Limit distributions for multitype branching processes of <span class="mathjax-formula">$m$</span>-ary search trees

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www.imstat.org/aihp 2014, Vol. 50, No. 2, 628–654

DOI:10.1214/12-AIHP518

Limit distributions for multitype branching processes of m-ary search trees

Brigitte Chauvin

^a

, Quansheng Liu

^b

and Nicolas Pouyanne

^a

aUniversité de Versailles–St-Quentin, Laboratoire de Mathématiques de Versailles, CNRS UMR 8100, 45 avenue des Etats-Unis, 78035 Versailles CEDEX, France

bUniversité de Bretagne-Sud, Laboratoire de Mathématiques de Bretagne Atlantique, CNRS UMR 6205, Campus de Tohannic, BP573, 56017 Vannes, France

Received 1 December 2011; revised 20 July 2012; accepted 20 July 2012

Abstract. Letm≥3 be an integer. The so-calledm-ary search treeis a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well- known phase transition: whenm≤26, the asymptotic behavior of the process is Gaussian, but form≥27 it is no longer Gaussian and a limitW^DT of a complex-valued martingale arises.

In this paper, we consider the multitype branching process which is the continuous time version of them-ary search tree.

This process satisfies a phase transition of the same kind. In particular, whenm≥27, a limitW of a complex-valued martingale intervenes in its asymptotics. Thanks to the branching property, the law ofW satisfies asmoothingequation of the typeZ=^L e⁻^λT(Z⁽¹⁾+ · · · +Z^(m)), whereλis a particular complex number,Z^(k)are independent complex-valued random variables having the same law asZ,T is aR₊-valued random variable independent of theZ^(k), and=^Ldenotes equality in law. This distributional equation is extensively studied by various approaches. The existence and uniqueness of solution of the equation are proved by contraction methods. The fact that the distribution ofW is absolutely continuous and that its support is the whole complex plane is shown via Fourier analysis. Finally, the existence of exponential moments ofWis obtained by consideringWas the limit of a complex Mandelbrot cascade.

Résumé. Soitm≥3 un entier. Très populaire en informatique fondamentale, l’arbrem-aire de rechercheest une chaîne de Markov à temps discret qui modélise de célèbres algorithmes de tri et de recherche de données. Ce processus aléatoire vérifie une transition de phase bien connue : lorsquem≤26, le comportement asymptotique du processus est gaussien. En revanche, lorsquem≥27, il n’est plus gaussien et fait apparaître la limiteW^DT d’une martingale à valeurs complexes.

Dans cet article, on considère le processus de branchement multitype qui est le plongement en temps continu de l’arbrem-aire de recherche. Ce processus fait l’objet d’une transition de phase du même type. En particulier, lorsquem≥27, son asymptotique s’exprime à l’aide de la limiteWd’une martingale complexe. Grâce à la propriété de branchement, la loi deWest solution d’une équation en distribution du typeZ=^Le⁻^λT(Z⁽¹⁾+ · · · +Z^(m))oùλest un nombre complexe particulier, lesZ^(k)sont des variables aléatoires complexes indépendantes dont la loi est celle deZ,T est une variable aléatoire réelle positive indépendante desZ^(k), et

=Ldésigne l’égalité en distribution. On étudie cette équation en loi par des approches variées. L’existence et l’unicité de solutions sont prouvées par des méthodes de contraction. L’absolue continuité deW et le fait que son support soit le plan complexe tout entier sont démontrés par analyse de Fourier. Enfin, on obtient l’existence de moments exponentiels en considérantWcomme la limite d’une cascade de Mandelbrot à valeurs complexes.

MSC:Primary 60C05; secondary 60J80; 05D40

Keywords:Martingale; Characteristic function; Embedding in continuous time; Multitype branching process; Smoothing transformation; Absolute continuity; Support; Exponential moments

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1. Introduction

Consider a continuous time multitype branching process(X(t), t≥0). Types are seen as colors of particles and there arem−1 colors, wherem≥3 is an integer. The reproduction of the process is given by a particular matrixR(written in (2.1)), and any particle of colourjlives a random time of exponential distribution with parameterj. Such a classical process is considered for example in Athreya and Ney [1] or Janson [12] and it is precisely defined in Section2.

When it is stopped at thenth jump time, this process is nothing but the composition vector process(X^DT_n , n≥0) say, of the so-calledm-ary search tree, which is an important algorithmic structure in computer science.

The continuous time random process(X(t), t≥0)exhibits a phase transition: whenm≤26, the random vector X(t )has a Gaussian behaviour when t tends to infinity. This fact, a consequence of classical results on branching processes, has been known for a long time. Details are recalled in the beginning of Section4.

Whenm≥27, inspired by the methods used for a two-color Pólya urn in [5], we first prove in Section4.1that X(t )admits the following asymptotic expansion:

X(t )=e^tξ v₁

1+o(1) +2

e^λ²^tW v₂

1+o(1) +o

e^σ²^t a.s.,

whereλ2is a particular complex number having a real partσ2 in]¹₂,1[,ξ is a Gamma distributed random variable andWaC-valued one andv₁, v₂are linearly independent vectors defined in (2.4).

We are interested in the limit random variablesW_k, k=1, . . . , m−1, each corresponding toX_k(t)which denotes the processX(t )when it starts from one particle of colork. Using the branching property, a system of dislocation equations is written for the random vectors X_k(t)in Section5.1. A system of fixed point equations satisfied by the corresponding limit laws is then derived in Section5.2. In particular, the complex-valued random variable W₁is a solution of the fixed point equation

Z=^Le⁻^λ²^T

Z⁽¹⁾+ · · · +Z^(m)

, (1.1)

where{Z^(k):k≥1}are independent copies ofZ,T =τ(1)+ · · · +τ(m−1),{τ(j ):j≥1}are random variables independent of each other and independent of{Z^(k)}, eachτ_{(j )}has distributionExp(j )(we denote byExp(j )the exponential distribution of parameterj: it has densityx→je⁻^{j x}on]0,∞[).

Further properties ofW₁are derived from a thorough study of Eq. (1.1). We first show in Theorems6.2and6.4 that Eq. (1.1) admits a unique square-integrable solution having a given mean. In particular, this implies that Eq. (1.1) characterizes the distribution of W₁. This result is proven by two contraction methods applied to the corresponding smoothing transformations. The first one deals with suitable spaces of probability measures where the classical Wasserstein metric is adapted to the complex field; it leads to Theorem6.2. The second contraction method, that gives a proof for Theorem 6.4, consists in working on Fourier transforms of solutions and provides a somewhat simpler proof. Furthermore, this second method gives a result of existence and uniqueness for solutions of the convolution equation

Φ(t )= _+∞

0

Φ^m(t−u)fT(u)du, t∈C,

in a convenient space of functions, wheref_T denotes the density ofT (see Remark6.6).

Once the characterization ofW₁by Eq. (1.1) is proven, it suffices to derive properties of solutions of this distributional equation. We show in this way the following results on the law ofW₁.

Theorem 1.1. Whenm≥27,the complex-valued random variableW1admits a density and its support is the whole complex plane.Its Fourier transform satisfies

Eeⁱ^t,W¹=O

|t|⁻^a when|t| → +∞,for somea >1.

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This theorem is a direct consequence of Theorem7.1that provides such properties for solutions of (1.1) admitting a nonzero mean. Our proof consists in showing successively that the characteristic function of any solution has modulus equal to 1 only at the origin, that it tends to zero at infinity, and finally that it is of order O(|t|⁻^a)as |t| → ∞ for somea >1, so that it is square-integrable onC. In the approach we need to prove a nonlattice property of Eq. (1.1) viaGelfand–Schneider theorem, using the algebraicity ofλ₂(see proof of Lemma7.4).

Theorem 1.2. Whenm≥27,the random variableW₁admits exponential moments in a neighbourhood of the origin of the complex plane.IfL1(z)=Ee^zW¹ denotes its Laplace series,thenL1is holomorphic near0and,after a change of variable,the functionz→ −^ρ_zL1(z⁻^λ²)is a solution of the differential equation

y^(m⁻¹⁾=y^m.

Theorem1.2is immediately derived from Theorems8.1and8.4just as Theorem1.1was derived from Theorem7.1.

To prove Theorems8.1and8.4, we consider a solution of (1.1) as the limit of a complex Mandelbrot cascade. The results are a consequence of fine analytical properties of the Fourier transform of the limit variable.

The paper is organized as follows. The continuous time multitype branching process is defined in Section2. Its relation with them-ary search tree is detailed in Section3, while Section4is devoted to the asymptotics ofX(t )and to its connection to the corresponding discrete time process. In Section5, we use the branching property of the process to show that the martingale limits of the continuous time process are related by a system of equations in law so that the fixed point equation (1.1) emerges. These first four sections constitute the first part of the paper.

The second part of the paper consists in putting the focus on Eq. (1.1) that turns out to characterize the distribution ofW₁so that all results on solutions provide results onW₁. In Section6we define the natural smoothing transform associated with Eq. (1.1) and we show that it defines a contraction in the space of square-integrable probability measures with given mean. Results on the support and on absolute continuity of solutions are obtained in Section7.

Finally, Section8is devoted to the exponential moments and the Laplace series of solutions.

2. Definition of the branching process

In this section we introduce the definition of the continuous time multitype branching process(X(t)), and present the spectral decomposition of its transition matrix.

2.1. Infinitesimal generator

In the whole paper, the underlying vector space isR^m⁻¹or sometimesC^m⁻¹. LetRbe the following square matrix of orderm−1:

R=

⎛

⎜⎜

⎝

−1 1−1 1

−1. ..

. .. 1

m −1

⎞

⎟⎟

⎠

, (2.1)

and fork=1, . . . , m−1, letw_kbe thekth row vector ofR: when 1≤k≤m−2, thekth coordinate ofw_kequals−1, the(k+1)th equals 1 and all the others are 0;w_m₋₁hasmas first coordinate,−1 as last one, and 0 for all others.

LetGbe the operator defined on functionsf fromC^m⁻¹to any real or complex vector space by the following formula: for any vectorvinC^m⁻¹,

G(f )(v)=

m−1 k=1

kl_k(v)

f (v+w_k)−f (v)

, (2.2)

wherelk are the coordinate forms:lk(x1, . . . , xm−1)=xk.

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Definition 2.1. The right-continuous processX=(X(t), t≥0)is the only continuous time Markov process with state spaceR^m⁻¹havingGas infinitesimal generator.

Equivalently, Xis a continuous time multitype branching process with m−1 types (or colors), havingR as reproduction matrix. Thekth coordinate of the vectorX(t ), namelyl_k(X(t)), is the number of particles of colorkat timet. A particle of colorklives a random exponential time with parameterk; when it dies, it reproduces one particle of colork+1 ifk=1, . . . , m−2, andmparticles of color 1 ifk=m−1.

This branching continuous time process can be thought as the embedded process of a discrete Markov chainX^DT = (X_n^DT)_n_∈Nwhich is a Pólya-type discrete Markov chain associated with the node process of anm-ary search tree, an important algorithmic structure in computer science. This connection is detailed in Section3.

2.2. Spectral decomposition

LetR_Gbe the matrix ofG’s restriction to linear forms in the canonical basis(l_k)₁_≤_k_≤_m₋₁. One immediately checks that

R_G=

⎛

⎜⎜

⎝

−1 1

−2 2

−3. ..

. .. m−2

m(m−1) −(m−1)

⎞

⎟⎟

⎠ ,

where an empty entry means a zero entry. It has been established in many papers – see for example Mahmoud [16], Chern and Hwang [6] or [4] – and it can be easily checked thatR_G’s (unitary) characteristic polynomial is

χ_R_G(λ)=

m−1 k=1

(λ+k)−m! = (λ+m)

(λ+1) −m!, (2.3)

where denotes Euler’s Gamma function. All eigenvalues are simple, 1 being the one having the largest real part. If m=3, the second eigenvalue is−3. Whenm≥4, all eigenvalues different from 1 are nonreal, except−(m+1)when mis odd.

Whenm≥4, in the whole paper,λ2will denoteχR_G’s root having the second largest real part namedσ2and a positive imaginary part namedτ₂.

The famous phase transition onm-ary search trees already mentioned in the introduction is due to the fact that (λ₂) >1/2 if and only if m≥27.

See for example [4]. The assumption(λ2) >1/2 will be frequently used in the sequel.

We adopt the following notations:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

∀n∈Z≥0, z

n

= (z+1)

n! (z−n+1)=z(z−1)· · ·(z−n+1)

n! ;

H_m(z)=

1≤k≤m−1

1 z+k; u1(x1, . . . , xm−1)=

1≤k≤m−1

kxk; u₂(x₁, . . . , x_m₋₁)=

1≤k≤m−1

λ2+k−1 k−1

x_k; v1= 1

H_m(1) 1

k(k+1)

1≤k≤m−1

; v₂= 1

H_m(λ₂) 1

k_λ₂₊_k

k

1≤k≤m−1

.

(2.4)

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The linear formsu₁andu₂are eigenvectors ofG, namelyG(u₁)=u₁andG(u₂)=λ₂u₂. The vectorsv₁andv₂are left eigenvectors ofRG, respectively associated with the eigenvalues 1 andλ2. They satisfyu1(v1)=u2(v2)=1 and u₁(v₂)=u₂(v₁)=0. These spectral data had already been essentially computed in [4] and [18]. Note that, sinceλ₂is not real,u2andv2have nonreal coordinates. Note also thatv2(for a complex vectorvwe denote byvits conjugate vector composed of the complex conjugates of the components ofv) is an eigenvector ofR_GlinearlyC-independent fromv2, that(v1, v2, v2)can be completed to provide a basis of complex eigenvectors ofRG, and that its dual basis is of the form(u₁, u₂, u₂, . . .). In particular,u₂(v₂)=0.

3. m-ary search trees and embedding

In this section we present the connection betweenm-ary search trees and the multitype branching process defined in Section2. This connection is the classical embedding of a discrete time Markov chain into a continuous time Markov process.

3.1. m-ary search trees

We define here a discrete time Markov chainX^DT =(X^DT_n , n≥0)with values inN^m⁻¹\ {0}. Theith coordinate of X^DT_n is denoted by X_n⁽ⁱ⁾and has a “physical” meaning detailed hereafter. The Markov chainX^DT is a random walk defined by an initial vectorX^DT₀ inN^m⁻¹\ {0}and by the following transition probabilities:∀v∈N^m⁻¹\ {0},

∀k=1, . . . , m−1,

q(v, v+w_k)= klk(v) _m₋₁

j=1 j l_j(v), (3.1)

where the increment vectorsw_k are given in Section2.1andl_k(v)denotes thekth coordinate of the vectorv.

Classically (see Norris [17] and for a synthetic exposition Bertoin [3]), this discrete time Markov chain is embedded in continuous time using a “Poissonization” of the time: givenX^DT, one can recoverX=(X(t), t≥0)as follows.

At time 0,X(0)=X₀^DT. For any vectorv∈R^m⁻¹, define¹ q(v):=

m−1 k=1

kl_k(v).

Letτ₁be a random time exponentially distributed with parameterq(X^DT₀ ). For any timet∈ [0, τ1[, letX(t )=X(0)= X₀^DT. At timeτ₁,Xjumps fromv=X(0)tov+w_kwith probability given by formula (3.1). More generally, letτ₀=0 and for anyn≥1, define thenth jumping timeτ_nby

τn=

n−1

i=0

_i q(X_i^DT),

where_i are independent random variables having the same exponential distribution with parameter 1. Let X(t )=X(τ_n)=X_n^DT ∀t∈ [τ_n, τ_n₊1[.

At timeτn+1,X jumps fromv=X(τn)tov+wk with probability given by formula (3.1). It is easy to see that this embedded processX(t )is the same one as the branching process defined in Section2.1.

WhenX^DT₀ =(1,0, . . . ,0), eachX_n⁽ⁱ⁾, i=1, . . . , m−1, can be seen as the number of nodes of typeiin a treeT_n: the sequence(Tn, n≥0)is a sequence of randomm-ary trees which grow by successive insertions of keys in their leaves. Each node of these trees contains at mostm−1 keys. Keys are i.i.d. random variablesx_i, i≥1, with any diffusive distribution on the interval[0,1]. The treeT_n, n≥0, is recursively defined as follows:T0is reduced to an empty node-root;T₁is reduced to a node-root which containsx₁,T₂is reduced to a node-root which containsx₁and x₂, . . . , T_m₋₁has a node-root containingx₁, . . . , x_m₋₁. As soon as the (m−1)th key is inserted in the root,mempty

1Note thatq=u1whereu1was defined by (2.4).

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subtrees of the root are created, corresponding from left to right to themordered intervalsI₁= ]0, x(1)[, . . . , I_m= ]x_(m₋₁₎,1[, where 0< x₍₁₎ <· · ·< x_(m₋₁₎<1 are the ordered m−1 first keys. Each following key x_m, . . . , is recursively inserted in the subtree corresponding to the unique intervalI_j to which it belongs. As soon as a node is saturated,mempty subtrees of this node are created.

For eachi= {1, . . . , m−1}andn≥1,X_n⁽ⁱ⁾is the number of nodes inT_nwhich containi−1 keys (andigaps or free places) after insertion of thenth key; such nodes are named nodes of typei. We don’t worry about the number of saturated nodes. The vectorX^DT_n is called the composition vector of them-ary search tree. It provides a model for the space requirement of the algorithm. One can refer to Mahmoud’s book [16] for further details on search trees.

Notice that, in this dynamics, the insertion of a new key isuniformon thegaps, as can be seen on the transition probabilities (3.1).

3.2. Embedding

The embedding properties are summarized in the following lemma.

Lemma 3.1.

(1) For anyn≥1,the distribution ofτ_n−τ_n₋₁isExp(n−1+N₀),whereN₀is the number of free places inX(0):

N₀=u₁(X(0)).

(2) the processes(τ_n)_n_≥₁and(X(τ_n))_n_≥₁are independent.

(3) the processes(X(τ_n))_n_≥₁and(X^DT_n )_n_≥₁have the same distribution.

Proof. Part (1) is a consequence of the fact that the minimum ofkindependentExp(1)-distributed random variables isExp(k)-distributed, and that the total number of free places at timeτ_nequalsn−1+N₀.

Part (2) is the classical independence between the jump chain and the jump times in such Markov processes. The initial states and evolution rules of both Markov chains in discrete time and in continuous time are the same ones, so

that Part (3) holds.

Convention. From now on,thanks to Part(3)of Lemma3.1,we will as usual suppose that the discrete time process and the continuous time process are built on thesameprobability space on which

X(τn)

n≥1= X^DT_n

n≥1 a.s. (3.2)

Remark. The important benefit we get with the embedding is the independence in the continuous time process.This independence is the key point for the dislocation equations later on.

4. Asymptotics and martingale connection

In this section we present the asymptotic behaviour of the continuous time multitype branching process(X(t))_t in three principal directions and its connection with the discrete time process(X_n^DT)defined in Section3.1.

4.1. Asymptotics of the continuous time branching process

Whenm≤26, the random vectorX(t )satisfies a Gaussian asymptotics whent tends to infinity: firstly, the random vector e⁻^tX(t )converges almost surely toξ v₁whereξ is a positive random variable andv₁ a deterministic vector (in fact,v₁is defined by formula (2.4) and the proof of Theorem4.1shows thatξ isGamma-distributed and that this convergence is valid in any L^p,p≥1). Secondly,X(t )can be decomposed as the sumX(t )=X₁(t)+X_I(t)of two random vectors, whereX1is proportional tov1,X_I belongs to some fixed supplementary subspace, e⁻^tX1(t)→ξ v1

almost surely and e⁻^{t /2}XI(t)convergesin distributionto√

ξ N whereN is a centered Gaussian vector independent ofξ. For a statement and a proof of these facts, one can refer to Theorem 3.1 and Example 7.8 in Janson [12].

Whenm≥27, using the notations of Section2and especially the formula (2.4), the random vectorX(t )admits a 3-dimensional almost sure expansion ast goes to infinity, described in Theorem4.1. Denote by(v2)(resp.(v2))

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the vector ofR^m⁻¹made of the real (resp.imaginary) parts ofv₂’s coordinates. Then, geometrically speaking, The- orem4.1gives the principal part of X(t )’s coordinates along the three linearly independent vectorsv1,(v2)and (v2), the projection ofX(t )on a supplementary subspace being almost surely negligible.

Notice that these results had been partially proven in a more general frame by Athreya and Ney [1] and adapted to this particular branching process by Janson [12]. More precisely, sinceR_Gis diagonalizable onC, choose a basis (v_λ)_λ_∈_Sp(R_G₎of complex eigenvectors ofR_Gand name its dual basis(u_λ)_λ_∈_Sp(R_G₎. Then, the spectral decomposition C^m⁻¹=

Cv_λgives rise to the corresponding projectionsu_λv_λon all eigenlines. With this material, Janson’s result can be stated in the following way: for anyλ∈Sp(RG),

(i) if(λ) >1/2, e⁻^λtu_λ(X(t))converges almost surely to some (nonnormal) random variable. In particular, let ξ=limt→+∞e⁻^tu₁(X(t));

(ii) if(λ) <1/2, e⁻^{t /2}u_λ(X(t))converges in law to some product√

ξ N whereN is a centered normal distribution independent ofξ (note that(λ)=1/2 never happens).

Nevertheless, the global almost sure and L^premainder3(t)in Theorem4.1is a new result.

Theorem 4.1 (Asymptotics of continuous time process). Suppose thatm≥27.Then,with the notations of Section2 and especially the formulae(2.4),ast tends to infinity,

X(t )=e^tξ v1

1+ε1(t) +2

e^λ²^tW v2

1+ε2(t)

+e^λ²^t3(t), (4.1)

where

• ξ is a positiveGamma-distributed random variable with expectation N₀=u₁(X(0)) (total weighted number of particles at time0),

• W is a complex-valued random variable that admits moments of all ordersp≥1and whose expectation equals u₂(X(0)),

• the real-valued random variablesε1(t)andε2(t)and the real randomvector3(t)tend to0 ast tends to+∞, almost surely and in anyL^p-space,p≥1.

In other words, if one denotes byϕ any argument of the complex numberW, the trajectory of the random vector X(t ), projected in the 3-dimensional real vector space spanned by the vectors((v₂),(v₂), v₁)is almost surely asymptotic to the (random) spiral

⎧⎨

⎩

x(t)=2|W|e^σ²^tcos(τ2t+ϕ), y(t)= −2|W|e^σ²^tsin(τ2t+ϕ), z(t )=ξe^t,

drawn on the (random) revolution surface 4|W|²z^2σ²=ξ^2σ²

x²+y² , whenttends to infinity. See Fig.1

In the whole paper,W denotes our hero, namely the limit complex-valued random variable that appears inX(t )’s expansion, as in Theorem4.1.

Proof of Theorem4.1. Denote byAthe endomorphism ofR^m⁻¹having^tR_Gas matrix in the canonical basis. Let also M(t )=exp(−tA)X(t), for anyt≥0. By standard arguments from multitype branching process theory,(M(t))t≥0

is a vector-valued martingale. Sincem≥27, the real part ofλ2belongs to]1/2,1[so that the projected martingales u₁(M(t))andu₂(M(t))converge in L^pfor anyp≥1. For proofs of these results, see for example Athreya and Ney [1] or Janson [12] (especially Lemma 10.2 of Janson’s paper for the L^p-boundedness,X being here anirreducible process in the sense of [12]). The random variablesξ andWare respectively defined by

ξ=limt→+∞u1

e⁻^t^AX(t )

=limt→+∞e⁻^tu1

X(t ) , W=limt→+∞u2

e⁻^t^AX(t )

=limt→+∞e⁻^λ²^tu2

X(t )

. (4.2)

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Fig. 1. Spiral in the vector space spanned by((v2),(v2), v1)to whichX(t)is a.s. asymptotic.

An alternative proof of the L^p convergence can be made using the techniques of [19], as developed in [5] for two- colour urn processes. In particular,ξ’s distribution is attained by explicit computation of its moments: for any non- negative integerp, an elementary computation shows directly from (2.2) that the (so-calledreduced) polynomial

Q:=u1(u1+1)(u1+2)· · ·(u1+p−1)

is an eigenvector forX’s infinitesimal generatorG, associated with the eigenvaluep. ThusEQ(X(t))=e^ptQ(X(0)) for anyt. Besides, because of (4.2),Q(X(t ))=e^ptξ^p(1+o(1))asttends to infinity, almost surely and in L¹. Finally, the last two equalities provide

Eξ^p=Q X(0)

= (N₀+p) (N₀) .

This shows that the law ofξ is a Gamma distribution with parameterN0since a Gamma distribution is completely determined by its moments. The matrixRGis diagonalizable onCsince all roots of its characteristic polynomial are simple (see (2.3)). Extending notations (2.4), let(uλ)_λ_∈_Sp(_A₎be a basis of linear forms, eachuλbeing an eigenvector ofGassociated with the (complex) eigenvalueλ. Let also(vλ)_λ_∈_Sp(_A₎be the dual basis of(uλ)_λ_∈_Sp(_A₎, eachvλbeing thus a vector that satisfiesuλ(vμ)=δλ,μ(Kronecker’s notation). Note that one can chooseuλ₂=u2and, consequently, vλ₂=v2(cf. notations (2.4)).

For anyt≥0, split the spectral decomposition of the vectorX(t )with respect toGinto four terms:

X(t )=

λ∈Sp(A)

u_λ X(t )

·v_λ=X₁(t)+X₂(t)+X₃(t)+X₄(t),

where

⎧⎪

⎪⎨

⎪⎪

⎩

X1(t)=u1

X(t ) v1, X₂(t)=u_λ₂

X(t )

v_λ₂+u_λ

2

X(t ) v_λ

2, X₃(t)=

1/2<λ<λ₂u_λ X(t )

v_λ, X4(t)=

λ<1/2uλ

X(t ) vλ.

Note that this partition of Sp(A)is valid because ¹₂ is not an eigenvalue ofAas can be checked from (2.3). We deal separately with these four components ofX(t ). Defineε3byε3(t)=X3(t)+X4(t), for anyt≥0.

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•The formulae (4.2) provide directly the asymptotics

⎧⎪

⎨

⎪⎩

X₁(t)= e^tξ+o

e^t v₁, X2(t)=

e^λ²^tW+o e^λ²^t

v2+

v2

=2

v₂ , leading to the first two terms of the expansion (4.1).

•Suppose thatλis an eigenvalue ofAsuch that ¹₂<λ <λ2. Then, with the same general arguments as in the very beginning of the proof, it can be seen that

u_λ M(t )

=e⁻^{t λ}u_λ X(t )

and that(u_λ(M(t)))_t_≥₀is a convergent martingale, bounded in any L^p,p≥1. In particular,u_λ(X(t))=o(e^λ²^t)ast tends to infinity, almost surely and in any L^p,p≥1. This shows thatX3(t)is o(e^λ²^t)whent→ +∞.

•It remains to deal with the small eigenvalues, namely with allλsuch thatλ <¹₂.

Lemma 4.2. Suppose thatλis an eigenvalue such thatλ <¹₂ and letη >0.Then, e⁻^(1/2⁺^η)tu_λ(X(t))is bounded almost surely and in anyL^p-space,p≥1.

The proof of this lemma is given just hereafter. Therefore, ifλ <¹₂, then e⁻^λ²^tu_λ

X(t )

=e^(1/2⁺^η⁻^λ²^)t

e⁻^(1/2⁺^η)tu_λ X(t )

t−→→∞0

almost surely as soon as 0< η <λ2−¹₂. Suchηexist because λ2>¹₂. This shows thatX4(t)is o(e^λ²^t)when t→ +∞. The same argument holds for the L^pconvergence, making the proof complete.

Proof of Lemma4.2. The main idea consists in taking advantage of the following fact: whentbelongs to the interval [τ_n, τ_n₊₁[, the vectorX(t )remains equal toX_n^DT. This being considered, we make use of the moment bounds of the discrete time process that can be found in [19] (Theorem 3.4(1)): whenλ <¹₂,

∀p≥1,∀ε >0, Eu_λ

X^DT_n ^p=O

n^p(1/2⁺^ε)

, n→ +∞. (4.3)

•Almost sure bound: we prove that

C→+∞lim P

∃t >0,e⁻^(1/2⁺^η)tu_λX(t )> C

=0, (4.4)

which suffices to get the almost sure boundedness. LetC >0,η >0 and letλbe an eigenvalue such thatλ <¹₂. The jump timeτ_ntends almost surely to+∞which is a classical result that can be deduced from Lemma3.1, so that

P

∃t >0,e⁻^(1/2⁺^η)tu_λX(t )> C

≤

n≥0

P

∃t∈ [τn, τn+1[,e⁻^(1/2⁺^η)tuλ

X(t )> C .

SinceX(t )=X^DT_n for anyt∈ [τn, τn+1[, this leads to P

∃t >0,e⁻^(1/2⁺^η)tu_λX(t )> C

≤

n≥0

Pu_λ

X^DT_n > Ce^(1/2⁺^η)τⁿ .

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Conditioning with respect toτ_n, using Markov inequality and the fact thatτ_n andX_n^DT are independent, one gets successively, for anyp≥1:

P

∃t >0,e⁻^(1/2⁺^η)tu_λX(t )> C

≤

n≥0

E Pu_λ

X^DT_n > Ce^(1/2⁺^η)τⁿ|τ_n

≤

n≥0

E

E|u_λ(X_n^DT)|^p C^pe^p(1/2⁺^η)τⁿ

= 1 C^p

n≥0

Euλ

X_n^DT^pE

e⁻^p(1/2⁺^η)τⁿ . The density of thenth jump timeτ_nis the function

u∈R−→ne⁻^u

1−e⁻^un−1

1_R₊(u),

so that its Laplace transform can be elementarily computed: for anys≥0, E

e⁻^sτⁿ

= n! (s+1)

(s+1+n)∼ (s+1)n⁻^s, n→ +∞. Together with (4.3), this leads to:∀η >0,∀ε >0,∀p≥1,

Eu_λ

X^DT_n ^pE

e⁻^p(1/2⁺^η)τⁿ

=O 1

n^p(η⁻^ε)

, n→ +∞,

which is the general term of a convergent series as soon as one takesε < ηandp >_η₋¹_ε. Finally, lettingC tend to infinity shows (4.4).

•Bound in L^p-space: letp≥1 andt >0. Then, e⁻^(1/2⁺^η)tu_λX(t )^p

p=e⁻^(1/2⁺^η)ptEu_λX(t )^p.

Using the relation with the discrete time process(X^DT_n )n, one has successively e⁻^(1/2⁺^η)tuλ

X(t )^p

p=e⁻^(1/2⁺^η)pt

n≥0

E

1τ_n≤t <τ_n₊₁uλ

X(t )^p

=e⁻^(1/2⁺^η)pt

n≥0

E

1τ_n≤t <τ_n₊₁uλ

X^DT_n ^p

=e⁻^(1/2⁺^η)pt

n≥0

E(1τ_n≤t <τ_n+1)Eu_λ

X^DT_n ^p ,

where the last equality holds due to the independence betweenτnandX_n^DT. Besides,τnandτn+1−τnare independent andτ_n₊₁−τ_nisExp(n+N₀)-distributed (see (3.1)), so that, using the density ofτ_nwritten above, one gets

E(1τn≤t <τn+1)=E

1t≥τnE(1τn+1−τn≥t−τn|τ_n)

=E

1t≥τ_ne⁻⁽ⁿ⁺^N⁰^)(t⁻^τⁿ⁾

= _t

0

e⁻⁽ⁿ⁺^N⁰^)(t⁻^u)ne⁻^u

1−e⁻^un−1

du

≤ne⁻⁽ⁿ⁺^1)t t

0

e^u−1n−1

e^udu=

1−e⁻^tn

e⁻^t. Thus,

e⁻^(1/2⁺^η)tu_λX(t )^p

p≤e⁻^te⁻^(1/2⁺^η)pt

n≥0

1−e⁻^tn

Eu_λ

X^DT_n ^p .

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Let nowε >0. On one hand, (4.3) implies that Eu_λ

X_n^DT^p

=O

n^p(1/2⁺^ε) .

On the other hand, Stirling’s formula applied to generalized binomial coefficients yields classically that for anyα∈C, zⁿ

(1−z)⁻^α⁻¹= n^α (α+1)

1+O

1 n

,

where the notation[zⁿ]A(z)means the coefficient ofzⁿin the power expansion ofA(z)at the origin. Consequently, Eu_λ

X_n^DT^p

=O zⁿ

(1−z)⁻¹⁻^p(1/2⁺^ε) .

This implies that for anyε >0, there exists a constantC_εsuch that for anyt >0, e⁻^(1/2⁺^η)tu_λX(t )^p

p≤C_εe⁻^te⁻^(1/2⁺^η)pt 1−

1−e⁻^t₋1−p(1/2+ε)

=C_εe⁻^{pt (η}⁻^ε).

It suffices to takeε=η/2 to conclude that the L^p-norm of e⁻^(1/2⁺^η)tu_λ(X(t))is bounded above.

Remark 4.3. The distribution ofWis infinitely divisible,because it is the limit of infinitely divisible ones,obtained by scaling and projection of infinitely divisible ones.Indeed,in finite time,for anyx₀∈R^m⁻¹,denote by(X_x₀(t), t≥0) the process(X(t), t≥0)defined in Section2.1starting from initial statex₀.By the branching property

Xx₀(t)= [^L n]Xx₀/n(t),

where the notation[n]Xdenotes the sum ofnindependent copies of the random variableX.The infinite divisibility of Whad already been noticed by Janson([12],proof of Theorem3.9).

4.2. Martingale connection

In this subsection, we use the embedding equality (3.2) to deduce connections between the asymptotic behaviours of X_n^DT whenn→ +∞andX(t )whent→ +∞.

The Markov chain(X_n^DT)_nexhibits a well-known phase transition of the same kind as the continous time process:

whenm≤26, with notations of Section2.4,n⁻^1/2(X_n^DT−nv1)converges in law to a centered Gaussian vector (see Mahmoud’s book [16]). Form≥27, it has been proved in [4] and [18] that

X^DT_n =nv₁+2

n^λ²W^DTv₂ +o

n^σ²

a.s. and inL^p,∀p≥1, (4.5)

wherev₁, v₂are the deterministic vectors defined in (2.4),W^DT is the limit of a complex-valued martingale. Moreover, W^DT admits moments of all orders that can be recursively calculated and satifiesW^DT =limn→∞n⁻^λ²u2(X^DT_n ) almost surely.

Proposition 4.4. The following two assertions hold:

n→+∞lim ne⁻^τⁿ=ξ a.s.and inL^p,∀p≥1, (4.6)

W=ξ^λ²W^DT a.s.withξandW^DT independent. (4.7)

The equality (4.7), commonly referred to asmartingale connection, establishes the link betweenWandW^DT. In this way, the results onWin the present paper can be seen as a first step to a better knowledge ofW^DT distribution.

Proof of Proposition 4.4. We first prove (4.6). Applying the first projection to the embedding equality (3.2), we obtain that

u1

X(τn)

=u1

X_n^DT a.s.,

whereu1has been defined in (2.4). This is the total number of free places at timeτn, and is equal ton−1+N0=