Asymptotics in Knuth's parking problem for caravans

(1)

HAL Id: hal-00004229

https://hal.archives-ouvertes.fr/hal-00004229

Submitted on 10 Feb 2005

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Asymptotics in Knuth’s parking problem for caravans

Jean Bertoin, Grégory Miermont

To cite this version:

Jean Bertoin, Grégory Miermont. Asymptotics in Knuth’s parking problem for caravans. Random

Structures and Algorithms, Wiley, 2006, 29 (1), pp.38–55. �10.1002/rsa.20092�. �hal-00004229�

(2)

ccsd-00004229, version 1 - 10 Feb 2005

Jean Bertoin

∗

& Grégory Miermont

†

10th February 2005

Abstra t

We onsiderageneralizedversionofKnuth'sparkingproblem,inwhi h aravans

onsisting of a random number of ars arrive at random on the unit ir le. Then

ea h arturns lo kwiseuntilitnds afreespa eto park. Extendinga re ent work

by Chassaing and Lou hard [8 ℄, we relate the asymptoti s for the sizes of blo ks

formed by o upied spots withthe dynami s of the additive oales ent. A ording

to thebehaviorof the aravan's size tail distribution, several qualitatively dierent

versionsof eternaladditive oales ent areinvolved.

Keywords: Parkingproblem,additive oales ent,bridgeswithex hangeablein rements.

M.S.C. ode: 60F17,60J25.

∗

LPMA ,UniversitéParis6,175rueduChevaleret,F-75013Paris

†

(3)

1 Introdu tion

The original parking problem of Knuth an be stated as follows. Consider a parking lot

with

n

spa es, identied with the y li group

Z

/nZ

. Initiallythe parking lot is empty,

and

m ≤ n

arsinaqueuearriveonebyone. Car

i

triestoparkonauniformlydistributed

spa e

L

i

among the

n

possible, independently of other ars, but if the spa e is already o upied, then it tries pla es labeled

L

i

+ 1, L

i

+ 2, . . .

until it nally nds a free spot to park. As ars arrive, blo ks of onse utive o upied spots are forming. It appears

that a phase transition o urs at the stage where the parking lot is almost full, more

pre isely when the number of free spots is of order

√

n

. Indeed, while the largest blo k of o upied spots is of order

log m

with high probability as long as

√

m = o(n − m)

, a

blo kof size approximately

n

ispresent(whilethe othersare of orderatmost

log n

)with high probability when

n − m = o(

√

m)

. In the meanwhile, pre isely when

n − m

is of order

λ

√

_m

with

λ > 0

,a lusteringphenomenon o ursas

λ

de ays. Thebehaviorofthis

lusteringpro esshas been studiedpre iselyby Chassaingand Lou hard [8℄. Itturnsout

thatthepro ess oftherelativesizesofo upiedblo ks isrelatedtotheso- alledstandard

additive oales ent[10, 1℄.

The model originatesfrom a probleminComputer S ien e: spa es inthe parking lot

should be thought of as elementary memory spa es, ea h of whi h an be used to store

elementarydata ( ars). Roughly, our aiminthis work is toinvestigate the more general

situationwhereone wantsto storelarger les, ea h requiringseveral elementarymemory

spa es. In other words, single ars are repla ed by aravans of ars, i.e. several ars may

arrivesimultaneouslyatthesame spot. Inthis dire tion,itwillbe onvenient to onsider

a ontinuous version of the problem, that goesas follows. Let

p

1 , . . . p

m

be a sequen e of positive real numbers with sum

1

, and

s

1 , . . . , s

m

,

m

distin tlo ationson the unit ir le

T

_{:= R/Z}

. Imagine that

m

drops of paint with masses

p

1 , . . . , p

m

, fall su essively at

lo ations

s

1 , . . . , s

m

. Ea htime a drop of paintfalls, we brushit lo kwise in su h away that the resultingpainted portion of

T

is overed by a unit density of paint. Soat ea h step the drop of paintis used to over anew portion of the ir le and the total length of

thepaintedpartof the ir lewhen

i ≤ m

dropshavefallenis

p

1 + · · · + p

i

. Inthissetting, dropsofpaintplay the roleof aravans, andthe paintedportionofthe ir le orresponds

too upied spotsin the parking lot.

More pre isely, we onsider anin reasing sequen e

(A

0 , . . . , A

m

)

of open subsetsof

T

, startingfrom

A

0 = ∅

and endingat

A

m

= T

, whi h an be thought of as the su essive painted portions of the ir le. Given

A

i

and the lo ation

s

i+1

from where the

i + 1

-th dropofpaintwillbebrushed, wepaintasmany spa easpossibletotherightof

s

i+1

with the quantity

p

i+1

of paint, without overing the already painted parts, i.e. the blo ks of

A

i

. Alternatively,webreakthe

i + 1

-th aravanintoseveral pie es,sothattollasmu h as possible the holes left by

T

\ A

i

after

s

i+1

, when readingin lo kwise order. The last ar topark arrives atsome lo ation

t

i+1

, and we let

A

i+1

be the union of

A

i

and the ar between

s

i+1

and

t

i+1

,see Figure1. More formaldenitions will ome inSe t. 2.

In parti ular,

A

i

is a disjoint union of open intervals and

Leb(A

i

) = p

1 + . . . + p

i

. Let

Λ

p

_{(i)(= Λ(p}

1 , . . . , p

m

, s

1 , . . . , s

m

, i))

be the sequen e ofthe Lebesgue measures of the

onne ted omponents of

A

i

, ranked by de reasing order. It will be onvenient to view

Λ

p

_(i)

as aninnite sequen e, by ompletingwith an innitenumber of zero terms.

(4)

A

i

s

i+1

t

i+1

A

i

A

i

A

i

i + 1

-th aravan

p

i+1

Figure1: Arrival,splittingand parking of the

i + 1

-th aravan inthe pro ess

nite expe tation

µ

1 = E[ℓ]

. We say that

ℓ ∈ D

2

whenever

ℓ

has a nite se ond moment

µ

2 = E[ℓ

2 ]

. For

α ∈ (1, 2)

,we say that

ℓ ∈ D

α

whenever

P

_{(ℓ > x) ∼}

x→∞

cx

−α

(1)

for some

0 < c < ∞

. This implies that

ℓ

is in the domain of attra tion of a spe trally

positive stable random variable with index

α

, and we stress that our results an be extended under this more generalhypothesis; (1)is only intended tomake things easier.

We suppose from now onthat

ℓ ∈ D

α

for some

α ∈ (1, 2]

, and take a random iidsample

ℓ

1 , ℓ

2 , . . .

of variables distributed as

ℓ

, and independently of this sequen e, iid uniform

randomvariables on

[0, 1)

,

U

1 , U

2 , . . .

. For

ε > 0

, set

T

ε

= inf{i : ℓ

1 + . . . + ℓ

i

≥ 1/ε},

so by the elementary renewal theorem,

T

ε

∼ 1/(εµ

1 )

. Then introdu e the sequen e

(ℓ

∗

i

, 1 ≤ i ≤ T

ε

)

dened by

ℓ

∗

_i

= ℓ

i

for

1 ≤ i ≤ T

ε

− 1

and

ℓ

∗

T

ε

= ε

−1

− (ℓ

1 + . . . + ℓ

T

ε

−1

),

sothe terms of

ℓ

∗

sum to

1/ε

.

Following Chassaing and Lou hard [8℄, we are interested in the formation of

ma ro-s opi painted omponents inthe limitwhen

ε

tends to

0

, attimes lose to

T

ε

, i.e. when the ir le isalmost entirelypainted. Spe i ally,we let

X

(ε)

(t) = Λ

p

(T

ε

− ⌊tε

−1/α

⌋) ,

t ≥ 0 ,

for

Λ

p

dened asabovewiththe data

m = T

ε

, p

i

= εℓ

∗

i

, s

i

= U

i

. Observethat

T

ε

−[tε

−1/α

_]

de reaseswhen

t

in reases, andtherefore,inordertoinvestigatethe formationofpainted omponents, we should onsider the pro ess

X

(ε)

_{(t), t ≥ 0}

ba kwards in time. This is

whatwe shall do inTheorem 1, using the exponentialtime hange

t → e

−t

(5)

Before des ribing our main result, let us rst re all some features of the additive

oales ent. The additive oales ent

C

is a Markov pro ess with values in the innite ordered simplex

S =

(

s

= (s

1 , s

2 , . . .) : s

1 ≥ s

2 ≥ . . . ≥ 0,

∞

X

i=1

s

i

≤ 1

)

endowed withthe uniformdistan e, whose evolution isdes ribed formallyby: given that

the urrent state is

s

, two terms

s

i

and

s

j

,

i < j

, of

s

are hosen and mergeintoa single term

s

i

+ s

j

(whi h implies some reordering of the resultingsequen e) at a rate equalto

s

i

+ s

j

. A version

(C(t), t ∈ R)

ofthis pro ess dened for times des ribingthe whole real axisis alled eternal. We refer to[1, 3℄for ba kground.

Asshownin[4℄,eternaladditive oales ents anbeen odedby ertainbridges.

Spe if-i ally, let

B = (B(x), 0 ≤ x ≤ 1))

be a àdlàg real-valued pro ess with ex hangeable in rements, su h that

B(0) = B(1) = 0

. Suppose further that

B

has innite variation and no negative jumps a.s. Then

B

attains its overall inmum at a unique lo ation

V

(whi hisuniformlydistributedon

[0, 1]

),and

B

is ontinuousat

V

. Considertheso- alled Vervaat transform whi hmaps the bridge

B

into anex ursion

E

dened by

E(x) = B(V + x) − B(V )

for

0 ≤ x ≤ 1,

wherethe addition

V + x

is modulo

1

. Finally,welet for

t ≥ 0

E

(t)

x

= E(x) − tx

for

0 ≤ x ≤ 1,

and introdu e

F(t)

as the random element of

S

dened by the ranked sequen e of the lengthsofthe onstan yintervalsofthepro ess

E

(t)

_{= (inf}

0≤y≤x

E

(t)

(y), 0 ≤ x ≤ 1)

. Here, a onstan y interval means a onne ted omponent of the omplement of the support of

the Stieltjes measure

d(−E)

. Finally,if we dene

C(t) = F(e

−t

₎

,then

C

= (C(t), −∞ <

t < ∞)

is aneternal additive oales ent(see Se tion6.1for omments and details).

In this work, eternaladditive oales ent asso iated to ertain remarkable bridgeswill

play a key role. More pre isely, we write

C

(2)

_{= (C}

(2)

_{(t), −∞ < t < ∞)}

for the eternal

additive oales ent

C

onstru ted above when

B = B

(2)

is a standard Brownian bridge;

sothat

C

(2)

is the so- alledstandard additive oales ent( f. [4,1℄). Next, for

1 < α < 2

, we denote by

σ

(α)

_{= (σ}

(α)

_{(t), t ≥ 0)}

a standard spe trally positive stable Lévy pro ess

withindex

α

,that is

σ

(α)

has independentand stationaryin rements,nonegative jumps,

and

E

_(exp(−λσ

(α)

_{(t))) = exp(tλ}

α

_{) ,}

for all

λ ≥ 0.

We allstandard stableloop 1

of index

α

the pro ess

B

(α)

dened by

B

(α)

_{(x) = σ}

(α)

_{(x) − xσ}

(α)

_(1),

for

0 ≤ x ≤ 1.

(2) We nally write

C

(α)

_{= (C}

(α)

_{(t), −∞ < t < ∞)}

for the eternal additive oales ent

C

onstru tedabove when the bridge

B

isthe standard stable loopof index

α

.

We are now able to state our main result.

1

We all

B

(α)

a loop and nota bridge to avoid apossible onfusion: even though

B

(α)

starts from

0

, endsat

0

andhasex hangeablein rements,itdoesnothavethesamelawasthestablepro ess

σ

(α)

onditionedon

σ

(α)

_{(1) = 0}

!

(6)

Theorem 1 The pro ess

(X

(ε)

_{(t), 0 ≤ t < T}

ε

)

onverges as

ε ↓ 0

in the sense of weak onvergen e of nite-dimensional distributions to some pro ess

X

= (X(t), 0 ≤ t < ∞)

. The exponential time- hanged pro ess

(X(e

−t

_{), −∞ < t < ∞)}

is an eternal additive

oales ent; more pre isely:

(i)When

α = 2

,

(X(e

−t

_{), −∞ < t < ∞)}

is distributed as

(C

(2)

(t +

1

2 log(µ

2 /µ

1 ) − log µ

1 ), −∞ < t < ∞).

(ii)When

1 < α < 2

,

(X(e

−t

_{), −∞ < t < ∞)}

is distributed as

(C

(α)

(t +

1 α

log

Γ(2 − α)c

(α − 1)µ

1 − log µ

1 ), −∞ < t < ∞).

It might be interesting to dis uss further the role of the parameter

α

and the inter-pretationintermsof phasetransition. Asitwasalready mentioned, therenewaltheorem

entailsthanthenumberofdropsofpaintneededforthe omplete overingis

T

ε

∼ 1/(εµ

1 )

, aquantity whi hisnotsensitiveto

α

. Itiseasytoshowthatforevery

a < 1

,thereareno ma ros opi painted omponents whenonly

[aT

ε

]

dropsof paint have fallen,sothe phase transition(i.e. the number of drops whi h is needed for the appearan e of ma ros opi

omponents) o urs for numbers lose to

T

ε

. More pre isely, the regime for the phase transition is of order

T

ε

− ε

−1/α

; so the phase transition o urs loser to

T

ε

when

α

is larger. Wewould like alsotostressthat one-dimensional distributionsof the limiting

ad-ditive oales ent pro ess

X

depend on

α

, but not itssemigroup whi h isthe samefor all

α ∈ (1, 2]

. Aheuristi explanationmightbethefollowing: thenumberofdrops neededto

ompletethe overing on ethe phasetransitionhaso urredis toosmall(oforder

ε

−1/α

)

to observe signi ant dieren es in the dynami s of aggregation of ma ros opi painted

omponents.

Remark. Our model bears some similarity with another parking problem onthe ir le,

where drops of paints fall uniformly on the ir le and then are brushed lo kwise, but

where overlaps are now allowed (some points may be overed this way several times),

allitthe random overing of an interval problem. However, asshowed in [6℄, this last

modelhas very dierent asymptoti s fromthose of the parking problem, as it turns out

thattherandom overingofanintervalisrelatedtoKingman's oales entratherthanthe

additive oales ent. A sharedfeature is thatthe phase transitionof the random overing

problemappears also when the ir le is almost ompletely overed, but for example the

dierent fragments are ultimately nitein number ratherthan innite.

We alsomention yetanother parking problem,rst onsidered by Rényi (see [15, 9℄).

In an beformulated as follows: aravans with size

ε

are pla ed on

T

(the original work rather onsiders

(0, 1)

)one afteranother, but the lo ations

s

i

where ars parkare hosen uniformlyamongspa esthatdonot indu eoverlapsandsplittingof aravans, i.e.sothat

the lengthof the ar from

s

i

to

t

i

isexa tly

ε

. This isdone untilnoun overed sub-ar of

T

with size

≥ ε

remains. This pro ess does not involve oales ing blo ks of ars, and one isratherinterested inthe propertiesof the randomnumberof ars that are able topark.

Themethodin[8℄reliesonanen oding parkingfun tionwhi hisshown tobe

asymp-toti ally related to a fun tion of standard Brownian bridge, and a representation of the

(7)

spirit to that of [8℄, and uses the representation of eternal additive oales ent that we

presented above; we briey sket h it here. First, we en ode the pro ess

X

(ε)

by a bridge

withex hangeablein rementsinSe t.2. InSe t.3,weshowthat thisbridge onvergesto

some bridge with ex hangeable in rements that an be represented in terms of the

stan-dard Brownian bridge (for

α = 2

) or the standard stable loop(for

1 < α < 2

). Theorem 1then follows readily.

Therestofthisworkisorganizedasfollows. InSe tion2weprovidearepresentationof

thepainted omponentsintermsofabridgeanditsVervaat'stransform. The onvergen e

ofthese bridgeswhen

ε

tends to

0

isestablished inSe tion3,and that ofthe sequen e of thesizesofthepainted omponentsinSe tion4. Se tion5isdevotedtoabriefdis ussion

of the analogous dis rete setting (i.e. Knuth's parking for aravans), and nally some

omplements are presented in Se tion6.

2 Bridge representation

We develop a representation of the parking pro ess with the help of bridges with

ex- hangeable in rements, whi his ru ial toour study.

Let us rst give the proper denition the of sequen e

(∅ = A

0 , . . . , A

m

)

of the In-trodu tion. We identify the ir le

T

with

[0, 1)

and write

p

T

: R → T

for the anoni al proje tion. If

A

isameasurablesubsetof

T

(identiedwith

[0, 1)

),let

F

A

beitsrepartition fun tiondenedby

F

A

(x) = Leb([0, x]∩A)

for

0 ≤ x < 1

,where

Leb

isLebesguemeasure. Also, extend

F

A

on the whole real line with the formula

F

A

(x + 1) = F

A

(x) + F

A

(1−)

. Given

A

i

forsome

0 ≤ i ≤ m − 1

, let

t

i+1

= inf{x ≥ s

i+1

: F

A

i

(x) + p

i+1

− (x − s

i+1

) ≤ F

A

i

(s

i+1

)}.

Noti e that the ar

p

T

((s

i+1

, t

i+1

))

oriented lo kwise from

s

i+1

to

p

T

(t

i+1

)

has length

t

i+1

− s

i+1

≥ p

i+1

. Then let

A

i+1

be the interior of the losure of

p

T

((s

i+1

, t

i+1

)) ∪ A

i

. Thepointintaking the losureand thenthe interioristhat we onsider thattwopainted

onne ted omponents of

T

that are at distan e

0

onstitute in fa t a single painted onne ted omponent.

Dene

h

p

i+1

(x) = F

A

i

(x) − F

A

i

(s

i+1

) + p

i+1

− (x − s

i+1

)

s

i+1

≤ x ≤ t

i+1

,

and

h

p

i+1

(x) = 0

in

[t

i+1

, s

i+1

+ 1)

, so

h

p

i

is a àdlàgfun tion (right- ontinuous with left-limits) on

[s

i+1

, s

i+1

+ 1)

. Consider it as a fun tion on

T

by letting

h

p

i+1

(x) = h

p

i+1

(y)

where

y

is the element of

[s

i+1

, s

i+1

+ 1) ∩ p

−1

T

(x)

. The quantity

h

p

i+1

(x)

an be thought of asthe quantity of ars of the

i + 1

-th aravanthat try topark at

x

. See Figure2.

We onsider the prole

H

p

i

=

i

X

j=1

h

p

j

(3)

of the parking at step

0 ≤ i ≤ m

, so

H

p

i

(x)

is the total quantity of ars that have tried (su essfully or not) to park at

x

(with the onvention that

H

p

i

(1) = H

p

i

(0)

) before the

(8)

H

p

i

A

i

A

i

A

i

A

i

p

i+1

s

i+1

t

i+1

h

p

i+1

Figure2: The fun tion

h

p

i+1

(thi kline) orrespondingtothe

i + 1

-th aravanof Figure1. Theblo ksof

A

i

are represented undertheaxis,and thedashedlinesrepresenttheprole

H

p

i

(itgivesmoreinformationthan

A

i

alone). Thebra ketunderthegureindi ateshow

A

i+1

is obtained by formation of a new blo k omprising the blo ks of

A

i

between

s

i+1

and

t

i+1

Lemma 1 For

1 ≤ i ≤ m

,

(i)the set

A

i

is the interior of the support of

H

p

i

. (ii)

H

p

i

(t

i

−) = 0

. (iii)

H

p

i

jumps at times

s

1 , . . . , s

i

with respe tive jump magnitudes

p

1 , . . . , p

i

, and has a drift with slope

−1

on its support.That is, if

[v, v

′

_{] ⊆ supp (H}

p

i

)

,

H

p

i

(x + v) = H

p

i

(v−) − x +

i

X

j=1

p

j

1

{v≤s

j

≤v+x}

0 ≤ x ≤ v

′

− v.

Proof. Properties (i) and (iii) are easily shown using a re ursion on

i

and splitting the behavior of

h

p

i

on

A

i−1

and

A

i

\ A

i−1

. We give some details for (ii). For

i ≥ 1

, noti e that by denition

t

i

annot be a point of in rease of

F

A

i−1

, i.e. a point su h that

F

A

i−1

(t

i

− ε) < F

A

i−1

(t

i

) < F

A

i−1

(t

i

+ ε)

for every

ε > 0

. Therefore,

t

i

∈ A

/

i−1

and

h

p

j

(t

i

) = h

p

j

(t

i

−) = 0

for

j < i

. Sin e it follows by ontinuity of

F

A

i−1

that

h

p

i

(t

i

−) = 0

,

(ii)isproved.

Consider the bridge fun tion:

b

p

i

(x) = −x +

i

X

j=1

p

j

1

{x≥s

j

}

0 ≤ x < 1,

whi h starts from

b

p

i

(0) = 0

and ends at

b

p

i

(1−) = p

1 + . . . + p

i

− 1

. We extend

b

p

i

to a fun tion on

R

by setting

b

p

i

(x + 1) = b

p

i

(x) + b

p

i

(1−)

. For any

v ∈ [0, 1)

, it is easily seen

using (iii)inLemma 1that

H

p

i

(x + v) = H

p

i

(v−) + b

p

i

(x + v) −

0 ∧

inf

u∈[v,v+x]

(H

p

i

(v−) + b

p

i

(u))

.

Suppose

v

issu hthat

H

p

m

(v−) = 0

(here

H

p

m

(0−) = H

m

p

(1−)

), allsu hanumberalast

(9)

t

m

. On the other hand, by (i) in the same lemma, the support of

H

p

m

is the losure of

A

m

whi h has measure

1

,hen e itis

T

. By(iii),we on lude by letting

v = 0, v

′

_{↑ 1}

that

H

p

m

(x) = H

m

p

(0) + b

p

m

(x)

for

0 ≤ x < 1

, so for

x = t

m

−

,

H

p

m

(0) = −b

p

m

(t

m

−) = − inf b

p

m

ne essarilysin e

H

p

m

isnon-negative. This impliesthatthe lastempty spotsare those

v

's su h that

b

p

m

(v−) = inf b

p

m

. We hoose one of themby letting

V = inf{x ∈ [0, 1] : b

p

m

(x−) = inf

u∈[0,1]

b

p

m

(u)},

the rst lo ation when the inmum of

b

p

m

is rea hed. We have proved

Lemma 2 For any

0 ≤ x < 1, 1 ≤ i ≤ m

,

H

p

i

(x + V ) = b

p

i

(x + V ) −

inf

u∈[V,V +x]

b

p

i

(u).

Re allthatweareinterestedin

Λ

p

_(i)

,therankedsequen eofthelengthsoftheinterval

omponentsof

A

i

,where

A

i

anbeviewedasthepaintedportionofthe ir leafter

i

drops ofpainthavefallen,orthesetofo upiedspotsafterthe

i

-th aravanhasarrived. Lemma 1(i) enablesus toidentify

A

i

asthe interiorof supportof the fun tion

H

p

i

, and sin e the Lebesgue measure of the interval omponents of the interior of the supportof

H

p

i

is not ae tedby a y li shift,we re ord the followingsimple identi ation

Lemma 3 Forevery

i = 1, . . . , m

,

Λ

p

_(i)

oin ideswiththerankedlengthsof theintervals

of onstan y of the fun tion

x 7−→

inf

u∈[V,V +x]

b

p

i

(u) ,

x ∈ [0, 1].

3 Convergen e of bridges

We now onsider a res aled randomized version of the bridges introdu ed above. Let

B

(ε)

_{= ε}

−1+1/α

_b

p

m

, where

b

p

m

is obtained asabove withdata

m = T

ε

, p

i

= εℓ

∗

i

, s

i

= U

i

, and thesequantities are introdu ed inthe Introdu tion. Sofor

0 ≤ x ≤ 1

B

(ε)

_{(x) = −ε}

−1+1/α

x +

T

ε

X

i=1

ε

1/α

ℓ

∗

_i

1

{x≥U

i

}

= ε

1/α

T

ε

X

i=1

ℓ

∗

_i

(

1

{x≥U

i

}

− x),

be ause

ℓ

∗

1 + . . . + ℓ

∗

T

ε

= 1/ε

. Re all that

B

(2)

denotes the standardBrownianbridge, and

B

(α)

the standard stable loopwith index

α

asdened in(2).

Lemma 4 As

ε ↓ 0

, the bridge

B

(ε)

onverges weakly on the spa e

D

of àdlàgpaths en-dowedwithSkorokhod'stopology,toabridgewithex hangeablein rements

B = (B(x), 0 ≤

x ≤ 1)

. More pre isely:

(i) If

α = 2

then

B

isdistributed as

p

µ

2 /µ

1 B

(2)

.

(ii) If

α ∈ (1, 2)

, then

B

is distributed as

Γ(2 − α)c

(α − 1)µ

1

1 α

B

(α)

.

(10)

The proof of Lemma4(ii) willuse the followingwell-known representation:

Γ(2 − α)c

(α − 1)µ

1

1 α

B

(α)

_{(x) =}

∞

X

i=1

∆

i

1

{x≥U

i

}

− x

,

_{0 ≤ x ≤ 1,}

where

(U

i

, i ≥ 1)

is a sequen e of i.i.d. uniform

(0, 1)

r.v.'s,

(∆

i

, i ≥ 1)

is the ranked sequen e of the atoms of aPoisson measure on

(0, ∞)

with intensity

αcµ

−1

1 x

−1−α

dx

, and these two sequen es are independent. More pre isely, the series in the right-hand side

doesnot onverge absolutely, but is taken inthe sense

∞

X

i=1

∆

i

1

{x≥U

i

}

− x

= lim

n→∞

n

X

i=1

∆

i

1

{x≥U

i

}

− x

,

where the limit is uniform in the variable

x

, a.s. This representation follows immedi-ately from the elebrated Lévy-It de omposition, spe ied for the stable pro ess

σ

(α)

,

asthe pro ess of the jumps of the latter is a Poisson point pro ess on

R

+

with intensity

α(α−1)

Γ(2−α)

x

−1−α

_dx

. See alsoKallenberg [11℄.

Proof. Following Kallenberg [11℄, we represent the jump sizes of the bridge

B

(ε)

by the

randompoint measure

ψ

ε

=

T

ε

X

i=1

(ε

1/α

ℓ

∗

_i

)

2 δ

_ε

1 _/α

ℓ

∗

i

.

ByTheorem 2.3 in[11℄, we have to show:

if

α = 2

,then

ψ

ε

→ (µ

2 /µ

1 )δ

0 ,

(4) and if

α < 2

, then

ψ

ε

→ ψ :=

∞

X

i=1

∆

2 i

δ

∆

i

,

(5)

wherethe onvergen eisinlawwithrespe t totheweaktopologyonmeasures on

[0, ∞)

,

andin(5),

(∆

i

, i ≥ 1)

isthe rankedsequen e ofthe atomsofaPoisson measureon

(0, ∞)

with intensity

αcµ

−1

1 x

−1−α

dx

.

Case (i)is easier totreat. Indeed, noti e that the total mass of

ψ

ε

is

ψ

ε

(R

+

) = ε

T

ε

X

i=1

(ℓ

∗

i

)

2 = εT

ε

(ℓ

∗

T

ε

)

2 ₊

P

T

ε

−1

i=1

ℓ

2 i

T

ε

.

Sin e

ℓ

∗

i

≤ ℓ

i

, the law of large numbers gives

ψ

ε

(R

+

) → µ

2 /µ

1

. Now let

m

∗

_ε

:=

√

ε max

1≤i≤T

ε

ℓ

∗

_i

and

M

n

:=

√

ε max

1≤i≤n

ℓ

i

,

soto prove (4), it su es to showthat

m

∗

ε

→ 0

in probability. Noti e that

m

∗

ε

≤ M

T

ε

. Let

η > 0

and

K > µ

−1

1

. Then

P

_(m

∗

ε

> η) = P(m

∗

ε

> η, T

ε

≤ Kε

−1

) + P(m

∗

ε

> η, T

ε

> Kε

−1

)

≤ P(M

⌊Kε

−1

_⌋

> η) + P(T

_ε

> Kε

−1

).

(11)

The se ondterm onverges to

0

sin e

εµ

1 T

ε

→ 1

a.s. Forthe rst term, noti e that

P

_(M

_⌊Kε

−1

_⌋

≤ η) = (1 − P(ℓ > η/

√

ε))

⌊Kε

−1

_⌋

.

Taking logarithms and he king that

ε

−1

_P

_(ℓ

2 _{> η}

2 _{/ε) → 0}

as

ε ↓ 0

(whi h holds sin e

E

_[ℓ

2 _{] < ∞}

), we nallyobtain that

P

(M

⌊Kε

−1

⌋

≤ η) → 1

. This ompletes the proof of(4).

Now we turn our attention to (5). It su es to show that for every fun tion

f :

[0, ∞) → [0, ∞)

, say of lass

C

1

with bounded derivative

lim

ε→0

E

_(exp(−hψ

_ε

_{, f i)) = E (exp(−hψ, fi)) ;}

(6)

see for instan e Se tion II.3 in Le Gall [12℄. In this dire tion, re all from the lassi al

formulafor Poisson random measures that

E

_{(exp(−hψ, fi)) =}

αc

µ

1 Z

∞

0 (1 − exp(−y

2 _{f (y)))y}

−1−α

_dy.

To start with, we observe from the renewal theorem that

ε

1/α

_ℓ

∗

T

ε

onverges to

0

in probabilityas

ε → 0

,so in(6), wemay repla e

ψ

ε

by

ψ

′

_ε

=

T

_X

ε

−1

i=1

(ε

1/α

ℓ

i

)

2 δ

ε

1 _/α

ℓ

i

.

Next, forevery

a ≥ 0

,we onsider the randommeasure

ψ

ε,a

=

a/ε

X

i=1

(ε

1/α

_ℓ

i

)

2 δ

ε

1 /α

_ℓ

_i

.

Again,bythe(elementary)renewaltheorem,

εT

ε

→ µ

−1

1

inprobability,soforevery

η > 0

, the event

hψ

ε,µ

−1

1 −η

, f i ≤ hψ

′

ε

, f i ≤ hψ

ε,µ

−1

1 +η

, f i

(7)

has aprobability whi htends to

1

as

ε → 0

. Now

E

_(exp(−hψ

_ε,a

_{, f i)) = E exp(−f(ε}

1/α

_ℓ)(ε

1/α

_ℓ)

2 ₎

a/ε

_.

Takinglogarithms, we have to estimate

a

ε

E

1 − exp(−f(ε

1/α

_ℓ)(ε

1/α

_ℓ)

2 ₎

=

a

ε

Z

∞

0 ε

2/α

(2xf (ε

1/α

x) + ε

1/α

x

2 f

′

(ε

1/α

_{x)) exp(−(ε}

1/α

x)

2 f (ε

1/α

x))P(ℓ > x)dx

=

a

ε

Z

∞

0 (2yf (y) + y

2 f

′

_{(y)) exp(−y}

2 f (y))P(ℓ > y/ε

1/α

)dx.

By(1)anddominated onvergen e, weseethatthepre edingquantity onverges as

ε → 0

towards

ac

Z

∞

0 (2yf (y) + y

2 f

′

_{(y)) exp(−y}

2 f (y))y

−α

dx = αac

Z

∞

0 (1 − exp(−y

2 _{f (y)))y}

−1−α

_dx.

Taking

a = µ

−1

1 ± η

, using (7) and letting

η

tend to

0

, we see that (6) holds, whi h

(12)

4 CONVERGENCE OF

X

(ε)

11

4 Convergen e of

X

(ε)

Inthisse tion,wededu e Theorem1fromLemmas3,4. Re allthedenitionofthebridge

b

p

i

in Se tion2. For

i ≤ T

ε

, let

B

(ε)

i

be the bridge

ε

−1+1/α

_b

p

i

with data

p

j

= εℓ

∗

j

, s

j

= U

j

, so

B

(ε)

T

ε

= B

(ε)

. Letalso

V

ε

be the left-most lo ation of the inmum of

B

(ε)

,and

V

B

(ε)

(x) = B

(ε)

(x + V

ε

) − inf B

(ε)

,

0 ≤ x ≤ 1

the Vervaat transform of

B

(ε)

. By Lemma 3,

X

(ε)

_{(t) = Λ}

p

_(T

ε

− ⌊tε

−1/α

⌋)

oin ides with

the ranked sequen e of lengths of onstan y intervals of the inmum pro ess of

B

(ε)

_T

ε

−⌊tε

−1/α

⌋

(x + V

ε

) − inf B

(ε)

_,

0 ≤ x ≤ 1,

wherethe onstant

− inf B

(ε)

has no ee t and is addedfor future onsiderations.

Lemma 5 For every

t ≥ 0

, the dieren e

B

(ε)

_{(x) − B}

_T

(ε)

ε

−⌊tε

−1/α

⌋

(x) = ε

1/α

⌊tε

−1/α

_−1⌋

X

j=0

ℓ

∗

_T

_ε

_−i

1

{x≥U

Tε−i

}

0 ≤ x ≤ 1

onverges in probability for the uniform norm to the pure drift

x 7→ tµ

1 x

as

ε ↓ 0

. Proof. Re all from the renewal theorem that

ε

1/α

_ℓ

T

ε

→ 0

in probability as

ε ↓ 0

. Therefore, we might start the sum appearing in the statement from

j = 1

. Now, the sequen es

(ℓ

1 , . . . , ℓ

T

ε

−1

)

and

(ℓ

T

ε

−1

, . . . , ℓ

1 )

have the same distribution. Up todoing the

substitution, Lemma 5 for xed

s

is therefore a simple appli ation of the strong law of largenumbers. The on lusion is obtained by standard monotoni ity arguments.

As a onsequen e of Lemmas 4, 5, and the fa t that

s 7→ tµ

1 s

is ontinuous, the pro ess

B

_T

(ε)

ε

−⌊tε

−1/α

⌋

(x + V

ε

) − inf B

(ε)

_{= VB}

(ε)

_{(x) −}

_B

(ε)

_{(x + V}

ε

) − B

_T

(ε)

_ε

_−⌊tε

−1/α

_⌋

(x + V

ε

)

onverges in the Skorokhod spa e to

E

(tµ

1 )

= (E(x) − tµ

1 x, 0 ≤ x ≤ 1),

where

E(x) = B(x + V ) − inf B,

0 ≤ x ≤ 1

istheVervaat transformofthe limitingbridge

B

whi happearsinLemma4,

V

being the a.s. unique lo ation of its inmum. Now letting

E

(t)

be the inmum pro ess of

E

(t)

and

F(t)

be the de reasing sequen e of lengthsof onstan y intervalsof

E

(t)

,we have

Proposition 1 The pro ess

(X

(ε)

_{(t), t ≥ 0)}

onverges to

(F(µ

1 t), t ≥ 0)

in the sense of weak onvergen e of nite-dimensional marginals.

(13)

Proof. Thete hni alpointisthatSkorokhod onvergen eof

B

(ε)

T

ε

−⌊tε

−1/α

⌋

(x+V

ε

)−inf B

(ε)

to

E

(tµ

1 )

, though it does imply onvergen e of respe tive inmum pro esses, does not

a priori imply that of the ranked sequen e of lengths of onstan y intervals of these

pro esses. However, this onvergen e doeshold be ause forevery

t ≥ 0

,if

(a, b)

is su h a onstan y interval,then

E

(tµ

1 )

(x) > E

(tµ

1 )

(a)

for

x ∈ (a, b)

, a.s. See e.g. Lemmas4 and 7

in[5℄.

This proposition proves Theorem 1. Indeed, re all from Lemma 4 that

B = c

α

B

α

, where

c

2 =

p

µ

2 /µ

1

and for

1 < α < 2

c

α

=

Γ(2 − α)c

(α − 1)µ

1

1 α

.

Then plainly,

F(e

−t

_{) = C}

(α)

_{(t + log c}

α

)

, and hen e the limiting pro ess

X(e

−t

₎

is

dis-tributed as

F(µ

1 e

−t

_{) = C}

(α)

_{(t + log c}

α

− log µ

1 )

.

5 Related results for a dis rete problem

In situations involving parking problems, it may be more natural to onsider dis rete

parking lots, i.e.

Z

/nZ

instead of the unit ir le, and aravans with integer sizes, e.g. as in Knuth's original parking problem. Ea h aravan hooses a random spot, uniform

on

Z

/nZ

, and tries to park at that spot. Studying the frequen ies of blo ks of ars ts with our general framework by taking

ℓ

with integer values,

ε = 1/n

and

s

i

= ⌊nU

i

⌋/n

. Renameby

T

n

the former quantity

T

ε

(the numberof aravans). Let

e

B

(n)

(x) = n

1/α

T

n

X

i=1

ℓ

∗

i

n

1

{x≥⌊nU

i

⌋/n}

− x

0 ≤ x ≤ 1

B

(n)

(x) = n

1/α

T

n

X

i=1

ℓ

∗

i

n

1

{x≥U

i

}

− x

0 ≤ x ≤ 1,

so

B

(n)

would be

B

(1/n)

in the notationabove. The analogof Lemma 5is stilltrue when

repla ing

B

(n)

by

B

e

(n)

,withoutessential hangeintheproof. Thustoobtaintheverysame

on lusions asin the pre eding se tions, it su es to he k a result similar to Lemma4.

Namely,wemust prove that

B

e

(n)

_{→ B}

inthe Skorokhod spa eas

n → ∞

. Nowitiseasy to he k thata.s.,

| e

B

(n)

_{(x) − B}

(n)

_{(⌈nx−⌉/n)| ≤ n}

1/α

_/n

forevery

n ≥ 1, x ∈ [0, 1]

,be ause no

U

i

is rational a.s. Therefore, it su es to he k that

B

(n)

_{(⌈n · +⌉/n)}

onverges to

B

indistributionforthe Skorokhodtopologyon

D

. Uptousing Skorokhod's representation theorem, this is doneby taking

f

n

= B

(n)

and

κ

n

(x) = ⌈nx+⌉/n

in the next lemma.

Lemma 6 Let

(f

n

, n ≥ 1)

be a sequen e of fun tions onverging in

D

to

f

. For

n ∈ N

let also

κ

n

be a right- ontinuous non-de reasingfun tion (not ne essarily bije tive) from

[0, 1]

to

[0, 1]

, su h thatthe sequen e

(κ

n

)

onverges to the identity fun tionuniformly on

[0, 1]

. Then

f

n

◦ κ

n

→ f

in

D

.

Proof. First onsider the ase

f

n

= f

for every

n

. Fix

ε > 0

. Let

κ

−1

n

be the right- ontinuous inverse of

κ

n

dened by

(14)

It is easy to prove that

κ

n

(κ

−1

n

(x)−) ≤ x ≤ κ

n

(κ

−1

n

(x))

for every

x

. Sin e

f

is àdlàg,

one may nd

0 = x

0 < x

1 < . . . < x

k

= 1

su h that the os illation

ω(f, [x

i

, x

i+1

)) < ε

for

0 ≤ i ≤ k − 1

,where

ω(f, A) = sup

x,y∈A

|f(x) − f(y)|.

Sin e

κ

n

approa hestheidentity,for

n

largewemayassume

κ

n

(κ

−1

n

(x

i

)) < κ

n

(κ

−1

n

(x

i+1

)−)

for

0 ≤ i ≤ k − 1

. Dene atime- hange

λ

n

(i.e. anin reasingbije tion between

[0, 1]

and

[0, 1]

)byinterpolatinglinearlybetweenthe points

(0, 0), (κ

−1

n

(x

i

), x

i

), 1 ≤ i ≤ k −1, (1, 1)

.

Now let

x ∈ [0, 1]

. Suppose

κ

−1

n

(x

i

) ≤ x < κ

−1

n

(x

i+1

)

for some

0 ≤ i ≤ k − 1

, and

noti ethat

x

i

≤ κ

n

(κ

−1

n

(x

i

)) ≤ κ

n

(x) < κ

n

(κ

−1

n

(x

i+1

)−) ≤ x

i+1

. Therefore,

κ

n

(x)

belongs

to

[x

i

, x

i+1

)

aswell as

λ

n

(x)

by denition of

λ

n

, and

|f(κ

n

(x)) − f(λ

n

(x))| ≤ ω(f, [x

i

, x

i+1

)) ≤ ε.

Else,onemust have

x < κ

−1

n

(0)

or

x ≥ κ

−1

n

(1)

,andtheresultissimilar. Finally,doingthe samereasoningfor

ε = ε

n

onvergingto

0

slowlyenoughgivesthe existen eofsome time- hanges

λ

n

onvergingtotheidentityuniformlysu hthat

sup

x∈[0,1]

|f(κ

n

(x))−f(λ

n

(x))| ≤

2ε

n

, hen e giving onvergen e of

f ◦ κ

n

to

f

in the Skorokhod spa e.

In thegeneral ase,for every

n ≥ 0

let

λ

n

beatime- hangesu hthat

λ

n

onverges to the identity as

n → ∞

and

f

n

◦ λ

n

onverges to

f

uniformly. Take

κ

′

n

= λ

−1

n

◦ κ

n

. Then

f

n

◦ κ

n

− f ◦ κ

′

n

→ 0

uniformly,soitsu es toshow that

f ◦ κ

′

n

→ f

in

D

,whi his done

by the former dis ussion.

In parti ular, were overand extend a ertain number of resultsfrom [8℄.

6 Complements

Inthis se tion, wewould liketo provide some informationon the eternaladditive

oales- ents

C

(α)

for

1 < α < 2

, whi h appear in Theorem 1.

6.1 Mixture of extremes

To start with, we should like to spe ify the representation of

C

(α)

as a mixture of

so- alledextremeeternal additive oales ents ([3℄,[5℄). In this dire tion,letus rst onsider

asequen e

θ

= (θ

0 , θ

1 , θ

2 , . . .)

of non-negativenumbers satisfying

P

i≥0

θ

2 i

= 1

and either

θ

0 > 0

or

X

i≥0

θ

i

= ∞.

(8)

FollowingKallenberg [11℄ we asso iate to

θ

abridge with ex hangeable in rements

B

θ

(x) = θ

0 β(x) +

X

i≥1

θ

i

(

1

{x≥U

i

}

− x)

0 ≤ x ≤ 1

(9)

where

(U

i

, i ≥ 1)

denotes a sequen e of iid uniform variables and

β

is an independent standard Brownian bridge. We write

C

θ

for the eternal additive oales ent asso iatedto

the bridge

B = B

θ

as explained inthe Introdu tionand allsu h

C

θ

extreme.

A ording to [3, Theorem 15℄, every eternalversion of the additive oales ent

C

an beobtainedasamixingofshifted versionsof extremeeternaladditive oales ents

C

θ

(15)

C

an be expressed in the form

(C

θ

∗

(t − t

∗

_{), t ∈ R)}

with

θ

∗

_{, t}

∗

random. Equivalently,

C

anbeviewed asthe eternaladditive oales ent onstru tedinthe Introdu tionfrom the

bridgewith ex hangeablein rements

B = e

t

∗

B

θ

∗

. Asobserved by Aldousand Pitman[3℄,

the mixingvariables

θ

∗

, t

∗

an be re overed from the initialbehaviorof

C

:

e

t

∗

θ

∗

_i

= lim

t→−∞

e

−t

_C

i

(t)

and

e

2t

∗

= lim

t→−∞

e

−2t

∞

X

i=1

C

2 _i

(t) .

In the ase of the standard stable loop

B

(α)

with

1 < α < 2

, re all fromthe Lévy-It

de omposition that

θ

∗

0 = 0

and

(e

t

∗

θ

∗

1 , e

t

∗

θ

∗

2 , . . .) = (∆

1 , ∆

2 , . . .)

is the ranked sequen e

of the atoms of a Poisson random measure on

(0, ∞)

with intensity

α(α−1)

Γ(2−α)

x

−1−α

_dx

. In parti ular,

e

2t

∗

=

∞

X

i=1

∆

2 i

(10)

has the lawof a(positive)stable variable with index

α/2

and

θ

_i

∗

= ∆

i

/e

t

∗

,

i = 1, 2, . . .

(11)

is su h that the sequen e of squares

((θ

∗

1 )

2 , (θ

∗

2 )

2 , . . .)

is distributed a ording to the

Poisson-Diri hlet law

PD(α/2, 0)

;see Pitmanand Yor [14℄. We also stress that every oales ent

C

θ

an be obtained as a limit of appropriate

aravan parking problems, whi h are quite natural given the results of [3, 5℄. Pre isely,

suppose that asequen e ofprobabilities

p

n

_{= (p}

n

1 , . . . , p

n

m

n

)

satisfying

p

n

1 ≥ . . . ≥ p

n

m

n

> 0

isgiven, and satises

max

1≤i≤m

n

p

n

i

→

n→∞

0

and

σ(p

n

₎

−1

_p

n

i

→

n→∞

θ

i

i ≥ 1

(12)

forasequen e

θ

asdes ribed above,and where

σ(p) =

pP

m

i=1

p

2 i

when

p

= (p

1 , . . . , p

m

)

.

Forevery

n

,let

τ

n

beauniformpermutationon

{1, 2, . . . , m

n

}

. Considertheparking prob-lemwhere the aravans whi h try to park su essively have magnitudes

p

n

τ

n

(1)

, p

n

τ

n

(2)

, . . .

.

Let

U

1 , U

2 , . . .

be independent uniform

(0, 1)

random variables independent of

τ

n

, so we

may onsider the bridge with ex hangeable in rements

B

(n)

(s) = σ(p

n

)

−1

_{−x +}

m

n

X

i=1

p

n

_τ

_n

_(i)

1

{s≥U

i

}

!

,

_{0 ≤ s ≤ 1.}

Kallenberg'stheoremshowsthatundertheasymptoti assumptionson

p

n

,

B

(n)

onverges

indistribution tothe bridge

B

θ

dened above.

Now for

t ≥ 0

, let

I

n

t

= inf{i ≥ 1 :

P

m

n

j=i+1

p

n

τ

n

(j)

≤ t}

. The following analogue of Lemma5 holds.

Lemma 7 For every

t ≥ 0

, the pro ess

σ(p

n

)

−1

m

n

X

i=I

n

t

+1

p

n

τ

n

(i)

1

{s≥U

i

}

,

0 ≤ s ≤ 1

(16)

Proof. The key tothis lemma is toshow that

max

i≥I

n

t

σ(p

n

)

−1

p

n

_τ

_n

_(i)

_{→ 0}

(13)

in probability as

n → ∞

. The result is then obtained via the so- alled weak law of large numbers for sampling without repla ement: if

x

n

i

, 1 ≤ i ≤ n

is a sequen e with

sum

t

satisfying

max

1≤i≤n

x

n

i

→ 0

as

n → ∞

, and if

τ

n

is a uniform permutation

on

{1, . . . , n}

, then for every rational

r ∈ [0, 1]

,

P

n

i=1

x

n

τ

n

(i)

1

{r≥U

i

}

→ tr

in probabil-ity (in fa t in

L

2

). The result in probability remains true if

x

n

i

, 1 ≤ i ≤ n

is

ran-dom with sum

t

, and

max

1≤i≤n

x

n

i

→ 0

in probability. One on ludes that the pro- ess

(

P

n

i=1

x

n

τ

n

(i)

1

{s≥U

i

}

, 0 ≤ s ≤ 1)

onverges in probability to

(ts, 0 ≤ s ≤ 1)

for

the uniform norm by a monotoni ity argument. The lemma is then proved by letting

x

1 = σ(p

n

)

−1

p

n

_τ

_n

_(I

n

t

+1)

, x

2 = σ(p

n

₎

−1

_p

n

τ

n

(I

t

+2)

, . . . , x

m

n

−I

t

n

= σ(p

n

₎

−1

_p

n

τ

n

(m

n

)

, x

m

n

−I

t

n

+1

=

t −

P

m

n

i=I

n

t

+1

σ(p

n

₎

−1

_p

n

τ

n

(I

t

n

+1)

(note that this lastterm is

≤ σ(p

n

₎

−1

_p

n

τ

n

(I

t

n

)

, whi hgoesto

0

).

Solet usshow (13). Tothis end, let

0 < ρ < 1

, then

X

ρ

n

:=

P

m

n

i=⌊ρm

n

⌋

σ(p

n

₎

−1

_p

n

τ

n

(i)

→

∞

in probability,sin e

E[X

ρ

n

] ∼ σ(p

n

)

−1

(1 − ρ)

goes toinnity (noti e

σ(p) ≤ p

1

) while

E[(X

ρ

n

)

2 ] ∼ E[X

n

ρ

]

2

, as a simple omputation shows. Therefore,

I

n

t

∼ m

n

in probability. Consequently, for any

K ∈ N

, the quantity

P (τ

−1

n

(1) < I

t

n

, . . . , τ

n

−1

(K) < I

t

n

)

goes to

1

,

so

min

i≥I

n

t

τ

n

(i) → ∞

inprobability. But then,for any

ε > 0

, if

K

issu h that

θ

K

< ε/2

,

then

σ(p

n

₎

−1

_p

n

K

≤ ε

for

n

large. Up to taking

n

even larger, with probability lose to

1

,

τ

n

(i) ≥ K

for

i ≥ I

n

t

and therefore

max

i≥I

n

t

σ(p

n

₎

−1

_p

n

τ

n

(i)

≤ ε

, hen e (13).

Onededu es, asaroundthe proofof Proposition1,the following laim. Let

X

(n)

_{(t) =}

Λ

p

n

◦τ

n

_(I

n

t

)

be asabove with data

m = m

n

, p

n

τ

n

(i)

, 1 ≤ i ≤ m

n

, s

i

= U

i

. Then

Proposition 2 As

n → ∞

,undertheasymptoti regime(12),thepro ess

(X

(n)

_{(t), t ≥ 0)}

onverges in the sense of weak onvergen e of nite-dimensional marginals to the

time-reversed eternal additive oales ent

(C

θ

_{(− log(t)), t ≥ 0)}

.

6.2 On the marginal distributions

Itwouldalsobeinterestingtodeterminethemarginallawsofthefragmentation

F

(α)

_{(t) :=}

C

(α)

_{(− log t)}

. The task seems quite di ult if started from the des ription of

F

(α)

_(t)

in

termsoflengthsof onstan y intervalsof Vervaat transformofbridges,be auseex ursion

theoryseems powerless here,unlikein[13℄. In parti ular,the fa tthat the fragmentation

is based on stable loops and not stable bridges impedes the appli ation of results of

Miermont[13℄ onadditive oales ents based onbridges of ertainLévy pro esses.

Another way to start the exploration is to use the representation of fragmentation

pro esses

F

θ

_{(t) := C}

θ

(− log t)

des ribed in the pre eding se tion with the help of

Inho-mogeneousContinuumRandomTrees (ICRT) dis ussed in[3℄. Inparti ular, itiseasyto

obtainthe rst moment of a size-biased pi k 2

F

†

(t)

fromthe sequen e

F(t)

for any xed

t

, asfollows.

Let us re all the basi fa ts on the ICRT

(θ)

onstru tion of

F

θ

. The ICRT an be

viewed viaa sti k-breaking onstru tion asthe metri ompletion of the positive realline

2

Re allthatasize-biasedpi k

X

†

froma(random)positivesequen e

(X

i

, i ≥ 1)

withsum

0 < S < ∞

a.s.isarandomvariableoftheform

X

i

∗

,where

P (i

∗

_{= i|X}

(17)

R

₊

endowed with a non standard metri . Pre isely, suppose we are given the following independent randomelements:

•

A Poisson pro ess

{(U

i

, V

i

), i ≥ 1}

on the o tant

O

= {(x, y) : 0 < y < x} ⊂ R

2 +

, with intensity

θ

0 dxdy

1

O

, so in parti ular

{U

i

, i ≥ 1}

is a Poisson pro ess with intensity

θ

0 xdx

1

x≥0

,

•

A sequen e of independent Poisson pro esses

{ξ

i,j

, j ≥ 1}, i = 1, 2, . . .

with respe -tive intensities

θ

i

dx

1

x≥0

, i = 1, 2, . . .

.

We distinguish the points

(V

i

, i ≥ 1), (ξ

i,1

, i ≥ 1)

as joinpoints, while

(U

i

, i ≥ 1), (ξ

i,j

, i ≥

1, j ≥ 2)

are alled utpoints. If

η

is a utpoint, let

η

∗

be its asso iated joinpoint, i.e.

U

∗

i

= V

i

, ξ

i,j

∗

= ξ

i,1

. Bythe assumptionon

θ

,itisa.s. possible toarrangethe utpointsby in reasing order

0 < η

1 < η

2 < . . .

. We then onstru t a family

R(k), k ≥ 1

of redu ed trees asfollows. Cuttheset

(0, ∞)

intobran hes

(η

i

, η

i+1

]

,whereby onvention

η

0 = 0

. Let

R(1)

bethe segment

(0, η

1 ]

,endowed with theusualdistan e

d

1 (x, y) = |x − y|

. Then given

R(k), d

k

, we obtain

R(k + 1)

by adding the bran h

(η

k

, η

k+1

]

somewhere on

R(k)

, and we plant the left-end

η

k

on the joinpoint

η

∗

k

(sin e a.s.

η

∗

_{< η}

, the point

η

∗

k

is indeed an element of

R(k)

). Pre isely,

R(k + 1) = (0, η

k+1

]

and

d

k+1

(x, y) = d

k

(x, y)

if

x, y ∈ R(k)

,

d

k+1

(x, y) = |x − y|

if

x, y ∈ (η

k

, η

k+1

]

, and

d

k+1

(x, y) = x − η

k

+ d

k

(y, η

∗

k

)

if

x ∈ (η

k

, η

k+1

], y ∈ R(k)

. As the distan es

d

k

are ompatible by denition, this denes a

randommetri spa e

(0, ∞), d

su hthattherestri tionof

d

to

R(k)

is

d

k

,we allitsmetri ompletion

T

θ

the ICRT

(θ)

, its elements are alled verti es. The point

∅

= lim

n→∞

1/n

isdistinguished and alled the root.

One an see that

T

θ

is an

R

-tree, i.e. a omplete metri spa e su h that for any

x, y ∈ T

θ

there is a unique simple path

[[x, y]]

from

x

to

y

, whi h is isometri to the segment

[0, d(x, y)]

,i.e. isageodesi . Moreover, it anbeendowed withanaturalmeasure

µ

θ

whi histheweaklimitas

n → ∞

oftheempiri almeasures

n

−1

P

n

i=1

δ

η

i

. Thismeasure isnon-atomi and supported onleaves,i.e.verti es

x ∈ T

θ

su hthat

x /

∈ [[∅, y]] \ {y}

for

any

y ∈ T

θ

. Non-leafverti esformaset alled theskeleton. A se ondnaturalmeasure is

the Lebesgue measure

λ

on

T

θ

, i.e.the unique measure su h that

λ([[x, y]]) = d(x, y)

for any

x, y

, and this measure is supported on the skeleton.

Nowforea h

t

onsideraPoissonmeasureon

T

θ

withatoms

{x

t

i

, i ≥ 1}

,withintensity

tλ(dx)

,sothedierentpro essesare oupledinthe naturalway as

t

varies,i.e.

{x

t

i

, i ≥ 1}

in reases with

t

. These points dis onne t the tree intoa forest of disjoint onne ted tree omponents,orderthemas

F

θ

i

(t), i ≥ 1

by de reasingorderof

µ

θ

-mass. Thenthe pro ess

((µ

θ

_(F

θ

i

(t)), i ≥ 1), t ≥ 0)

of these

µ

θ

-masses has same law as

F

θ

. A size-biased pi k

from this sequen e of masses is then obtained as the

µ

θ

-mass of the tree omponent at

time

t

that ontains an independent

µ

θ

-sample, onditionally on

(T

θ

_{, µ}

θ

₎

. Therefore, if

F

θ

†

(t)

denotes su hasize-biasedpi k,

E[F

θ

†

(t)]

istheprobabilitythattwoindependent

µ

θ

-samples

X

1 , X

2

belongtothesametree omponentofthe uttree,i.e.thatnoatomofthe Poisson measure at time