁿ be an ordered (deterministic) rooted tree with n ∈ N vertices. Suppose that there exists a sequence (a_n)_n>1 of non-negative real numbers such that lim_n→∞a_n= 0 and max_v∈T_nd_n(v) = O(a⁻¹_n ). Let α := ¹₂ ¹_k +_k+1¹ and x₀ := a^α_n. Then, for q ∈ N and uniformly for all x ∈ [0, x0],

P(Gamma(k) > x)^Dⁿ^(v¹^,...,v^q⁾ = Γ(k, x) Γ(k)

Dn(v1,...,vq)

= (1 + O(a

1

n2k)) exp

−Dn(v1, . . . , vq)x^k k!

, where the vertices v₁, . . . , v_q∈ Tn.

Proof. Our claim can be shown along the lines of [12, Proof of Lemma 5.1].

Recall that for two sequences of non-negative real numbers (A_n)_n>1 and (B_n)_n>1 such that B_n > 0, one writes A_n = o(B_n) if lim_n→∞A_n/B_n= 0.

Lemma 6. Let Tn be an ordered (deterministic) rooted tree with n ∈ N vertices. Suppose that there exists a sequence (a_n)_n>1 of non-negative real numbers with lim_n→∞a_n = 0, lim_n→∞na^1/kn = ∞ and max_v∈T_nd_n(v) = O(a⁻¹_n ). Then the moments of K₁(Tn) are given by

n^−qa^−q/k_n E[K1(Tn)^q] = (1 + O(a

q

n2k))q!

Z 1 0

· · · Z 1

0

H¯_n,q(t_q) d ~t_q+ o(1),

where

H¯_n,q(t_q) :=

Z ∞ 0

Z x1

0

· · · Z xq−1

0

G_a

nVbn(t_q, x_q) exp −a^1/k_n

q

X

i=1

x_i

!

d ~x_q, for q ∈ N. (15)

Proof. For simplicity, we write X_q := K₁(Tn)^q for q ∈ Z>0 and note that X_q = X₁^q. For q ∈ N, we observe that

X_q = (X₁− 1 + 1)^q = (X₁− 1)^q+

q−1

X

p=0

q p

(X₁− 1)^p = (X₁− 1)^q+ Y_q.

where Y_q:= Pq−1 p=0

Pp l=0

q p

_p

l(−1)^p−lX_l. Recall that I_1,v is the indicator that v ∈ Tn is a 1-record defined in (4). By the previous identity, we have that

X_q = X

v1,...,vq∈Tn\{◦}

I_1,v₁· · · I_1,v_q + Y_q = q! X

v1,...,vq∈Tn\{◦}

JE (v¹, . . . , v_q)K + Y^q

(10)

where E (v₁, . . . , v_q) := {E_1,v_q < · · · < E_1,v₁ and v₁, . . . , v_q are all 1-records}; recall that E_1,v₁, . . . , E_1,v_q are independent random variables with an Exp(1) distribution. To see the last identity, note that each product I_1,v₁· · · I_1,v_q occurs q! times with indices permuted and for exactly one of these permutations we have that E_1,v_q < · · · < E_1,v₁.

Consider the simple case q = 2. Conditioning on E_1,v₂ = x₂ < E_1,v₁ = x₁, we see that v₁ and v₂ are both 1-records, if and only if, the following two events happen:

(i) the D_n(v₁) ancestors of v₁ are removed after time x₁;

(ii) the D_n(v₁, v₂) vertices which are ancestors of v₂ but not of v₁ are removed after time x₂.

Since x₂ < x₁, we note that the event (i) implies that the vertices which are both the ancestors of v₁ and v₂ are removed after x₁. Let g(x) := P(Gamma(k) > x) for x ∈ R+. Since the events (i) and (ii) are independent, we have

P (E (v¹, v₂)) = Z ∞

0

Z x1

0

g(x₁)^Dⁿ^(v¹⁾g(x₂)^Dⁿ^(v¹^,v²⁾e^−x¹^−x² dx₂ dx₁. (16) Recall that we are assuming k > 2. Otherwise, when k = 1, the above equality is not entirely correct since E (v₁, v₂) is impossible if v₂ is an ancestor of v₁; see [25, Lemma 4.3]

for details in the case k = 1.

By generalizing the previous argument to q ∈ N, we see that P(E (v1, . . . , v_q)) =

Z ∞ 0

Z x1

0

· · · Z xq−1

0

g(x₁)^Dⁿ^(v¹⁾· · · g(x_q)^Dⁿ^(v¹^,...,v^q⁾e⁻^P^qⁱ⁼¹^xⁱ d ~x_q

= Z x0

0

Z x1

0

· · · Z xq−1

0

g(x₁)^Dⁿ^(v¹⁾· · · g(x_q)^Dⁿ^(v¹^,...,v^q⁾e⁻^P^qⁱ⁼¹^xⁱ d ~x_q +

Z ∞ x0

Z x1

0

· · · Z xq−1

0

g(x₁)^Dⁿ^(v¹⁾· · · g(x_q)^Dⁿ^(v¹^,...,v^q⁾e⁻^P^qⁱ⁼¹^xⁱ d ~x_q

= A₁+ A₂,

where _q~x = (x_q, . . . , x₁) ∈ R^q+, x₀ = a^α_n and α = ¹₂ _k¹ +_k+1¹ . On the one hand, Theorem 5 implies that

A₂ 6 Z ∞

x0

g(x)^Dⁿ^(v¹⁾e^−x dx 6 g(x0)^Dⁿ^(v¹⁾= O

exp

− x^k₀ 2a_nk!

;

we have used our assumption max_v∈T_nd_n(v) = O(a⁻¹_n ). On the other hand, Theorem 5 also implies that

A₁ = (1 + O(a

1

n2k))^q Z x0

0

Z x1

0

· · · Z xq−1

0

G_n(v_q, x_q)e⁻^P^qⁱ⁼¹^xⁱ d ~x_q

= (1 + O(a

q

n2k)) Z ∞

0

Z x1

0

· · · Z xq−1

0

G_n(v_q, x_q)e⁻^P^qⁱ⁼¹^xⁱ d ~x_q+ A₃,

(11)

where v_q = (v₁, . . . , v_q) ∈ T^qn and A₃ = (1 + O(a

q

n2k)) Z ∞

x0

Z x1

0

· · · Z xq−1

0

G_n(v_q, x_q)e⁻^P^qⁱ⁼¹^xⁱ d ~x_q= O

exp

− x^k₀ 2a_nk!

this estimation can be deduced similarly as the one for the integral A2. Therefore, the previous estimations and Remark 4 allow us to conclude that E[Xq] equals to

(1 + O(a

q

n2k))q! X

v1,...,vq∈Tn\{◦}

Z ∞ 0

Z x1

0

· · · Z xq−1

0

Gn(vq, xq)e⁻^P^qⁱ⁼¹^xⁱ d ~xq

+ E[Yq] + o(n^qa^q/k_n ) (17)

= (1 + O(a

q

n2k))q!2^−q

Z 2(n−1) 0

· · ·

Z 2(n−1) 0

Z ∞ 0

Z x1

0

· · · Z xq−1

0

GdVne(t_q, x_q)e⁻^P^qⁱ⁼¹^xⁱ d ~x_qd ~t_q + E[Yq] + o(n^qa^q/k_n )

= (1 + O(a

q

n2k))q!n^q Z 1

0

· · · Z 1

0

Z ∞ 0

Z x1

0

· · · Z xq−1

0

GVbn(t_q, x_q)e⁻^P^qⁱ⁼¹^xⁱ d ~x_qd ~t_q+ + E[Yq] + o(n^qa^q/k_n );

note that if we had not excluded the root, we would not be able to write the sum as an integral. By making the change of variables x_i = a^1/kn w_i, for 1 6 i 6 q, we have that

E[Xq] = (1 + O(a

q

n2k))q!n^qa^q/k_n Z 1

0

· · · Z 1

0

H¯_n,q(t_q)d ~t_q+ E[Yq] + +o(n^qa^q/k_n ).

Finally, our claim follows by induction on q ∈ N and the assumption limn→∞na^1/kn =

∞.

We are now able to establish Theorem 3.

Proof of Theorem 3. First note that by condition (a) of Theorem 3 and (13), we have max_v∈T_nd_n(v) = sup_t∈[0,1]Ve_n(t) = O(a⁻¹_n ). Thus the conditions for Theorem 5 and Theo- rem 6 are satisfied.

Recall the functions ¯H_n,q and H_f,q defined in (15) and (10), respectively. Therefore, notice that we only need to show that

Z

[0,1]^q

H¯_n,q(t_q) d ~t_q → Z

[0,1]^q

H_f,q(t_q) d ~t_q, as n → ∞. (18)

The above convergence together with Lemma 6 implies that E[K¹(Tⁿ)^q] = O(n^qa^q/kn ) which clearly proves the first claim in Lemma 3. The second claim follows immediately from Theorem 2 and the method of moments.