Behavior near the extinction time in self-similar fragmentations I : the stable case

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www.imstat.org/aihp 2010, Vol. 46, No. 2, 338–368

DOI:10.1214/09-AIHP317

Behavior near the extinction time in self-similar fragmentations I: The stable case

Christina Goldschmidt

^a

and Bénédicte Haas

^b

aDepartment of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK. E-mail:goldschm@stats.ox.ac.uk bUniversité Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France. E-mail:haas@ceremade.dauphine.fr

Received 15 May 2008; revised 28 January 2009; accepted 13 February 2009

Abstract. The stable fragmentation with index of self-similarityα∈ [−1/2,0)is derived by looking at the masses of the subtrees formed by discarding the parts of a(1+α)⁻¹–stable continuum random tree below heightt, fort≥0. We give a detailed limiting description of the distribution of such a fragmentation,(F (t), t≥0), as it approaches its time of extinction,ζ. In particular, we show thatt^1/αF ((ζ −t)⁺)converges in distribution ast→0 to a non-trivial limit. In order to prove this, we go further and describe the limiting behavior of (a) an excursion of the stable height process (conditioned to have length 1) as it approaches its maximum; (b) the collection of open intervals where the excursion is above a certain level; and (c) the ranked sequence of lengths of these intervals. Our principal tool is excursion theory. We also consider the last fragment to disappear and show that, with the same time and space scalings, it has a limiting distribution given in terms of a certain size-biased version of the law ofζ.

In addition, we prove that the logarithms of the sizes of the largest fragment and last fragment to disappear, at time(ζ−t)⁺, rescaled by log(t), converge almost surely to the constant−1/αast→0.

Résumé. La fragmentation stable d’incice α∈ [−1/2,0)est construite à partir des masses des sous-arbres de l’arbre continu aléatoire stable d’indice(1+α)⁻¹obtenus en ne gardant que les feuilles situées à une hauteur supérieure àt, pourt≥0. Nous donnons une description détaillée du comportement asymptotique d’une telle fragmentation,(F (t), t≥0), au voisinage de son point d’extinction,ζ. En particulier, nous montrons quet^1/αF ((ζ −t)⁺)converge en loi lorsquet→0 vers une limite non triviale. Pour obtenir ce résultat, nous allons plus loin et décrivons le comportement asymptotique en loi, après normalisation, (a) d’une excursion du processus de hauteur stable (conditionnée à avoir une longueur 1) au voisinage de son maximum; (b) des intervalles ouverts où l’excursion est au-dessus d’un certain niveau; et (c) de la suite décroissante des longueurs de ces intervalles.

Notre outil principal est la théorie des excursions. Nous nous intéressons également au dernier fragment à disparaître et montrons, qu’avec les mêmes normalisations en temps et espace, la masse de ce fragment a une distribution limite construite à partir d’une certaine version biaisée deζ.

Enfin, nous montrons que les logarithmes des masses du plus gros fragment et du dernier fragment à disparaître, au temps (ζ−t)⁺, divisés par log(t), convergent presque sûrement vers la constante−1/αlorsquet→0.

MSC:60G18; 60G52; 60J25

Keywords:Stable Lévy processes; Height processes; Self-similar fragmentations; Extinction time; Scaling limits

1. Introduction

The subject of this paper is a class of random fragmentation processes which were introduced by Bertoin [5], called the self-similar fragmentations. In fact, we will find it convenient to have two slightly different notions of a fragmentation process. By aninterval fragmentation, we mean a process(O(t), t≥0)taking values in the space of open subsets of

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(0,1)such thatO(t )⊆O(s)whenever 0≤s≤t. We refer to a connected interval component ofO(t )as ablock. Let F (t )=(F₁(t), F₂(t ), . . .)be an ordered list of the lengths of the blocks ofO(t ). ThenF (t )takes values in the space

S₁^↓=

s=(s₁, s₂, . . .): s₁≥s₂≥ · · · ≥0, ∞ i=1

s_i≤1

.

We call the process(F (t), t ≥0)aranked fragmentation. A ranked fragmentation is calledself-similar with index α∈Rif it is a time-homogeneous Markov process which satisfies certainbranchingandself-similarityproperties.

Roughly speaking, these mean that every block should split into a collection of sub-blocks whose relative lengths always have the same distribution, but at a rate which is proportional to the length of the original block raised to the powerα. (Rigorous definitions will be given in Section2.) Clearly the sign ofαhas a significant effect on the behavior of the process. Ifα >0 then larger blocks split faster than smaller ones, which tends to act to balance out block sizes.

On the other hand, ifα <0 then it is the smaller blocks which split faster. Indeed, small blocks split faster and faster until they are reduced todust, that is blocks of size 0.

The asymptotic behavior of self-similar fragmentations has been studied quite extensively. In one sense, it is trivial, in thatF (t )→0 a.s. ast→ ∞, whatever the value of the indexα(provided the process is not trivially constant, i.e.

equal to its initial value for all timest). Forα≥0, rescaled versions of the empirical measures of the lengths of the blocks have law of large numbers-type behavior (see Bertoin [6] and Bertoin and Rouault [8]). Forα <0, however, the situation is completely different. Here, there exists an almost surely finite random timeζ, called theextinction time, when the state is entirely reduced to dust. The manner in which mass is lost has been studied in detail in [13]

and [14].

The purpose of this article is to investigate the following more detailed question whenαis negative:how does the processF ((ζ −t )⁺)behave ast→0?We provide a detailed answer for a particularly nice one-parameter family of self-similar fragmentations with negative index, called thestable fragmentations.

The simplest of the stable fragmentations is theBrownian fragmentation, which was first introduced and studied by Bertoin [5]. Suppose that(e(x),0≤x≤1)is a standard Brownian excursion. Consider, fort≥0, the sets

O(t )=

x∈ [0,1]: e(x) > t

and let F (t )∈S₁^↓ be the lengths of the interval components ofO(t ) in decreasing order. Then it can be shown that (F (t), t≥0)is a self-similar fragmentation with index−1/2. Miermont [20] generalized this construction by replacing the Brownian excursion with an excursion of the height process associated with the stable tree of index β ∈(1,2), introduced and studied by Duquesne, Le Gall and Le Jan [11,19]. The corresponding process is a self- similar fragmentation of indexα=1/β−1.

The behavior near the extinction time in the Brownian fragmentation can be obtained via a decomposition of the excursion at its maximum. We discuss this case in Section3. Abraham and Delmas [1] have recently proved a generalized Williams’ decomposition for the excursions which code stable trees. This provides us with the necessary tools to give a complete description of the behavior of the stable fragmentations near their extinction time, which is detailed in Section4. In every case, we obtain that

t^1/αF

(ζ−t )⁺ d

→F_∞ ast→0,

whereF_∞is a random limit which takes values in the set of non-increasing non-negative sequences with finite sum.

The limitF_∞is constructed from a self-similar functionH_∞onR, which itself arises when looking at the scaling behavior of the excursion in the neighborhood of its maximum. See Theorems4.1and4.2and Corollary4.3for precise statements.

In Corollary4.4, we also consider the process of thelast fragment, that is the size of the (as it turns out unique) block which is the last to disappear. We call this sizeF_∗(t)and prove that, scaled as before,F_∗((ζ−t )⁺)also has a limit in distribution ast→0 which, remarkably, can be expressed in terms of a certain size-biased version of the distribution ofζ.

Sections5–8are devoted to the proofs of these results.

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Finally, in Section9, we consider the logarithms of the largest and last fragments and show that logF1((ζ−t )⁺)

log(t) → −1/α and logF_∗((ζ−t )⁺)

log(t) → −1/α

almost surely ast→0. In fact, these results hold for a more general class of self-similar fragmentations with negative index.

We will investigate the limiting behavior ofF ((ζ −t )⁺)as t →0 for general self-similar fragmentations with negative indexαin future work, starting in [12]. In general, as indicated by the results for the logarithms ofF₁andF_∗ above, the natural conjecture is thatt^1/αis the correct re-scaling for non-trivial limiting behavior. However, since the excursion theory tools we use here are not available in general, we are led to develop other methods of analysis.

2. Self-similar fragmentations

DefineO(0,1)to be the set of open subsets of(0,1). We begin with a rather intuitive notion of a fragmentation process.

Definition 2.1 (Interval fragmentation). An interval fragmentation is a process(O(t), t≥0)taking values inO(0,1)

such thatO(t )⊆O(s)whenever0≤s≤t.

In this paper we will be dealing with interval fragmentations which derive from excursions. Here, anexcursion is a continuous functionf:[0,1] →R⁺such thatf (0)=f (1)=0 andf (x) >0 for allx∈(0,1). The associated interval fragmentation,(O(t), t≥0)is defined as follows: for eacht≥0,

O(t )=

x∈ [0,1]: f (x) > t . An example is given in Fig.1.

We need to introduce a second notion of a fragmentation process. We endow the spaceS₁^↓ with the topology of pointwise convergence.

Definition 2.2 (Ranked self-similar fragmentation). A ranked self-similar fragmentation(F (t), t≥0)with index α∈Ris a càdlàg Markov process taking values inS₁^↓such that

• (F (t), t≥0)is continuous in probability;

• F (0)=(1,0,0, . . .);

• Conditionally on F (t )=(x₁, x₂, . . .), F (t+s) has the law of the decreasing rearrangement of the sequences x_iF⁽ⁱ⁾(x_i^αs),i≥1,whereF⁽¹⁾, F⁽²⁾, . . .are independent copies of the original processF.

Letr:O(0,1)→S₁^↓be the function which to an open setO⊆(0,1)associates the ranked sequence of the lengths of its interval components. We say that an interval fragmentation isself-similarif it possesses branching and self- similarity properties which entail that(r(O(t)), t≥0)is a ranked self-similar fragmentation. See [2,5,7] for this and further background material.

Fig. 1. An interval fragmentation derived from a continuous excursion: the setO(t)is represented by the solid lines at levelt.

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Bertoin [5] has proved that a ranked self-similar fragmentation can be characterized by three parameters,(α, ν, c).

Here,α∈R;νis a measure onS₁^↓such thatν((1,0, . . .))=0 and _S↓

1(1−s₁)ν(ds) <∞; andc∈R⁺. The parameter αis theindex of self-similarity. The measureνis thedislocation measure, which describes the way in which fragments suddenly dislocate; heuristically, a block of mass msplits at rate m^αν(ds) into blocks of masses(ms1, ms2, . . .).

The real numbercis theerosion coefficient, which describes the rate at which blocks continuously melt. Note thatν may be an infinite measure, in which case the times at which dislocations occur form a dense subset ofR⁺. When c=0, the fragmentation is a pure jump process.

In the context of an interval fragmentation derived from an excursion, it is easy to see that the extinction time of the fragmentation is just the maximum height of the excursion:

ζ = max

0≤x≤1f (x).

In the examples we treat, this maximum will be attained at a unique point,x_∗. In this case, letO_∗(t)be the interval component ofO(t )containingx_∗at timet, and letF_∗(t)be its length, i.e.F_∗(t)= |O_∗(t)|. We call bothO_∗andF_∗ thelast fragment process.

3. The Brownian fragmentation

We begin by discussing the special case of the Brownian fragmentation. The sketch proofs in this section are not rigorous, but can be made so, as we will demonstrate later in the paper. Our intention is to introduce the principal ideas in a framework which is familiar to the reader.

Let(e(x),0≤x≤1)be a normalized Brownian excursion with length 1 and for eacht≥0 define the associated interval fragmentation by

O(t ):=

x∈ [0,1]: e(x) > t .

See Fig. 2 for a picture. The associated ranked fragmentation process,(F (t), t ≥0), has index of self-similarity α= −1/2, binary dislocation measure specified byν(s1+s2<1)=0 and

ν(s₁∈dx)=2

2πx³(1−x)³₋1/2

1_[1/2,1)(x)dx,

Fig. 2. The Brownian interval fragmentation with the open intervals which constitute the state at timest=0,0.15,0.53 and 0.92 indicated.

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and erosion coefficient 0. See Bertoin [5] for a proof. The extinction time of this fragmentation process is the maximum of the Brownian excursion. In particular, from Kennedy [18] we have

P(ζ > t)=2 ∞ n=1

4t²n²−1 exp

−2t²n²

, t≥0.

To the best of our knowledge, this is the only fragmentation for which the law ofζ is known. It is well known that the maximum ofeis reached at a unique pointx_∗∈ [0,1]a.s., and so the mass,F_∗(t), of the last fragment to survive is well defined.

There is a complete characterization of the limit in law of the rescaled fragmentation near to its extinction time.

Theorem 3.1 (Brownian fragmentation). IfOis the Brownian interval fragmentation then O((ζ −t )⁺)−x_∗

t²

→d O_∞ ast→0, whereO_∞= {x∈R: H_∞(x) <1},

H_∞(x)=R₊(x)1_{x≥0}+R₋(−x)1_{x<0}

andR₊andR₋are two independentBes(3)processes.

A full discussion of the topology in which the above convergence in distribution occurs is deferred until Section5.1.

A proof of this theorem is given in Uribe Bravo [23]. We give here a sketch, since a rigorous proof will be given in the wider setting of general stable fragmentations.

Sketch of proof of Theorem3.1. We decompose the excursioneat its maximumζ. Define

˜ e(x)=

ζ−e(x_∗+x), 0≤x≤1−x_∗, ζ−e(x−1+x_∗), 1−x_∗< x≤1.

Then by Williams’ decomposition for the Brownian excursion [22], Section VI.55, we have that˜eis again a standard Brownian excursion. Moreover, ift < ζthen

t⁻²

O(ζ−t )−x_∗

= y∈

0, (1−x_∗)t⁻² : e˜

yt²

< t

∪ y∈

−x_∗t⁻²,0 : e˜

1+yt²

< t .

Now by the scaling property of Brownian motion,(t⁻¹e(xt˜ ²),0≤x≤t⁻²)has the distribution of a Brownian excursion of lengtht⁻², which we will denote(b^t(x),0≤x≤t⁻²). So the above set has the same distribution as

x∈

0, (1−x_∗)t⁻²

: b^t(x) <1

∪ x∈

−x_∗t⁻²,0 : b^t

t⁻²+x

<1 .

Fixn∈R⁺. Ast→0, the length of the excursion goes to∞and(b^t(x),0≤x≤n)→^d (R₊(x),0≤x≤n), where (R₊(x), x≥0)is a 3-dimensional Bessel process started at 0 (see, for example, Theorem 0.1 of Pitman [21]). More- over, by symmetry,(b^t(t⁻²−x),0≤x≤n)→^d (R₋(x),0≤x≤n), whereR₋ is (in fact) an independent copy of R₊. Thus we obtain

t⁻² O

(ζ−t )⁺

−x_∗ d

→O_∞,

as claimed.

Theorem3.1enables us to deduce an explicit limit law for the associated ranked fragmentation. See Corollary4.3 for a precise statement and note that, as detailed at the end of Section4, the passage from the convergence of open sets to that of these ranked sequences is not immediate. We also have the following limit law for the last fragment,F_∗.

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Corollary 3.2. IfF is the Brownian fragmentation then F_∗((ζ−t )⁺)

t²

→d

2ζ π

2

ast→0, or,equivalently,

t F_∗

(ζ−t )⁺₋1/2 d

→ζ_∗,

whereζ_∗is a size-biased version ofζ,i.e.E[f (ζ_∗)] =E[ζf (ζ )]/E[ζ]for every test functionf.

Proof. LetT₊=inf{t≥0: R₊(t)=1}andT₋=inf{t≥0: R₋(t)=1}, whereR₊ andR₋ are the independent Bes(3)processes from the statement of Theorem3.1. Then by Theorem3.1,

F_∗((ζ−t )⁺) t²

→d T₊+T₋.

By Proposition 2.1 of Biane et al. [9], T₊+T₋=^d

2ζ π

2

and, moreover, if we defineY=

π2ζ thenY satisfies E

f (1/Y )

=E Yf (Y )

for any test functionf (in particular,Y has mean 1). Hence, E

f

π 2ζ

= 2

πE ζf (ζ )

=E ζf (ζ )

/E[ζ],

which completes the proof.

Remark 1. As noted by Uribe Bravo[23],the random variable(2ζ /π)²has Laplace transform2λ(sinh√

2λ)⁻².He also uses another result in Biane et al. [9]to show that the Lebesgue measure of the setO_∞has Laplace transform (cosh√

2λ)⁻².LetM(t )be the total mass of the fragmentation at timet,that is the Lebesgue measure ofO(t ).Then this entails that

M((ζ−t )⁺) t²

→d M_∞,

whereM_∞has Laplace transform(cosh√ 2λ)⁻².

Remark 2. The Bes(3)process encodes a fragmentation process with immigration which arises naturally when study- ing rescaled versions of the Brownian fragmentation near t=0 (see Haas[15]).This is closely related to our ap- proach:using Williams’ decomposition of the Brownian excursion,we obtain results on the behavior of the frag- mentation near its extinction time by studying the sets of{x∈ [0,1]: e(x) < t}for smallt.This duality between the behavior of the fragmentation near0and near its extinction time seems to be specific to the Brownian case.

4. General stable fragmentations

There is a natural family which generalizes the Brownian fragmentation: thestable fragmentations, constructed and studied by Miermont in [20]. The starting point is the stable height processes H with index 1< β≤2 which were

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introduced by Duquesne, Le Gall and Le Jan [11,19] in order to code the genealogy of continuous state branching processes with branching mechanismλ^β via stable trees. We do not give a definition of these processes here, since it is rather involved; full definitions will be given in the course of the next section. Here, we simply recall that it is possible to consider a normalized excursion ofH, saye, which is almost surely continuous on[0,1]. Whenβ=2, this is the normalized Brownian excursion (up to a scaling factor of√

2).

Once again, let O(t ):=

x∈ [0,1]: e(x) > t .

For 1< β <2, Miermont [20] proved that the corresponding ranked fragmentation is a self-similar fragmentation of indexα=1/β−1 and erosion coefficient 0. The dislocation measure is somewhat harder to express than that of the Brownian fragmentation. LetT be a stable subordinator of Laplace exponentλ^1/βand writeT_[_0,1_]for the sequence of its jumps before time 1, ranked in decreasing order. Then for any non-negative measurable functionf,

S1^↓

f (s)ν(ds)=β(β−1)(1−_β¹)

(2−β) E

T1f

T₁⁻¹T_[0,1] .

As we will discuss in Section5.2, there is a unique pointx_∗∈ [0,1]at whicheattains its maximum (this maximum is denotedζ to be consistent with earlier notation, so thatζ=e(x_∗)). So the size of the last fragment,F_∗, is well defined for the stable fragmentations. We first state a result on the behavior of the stable height processes near their maximum.

Theorem 4.1. Letebe a normalized excursion of the stable height process with parameterβand extend its definition toRby settinge(x)=0whenx /∈ [0,1].Then there exists an almost surely positive continuous self-similar function H_∞onR,which is symmetric in distribution(in the sense that(H_∞(−x), x≥0)=^d (H_∞(x), x≥0))and converges to+∞asx→ +∞orx→ −∞,and which is such that

t⁻¹ ζ −e

x_∗+t⁻^1/α· d

→H_∞ ast→0,

whereα=1/β−1.The convergence holds with respect to the topology of uniform convergence on compacts.

A precise definition ofH_∞is given in Section5.3. Intuitively, we can think of it as an excursion of the height process of infinite length, centered at its “maximum” and flipped upside down.

Theorem4.1leads to the following generalization of Theorem3.1.

Theorem 4.2 (Stable interval fragmentation). Let O be a stable interval fragmentation with parameter β and consider the corresponding self-similar functionH_∞introduced in Theorem4.1.Then

t^1/α O

(ζ−t )⁺

−x_∗ d

→

x∈R: H_∞(x) <1

ast→0.

The topology on the bounded open sets ofRin which this convergence occurs will be discussed in the next section.

Define S^↓:=

s=(s₁, s₂, . . .): s₁≥s₂≥ · · · ≥0, ∞ i=1

s_i<∞

.

We endow this space with the distance d_S_↓

s,s

=^∞

i=1

s_i−s_i.

We also have the following ranked counterpart of Theorem4.2: letF (t )be the decreasing sequence of lengths of the interval components of(O(t), t≥0)and, similarly, letF_∞be the decreasing sequence of lengths of the interval components of{x∈R: H_∞(x) <1}. ThenF_∞∈S^↓.

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Corollary 4.3 (Ranked stable fragmentation). Ast→0, t^1/αF

(ζ−t )⁺ d

→F_∞.

In particular, this gives the behavior of the total massM(t ):=_∞

i=1F_i(t)of the fragmentation near its extinction time.

Finally, as in the Brownian case, the distribution of the limit of the size of the last fragment can be expressed in terms of a size-biased version of the heightζ.

Corollary 4.4 (Behavior of the last fragment). Ast→0, t

F_∗

(ζ−t )⁺α d

→ζ_∗^α, (4.1)

whereζ_∗^αis a “(−1/α−1)-size-biased” version ofζ,which means that for every test-functionf, E

f (ζ_∗^α)

=E[ζ⁻^1/α⁻¹f (ζ )]

E[ζ⁻^1/α⁻¹] . (4.2)

Moreover,

(i) there exist positive constants0< A < Bsuch that exp

−Bt^1/(1⁺^α)

≤P(ζ_∗^α≥t )≤exp

−At^1/(1⁺^α) for alltsufficiently large;

(ii) for anyq <1−1/α, P(ζ_∗^α≤t )≤t^q

for allt≥0sufficiently small.

The proof of Theorem4.1is based on the “Williams’ decomposition” of the height functionHgiven by Abraham and Delmas [1], Theorem 3.2, and can be found in Section6. We emphasise the fact that uniform convergence on compacts of a sequence of continuous functions fn:R→Rtof:R→Rdoes not imply in general that the sets {x∈R: f_n(x) <1}converge to{x∈R: f (x) <1}. Take, for example,f constant and equal to 1 andf_n constant and equal to 1−1/n(see the next section for the topology we consider on open sets ofR). Less trivial examples show that there may exist another kind of problem when passing from the convergence of functions to that of ranked sequences of lengths of interval components: take, for example, a continuous functionf:R→R⁺which is strictly larger than 1 onR\[−1,1]and then consider continuous functionsf_n:R→R⁺such thatf_n=f on[−n, n],f_n=0 on[n+1,2n]. Clearly,f_nconverges uniformly on compacts tof, but the lengths of the longest interval components of{x∈R: f_n(x) <1}converge to∞.

However, we will see that the random functions we work with do not belong to the set of “problematic” counter- examples that can arise and that it will be possible to use Theorem4.1to get Theorem4.2and Corollary4.3. Pre- liminary work will be done in Section5, where an explicit construction of the limit functionH_∞via Poisson point measures is also given. Theorem4.2and Corollary4.3are proved in Section7. Then we will see in Section8that the limitζ_∗^α arising in (4.1) is actually distributed as the length to the powerαof an excursion ofH, conditioned to have its maximum equal to 1. It will then be easy to check that this is a size-biased version ofζ as defined in (4.2).

The bounds for the tailsP(ζ_∗^α≥t )will also be proved in Section8, as well as the following remark.

Remark. In the Brownian case(α= −1/2),the distribution ofζ (and consequently that of ζ_∗^α),is known;see Sec- tion 3.We do not know the distribution of ζ_∗^α explicitly whenα∈(−1/2,0).However,in this case,if we set,for λ≥0,

Φ(λ):=E exp

−λζ_∗^1/αα

=E[exp(−λζ^1/α)ζ⁻^1/α⁻¹] E[ζ⁻^1/α⁻¹] ,

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then it can be shown thatΦ satisfies the following equation:

Φ(λ)=exp

−

R⁺×[0,λ^−α]

1−e⁻⁽⁻^α/(α⁺¹⁾⁾⁻^1/α^r ⁰^t⁽¹⁻^Φ(v⁻^1/α^))v^1/α^dv

× −α

(1+α)²(¹₁⁺₊^2α_α )e⁻^r(⁻^αt/(α⁺¹⁾⁾^1+1/αr⁻^1/(α⁺¹⁾drdt

. (4.3)

Finally, we recall that the almost sure logarithmic results forF₁andF_∗will be proved in Section9.

5. Technical background

We start by detailing the topology on open sets which will give a proper meaning to the statement of Theorem4.2. We then recall some facts about height processes and prove various useful lemmas. Finally, we introduce the decomposition result of Abraham and Delmas [1], in a form suitable for our purposes.

5.1. Topological details

When dealing with interval fragmentations, we will work in the setOof bounded open subsets ofR. This is endowed with the following distance:

d_O(A, B)=

k∈N

2⁻^kd_H

A∩(−k, k)c

∩ [−k, k],

B∩(−k, k)c

∩ [−k, k] ,

whereS^cdenotes the closed complement ofS∈Oandd_His the Hausdorff distance on the set of compact sets ofR. ForA=R, letχ_A(x)=infy∈A^c|x−y|. If we define, forx∈ [−k, k],χ_A^k(x)=χ_A_∩₍₋_k,k)(x)=infy∈(A∩(−k,k))^c|x−y| then we also have

d_O(A, B)=

k∈N

2⁻^k sup

x∈[−k,k]

χ_A^k(x)−χ_B^k(x)

(see p. 69 of Bertoin [7]). The open sets we will deal with in this paper arise as excursion intervals of continuous functions. In particular, we will need to know that if we have a sequence of continuous functions converging (in an appropriate sense) to a limit, then the corresponding open sets converge.

Consider the spaceC(R,R⁺)of continuous functions fromRtoR⁺. Byuniform convergence on compacts, we mean convergence in the metric

d(f, g)=

k∈N

2⁻^k

sup

t∈[−k,k]

f (t )−g(t)∧1

.

The name is justified by the fact that convergence indis equivalent to uniform convergence on all compact sets.

Supposef ∈C(R,R⁺). We say thata∈R⁺ is alocal maximumof f if there exists∈Randε >0 such that f (s)=a and maxs−ε≤t≤s+εf (t )=a. Note that this includes the case wheref is constant and equal toa on some interval, even iff never takes values smaller thana. We define alocal minimumanalogously.

Proposition 5.1. Letf_n:R→R⁺be a sequence of continuous functions and letf:R→R⁺be a continuous function such thatf (0)=0,f (x)→ ∞asx→ +∞orx→ −∞.Suppose also that1is not a local maximum off and that Leb{x∈R: f (x)=1} =0.Define A= {x ∈R: f (x) <1}and A_n= {x ∈R: f_n(x) <1}.Suppose now thatf_n converges tof uniformly on compact subsets ofR.Thend_O(A_n, A)→0asn→ ∞.

Define

g(x)=sup

y≤x: f (y)≥1

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and

d(x)=inf

y≥x: f (y)≥1 . Then

χ_A(x)=

x−g(x)

∧

d(x)−x

. (5.1)

Defineg_nandd_nto be the analogous quantities forf_n. The proof of the following lemma is straightforward.

Lemma 5.2. Forx, y∈R, χA(x)−χA(y)≤ |x−y|. Moreover,

χ_A_n(x)−χ_A(x)≤maxg_n(x)−g(x),d_n(x)−d(x).

Proof of Proposition5.1. It suffices to prove thatd_H((A_n∩(−1,1))^c∩ [−1,1], (A∩(−1,1))^c∩ [−1,1])→0 as n→ ∞forf, f_n:[−1,1] →R⁺such thatf (0)=0, 1 is not a local maximum off, Leb{x∈ [−1,1]:f (x)=1} =0 andfn→f uniformly. In other words, we need to show that sup_x_∈[−_1,1_]|χ_A¹

n(x)−χ_A¹(x)| →0 asn→ ∞. Note that the appropriate definitions ofg(x)andd(x)in order to make (5.1) true forχ_A¹ are

g(x)=sup

y≤x: f (y)≥1

∨ −1 and

d(x)=inf

y≥x: f (y)≥1

∧1,

and we adopt these definitions for the rest of the proof.

Letε >0. Forr >0 let E_r^↑=

x∈(−1,1): x∈(a, b)such thatf (y) >1,∀y∈(a, b),|b−a|> r , E_r^↓=

x∈(−1,1): x∈(a, b)such thatf (y) <1,∀y∈(a, b),|b−a|> r .

These are the collections of excursion intervals of length exceedingr above and below 1. Take 0< δ < ε/2 small enough that Leb(E_δ^↑∪E^↓_δ) >2−ε/2 (we can do this since Leb{x∈ [−1,1]: f (x)=1} =0). SetR= [−1,1] \ (E^↑_δ ∪E_δ^↓). Each ofE_δ^↑andE_δ^↓is composed of a finite number of open intervals.

•We will first deal withE_δ^↑. On this set,χ_A¹(x)=0. Take hereafter 0< η < δ/6 and let E_δ,η^↑ =

x∈E^↑_δ: (x−η, x+η)⊆E_δ^↑ .

Then inf_x_∈_E↑ δ,η

f (x) >1. Sincef_n→f uniformly, it follows that there existsn₁such thatf_n(x) >1 for allx∈E^↑_δ,η whenevern≥n1. Then forn≥n1, we haveχ_A¹

n(x)=0 for allx∈E_δ,η^↑ . Since|χ_A¹

n(x)−χ_A¹

n(y)| ≤ |x−y|, it follows thatχ_A¹

n(x) < ηfor allx∈E_δ^↑. So sup_x_∈_E↑

δ |χ_A¹(x)−χ_A¹

n(x)|< ηwhenevern≥n₁.

•Now turn toE_δ^↓. As before, define E_δ,η^↓ =

x∈E^↓_δ: (x−η, x+η)⊆E_δ^↓ .

Then sup_x_∈_E↓ δ,η

f (x) <1. Sincef_n→f uniformly, it follows that there existsn₂such thatf_n(x) <1 for allx∈E_δ,η^↓ whenevern≥n2. Now, for each excursion below 1, there exists a left end-pointgand a right end-pointd. For allx

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in the same excursion,g(x)=gandd(x)=d. Suppose first that we haveg= −1,d=1 (in this case we say that the excursiondoes not touch the boundary). Since 1 is not a local maximum off, there must existzg< gandzd> d such that|z_g−g|< η,|z_d−d|< η,f (z_g) >1 andf (z_d) >1.

Suppose there areN_δexcursions below 1 of length greater thanδwhich do not touch the boundary. To excursioni there corresponds a left end-pointgi, a right end-pointdi and pointszg_i,zd_i, 1≤i≤Nδ. Write

E˜_δ^↓=

N_δ

i=1

(g_i, d_i) and E˜_δ,η^↓ = ˜E^↓_δ ∩E^↓_δ,η=

N_δ

i=1

(g_i+η, d_i−η).

Then min1≤i≤Nδ(f (z_g_i)∧f (z_d_i)) >1. Since f_n→f uniformly, there exists n₃ such that min1≤i≤Nδ(f_n(z_g_i)∧ f_n(z_d_i)) >1 for alln≥n₃. So forn≥n₂∨n₃and anyx∈ ˜E^↓_δ,η, by the intermediate value theorem, there exists at least one pointa_n(x)∈(g(x)−η, g(x)+η)such thatf_n(a_n(x))=1 and at least one pointb_n(x)∈(d(x)−η, d(x)+η)such thatfn(bn(x))=1. Sinceg(x)andd(x)are constant on excursion intervals, it follows that sup_x_{∈ ˜}_E↓

δ,η|gn(x)−g(x)|<

ηand sup_x_{∈ ˜}_E↓

δ,η|d_n(x)−d(x)|< ηforn≥n₂∨n₃. Hence, by Lemma5.2, sup

x∈ ˜E_δ,η^↓

χ_A¹(x)−χ_A¹

n(x)< η

whenevern≥n₂∨n₃. Since |χ_A¹(x)−χ_A¹(y)| ≤ |x−y| and|χ_A¹

n(x)−χ_A¹

n(y)| ≤ |x−y|, by using the triangle inequality we obtain that

sup

x∈ ˜E_δ^↓

χ_A¹(x)−χ_A¹

n(x)<3η.

It remains to deal with any excursions inE_δ^↓ which touch the boundary. Clearly, there is at most one excursion in E^↓_δ touching the left boundary and at most one excursion touching the right boundary. For these excursions, we can argue as before at the non-boundary end-points. At the boundary end-points, the argument is, in fact, easier since we have (by construction)χ_A¹(−1)=χ_A¹

n(−1)=0 andχ_A¹(1)=χ_A¹

n(1)=0. So, there existsn₄ such that for all n≥n₂∨n₃∨n₄,

sup

x∈E_δ^↓

χ_A¹(x)−χ_A¹

n(x)<3η.

•For anyx∈R, we haveχ_A¹(x)≤δ/2. Moreover, since Leb(E_δ^↑∪E^↓_δ) >2−ε/2, there must exist a pointz(x)∈R such that|z(x)−x|< ε/2 which is the end-point of an excursion interval (above or below 1) of length exceedingδ.

So for allx∈Rand allnwe have χ_A¹(x)−χ_A¹

n(x)≤χ_A¹(x)+χ_A¹

n(x)−χ_A¹

n

z(x)+χ_A¹

n

z(x)

−χ_A¹z(x)

≤δ/2+x−z(x)+ sup

y∈E^↑_δ∪E_δ^↓

χ_A¹

n(y)−χ_A¹(y)

< δ/2+ε/2+ sup

y∈E_δ^↑∪E_δ^↓

χ_A¹

n(y)−χ_A¹(y) (note that at the second inequality we use the continuity ofχ_A¹ andχ_A¹

n and the fact thatχ_A¹(z(x))=0).

•Finally, letn0=n1∨n2∨n3∨n4. Then sinceη < δ/6 andδ < ε/2, forn≥n0we have sup

x∈[−1,1]

χ_A¹(x)−χ_A¹

n(x)< ε.

The result follows.

The following lemma will be used implicitly in Section6.

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Lemma 5.3. Suppose thata >0,α∈Rand thatf:R→R⁺is a continuous function.Letg(t)=a^αf (at ).Then the function(a, f )→gis continuous for the topology of uniform convergence on compacts.

Proof. Letf_n:R→R⁺be a sequence of continuous functions withf_n→f uniformly on compacts. Suppose that a_nis a sequence of reals witha_n→a >0. SupposeK⊆Ris a compact set. Then we have

sup

t∈K

a^αf (at )−a_n^αf_n(a_nt )

≤sup

t∈K

a^αf (at )−a^α_nf (at )+sup

t∈K

a_n^αf (at )−a_n^αf (a_nt )+sup

t∈K

a_n^αf (a_nt )−a^α_nf_n(a_nt )

≤ sup

t∈aK

f (t )a^α−a^α_n+a^α_nsup

t∈K

f (at )−f (ant )+a_n^αsup

t∈K

f (ant )−fn(ant ).

The setaK is compact and sof is bounded on it; it follows that the first term converges to 0. Take 0< ε < a. Since a_n→a, there existsnsufficiently large that|a_n−a|< ε. DefineK˜ = {bt: t∈K, b∈ [a−ε, a+ε]}. ThenK˜ is also a compact set. The second term converges to 0 becausef is uniformly continuous onK. The third term is bounded˜ above by((a−ε)^α∨(a+ε)^α)sup_t_{∈ ˜}_K|f (t )−f_n(t)|and so, sincef_n→f uniformly on compacts, this converges

to 0.

Finally, we will need the following lemma in Section7.

Lemma 5.4. Letf:R→R⁺be a continuous function such that Leb

x∈R: f (x)=1

=0.

Supposefn:R→R⁺is a sequence of continuous functions that converges tof uniformly on compacts.Then,for all K >0,asn→ ∞,

Leb

x∈ [−K, K]: f_n(x) <1

→Leb

x∈ [−K, K]: f (x) <1 . Proof. LetK >0 and fixε >0. For allnsufficiently large

f (x)−ε≤f_n(x)≤f (x)+ε for allx∈ [−K, K], hence

Leb

x∈ [−K, K]: f (x) <1−ε

≤Leb

x∈ [−K, K]: fn(x) <1

≤Leb

x∈ [−K, K]: f (x) <1+ε .

Whenε→0, the left-hand side of this inequality converges to Leb{x∈ [−K, K]: f (x) <1}and the right-hand side

to Leb{x∈ [−K, K]: f (x)≤1}, which are equal by assumption.

5.2. Height processes

Here, we define the stable height process and recall some of its properties. We refer to Le Gall and Le Jan [19] and Duquesne and Le Gall [11] for background. (All of the definitions and results stated without proof below may be found in these references.)

Suppose that X is a spectrally positive stable Lévy process with Laplace exponent λ^β, β ∈ (1,2]. That is, E[exp(−λX_t)] =exp(tλ^β)for allλ, t≥0 and, therefore, if β∈(1,2), the Lévy measure of Xisβ(β−1)((2− β))⁻¹x⁻^β⁻¹dx,x >0. LetI (t ):=inf0≤s≤tX(s)be the infimum process ofX. For eacht >0, consider the time- reversed process defined for 0≤s < tby

Xˆ^(t)(s):=X(t )−X

(t−s)− ,

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and let(Sˆ^(t)(s),0≤s≤t )be its supremum, that isSˆ^(t)(s)=sup_u_≤_sXˆ^(t)(u). Then the height processH (t )is defined to be the local time at 0 ofSˆ^(t)− ˆX^(t).

It can be shown that the processHpossesses a continuous version (Theorem 1.4.3 of [11]), which we will implicitly consider in the following. The scaling property ofX is inherited byH (see Section 3.3 of [11]) as follows: for all a >0,

a^1/β⁻¹H (ax), x≥0d

=

H (x), x≥0 .

Whenβ=2, the height process is, up to a scaling factor, a reflected Brownian motion.

The excursion measure ofX−I away from 0 is denoted byN.In the following, we work under this excursion measure. LetE be the space of excursions; that is, continuous functions f:R⁺→R⁺such thatf (0)=0, inf{t >

0: f (t )=0}<∞and iff (s)=0 for somes >0 thenf (t )=0 for allt > s. The lifetime ofH∈Eis then denoted byσ, that is

σ:=inf

x >0: H (x)=0 . We define its maximum to be

H_max:= max

x∈[0,σ]H (x).

We will also deal with (regular versions of) the probability measuresN(·|σ=v),v >0 andN(·|Hmax=m),m >0.

The following proposition is implicit in Section 3 of Abraham and Delmas [1] and is also a consequence of Theo- rem9.1(ii) below.

Proposition 5.5. For anyv >0,underN(·|σ=v)there exists an almost surely unique pointxmaxat whichH attains its maximum,that is

xmax:=inf

x∈ [0, σ]: H (x)=Hmax

.

Note thate,ζ andx_∗(see Section4) have the distributions ofH,Hmaxandxmaxrespectively underN(·|σ=1).

First we note the tails of certain measures.

Proposition 5.6. For allm >0,

N(Hmax> m)=(β−1)^1/(1⁻^β)m^1/(1⁻^β), (5.2)

N(σ > m)=

(1−1/β)₋1

m⁻^1/β. (5.3)

Proof. For (5.2) see, for example, Corollary 1.4.2 and Section 2.7 of Duquesne and Le Gall [11]. It is well known (Theorem 1, Section VII.1 of [3]) that the right inverse processJ =(J (t), t ≥0)ofI defined byJ (t ):=inf{u≥ 0:−I (u) > t}is a stable subordinator with a Lévy measure(β(1−1/β))⁻¹x⁻¹⁻^1/βdx,x >0. SinceN(σ∈dm)is

equal to this Lévy measure, (5.3) follows.

Recall that α=1/β −1. We will, henceforth, primarily work in terms of α rather than β. We will make ex- tensive use of the scaling properties of the height function under the excursion measure. Form >0, letH^[^m^](x)= m⁻^αH (x/m). Note that ifH has lifetimeσ thenH^[^m^]has lifetimemσ and maximum heightm⁻^αH_max. Note also that(H^[^m^])^[^a^]=H^[^ma^], for allm, a >0. The following proposition, which summarizes results from Section 3.3 of Duquesne and Le Gall [11], gives a version of the scaling property for the height process conditioned on its lifetime.

Proposition 5.7. For any test functionf:E→Rand anyx, m >0,we have N

f H^[^m^]

|σ=x/m

=N

f (H )|σ=x .