theorem for a branching random walk First-and second-order expansions in the central limit C.R. Acad. Sci. Paris, Ser.I

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Probability theory

First- and second-order expansions in the central limit theorem for a branching random walk

Développements du premier et du second ordre dans le théorème central limite pour une marche aléatoire avec branchement

Zhiqiang Gao

^a

, Quansheng Liu

^b

^,

^c

aSchoolofMathematicalSciences,LaboratoryofMathematicsandComplexSystems,BeijingNormalUniversity,Beijing100875,PRChina bUniv.Bretagne-Sud,CNRSUMR6205,LMBA,campusdeTohannic,56000 Vannes,France

cChangshaUniversityofScienceandTechnology,SchoolofMathematicsandStatistics,Changsha 410004,China

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received2September2015

Acceptedafterrevision25January2016 PresentedbyJean-PierreKahane

Wegivethefirst- andsecond-orderasymptoticexpansions forthecentrallimittheorem about the distribution of particles in a branching random walk on the real line. In particular,ourfirst-orderexpansionrevealstheexactconvergencerateinthecentrallimit theorem;itextendsandimprovesaknownresultforthebranchingWienerprocess.

©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

ré s u m é

Nousdonnonslesdéveloppementsasymptotiques d’ordres unetdeuxdans lethéorème centrallimitesurladistributiondesparticulesdansunemarchealéatoireavecbranchement surladroite réelle. Enparticulier, ledéveloppement asymptotique d’ordreun révèle la vitesseexacte deconvergence duthéorème central limite, cequi étendet amélioreun résultatconnupourleprocessusdeWieneravecbranchement.

©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

Versionfrançaiseabrégée

Considéronsunemarchealéatoireavec branchementsurladroiteréelleR^{.Lemodèleestdéfinicommesuit. Autemps} n

=

^{0,uneparticuleinitiale}∅^{estsituéeenS}∅

=

⁰

∈

R^.Autempsn

=

^{1,elleestremplacéeparN}

=

^N∅nouvellesparticules

∅ⁱ

=

^{i de la génération 1, localisées en L}i

=

^L∅i

,

1

≤

ⁱ

≤

^{N. En général, chaque particule u}

=

^u1u₂

· · ·

^un de génération n, située en Su, est remplacée au temps n

+

^{1 par N}u nouvelles particules ui de génération n

+

^{1, localisée en S}ui

=

Su

+

^Lui

,

1

≤

ⁱ

≤

^Nu, où lesvariables aléatoires Nu et Lu sont respectivement de loi

{

^pi

}

i∈N sur N ^{etde loi G}

( · )

sur R^. Touteslesvariablesaléatoires Nu etLu,indexéesparlessuitesfiniesu,sontmutuellementindépendantes.

E-mailaddresses:[email protected](Z. Gao),[email protected](Q. Liu).

http://dx.doi.org/10.1016/j.crma.2016.01.021

1631-073X/©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

Ondésignepar Tn l’ensembledesindividusde lan^e génération,représentéspar dessuitesu d’entierspositifsde lon- gueur

|

^u

| =

^n;commed’habitude,laparticuleinitialeestreprésentéeparlasuitenulle∅^{(delongueur0) ;siu}

∈

Tn,alors ui

∈

Tn+¹ sietseulementsi1

≤

ⁱ

≤

^Nu.

Soit

Z_n

=

u∈Tn

δ

Su

la mesure de comptage à l’instant n : pour unepartie A de R, Zn

(

A

)

est lenombre desparticules de la génération n localiséesdans A.Alors

{

^Zn

(R)}

^{estunprocessusde}Galton–Watson.Ilestbienconnuquelasuite

Wn

=

^Zn

( R )

mⁿ avecm

=

^∞

i=0

ipi

,

n

≥

0

,

formeunemartingalenonnégativeparrapportàlafiltrationnaturelle.Supposonsque

m

>

1 et

∞

i=1

i

(

lni

)

^1+λpi

< ∞,

(1)

oùlavaleur de

λ >

0 serapréciséedansleshypothèses desthéorèmes.Sous(1),leprocessusde Galton–Watson

{

^Zn

(R)}

estsurcritiqueetlasuite

{

^Wn

}

^{convergepresquesûrement(p.s.)verssalimiteW,avec}E^W

=

^{1 parlethéorèmedeKesten–}

Stigum(voir[2]).

Noussommesintéresséspardespropriétésasymptotiquesdelamesurede comptage Zn,quidécritlaconfigurationdu systèmedeparticulesàl’instantn.Sousl’hypothèse(1)pourun

λ >

0 etlacondition

|

^x

|

^2dG

(

x

) < ∞

^{,KaplanetAsmussen} [11]ontmontrélethéorèmecentrallimitesuivant :pourtoutt

∈

R^,

1

mⁿZn

((−∞,

nl

+ √

n

σ

t

]) −−−→

^n→∞

p.s.

(

t

)

W

,

(2)

où

l

=

xdG

(

x

), σ

²

=

(

x

−

l

)

²dG

(

x

), (

t

) =

t

−∞

√

1 2

π

^e

−^t2/2dx

.

Nous allonsapprofondir ce théorème enconsidéront des développement limités. Enparticulier, nousdonnons la vitesse exactedelaconvergencedans(2).Lesrésultatsdépendentdedeuxmartingales

{

^N1,n

}

^et

{

^N2,n

}

^{,quiconcernentlesdévia-} tionsdespositions Su autourdeleurmoyennes,d’ordrepremieretsecond,centréesetnormalisées :

N_1,n

=

¹ mⁿ

u∈Tn

(

S_u

−

^nl

)

et N_2,n

=

¹ mⁿ

u∈Tn

(

S_u

−

^nl

)

²

−

ⁿ

σ

²

.

Pourlaconvergencedesmartingales,nousauronsbesoindelaconditiondemoment :

|

^x

|

^ηdG

(

x

) < ∞, η >

0

.

(3)

Proposition0.1.Lessuites

{

^N1,n

}

^et

{

^N2,n

}

^sontdesmartingalesparrapportàlafiltrationnaturelle.Lesdeuxmartingalesconvergent p.s.sousdeconditionssimples :

(i) si(1)alieupourun

λ >

1et(3)alieupourun

η >

2,alorsV1

:=

_n^lim_→∞^N1,nexistep.s.dansR^; (ii) si(1)alieupourun

λ >

2et(3)alieupourun

η >

4,alorsV2

:=

_n^lim_→∞^N2,nexistep.s.dansR^.

Pourledévoplopementasymptotique,nousauronsbesoindelaconditiondeCramér :

lim sup

|t|→∞

e^itxdG

(

x

)

<

1

.

(4) Lethéorèmesuivantconcernelecomportementasympotiqued’ordre1danslethéorèmecentrallimite.

Théorème0.2.Onsupposelesconditionsdemoment(1)et(3)pourcertains

λ >

8et

η >

12,etlaconitiondeCramér(4).Alorsp.s.

nousavons :pourtoutt

∈

R^,lorsquen

→ ∞

^, 1

mⁿZ_n

( −∞ ,

nl

+ √

n

σ

t

] = (

t

)

W

+ √

¹

nV

(

t

) +

^o

( √

¹ n

),

oùV(t

)

estlavariablealéatoiredéfiniepar(11).

Ce théorème révèle lavitesseexacte de convergencedansle théorèmecentral limite. Il étendet amélioreunrésultat connu pour leprocessusde Wieneravec branchement [4]. Ila été obtenupar Kabluchko[10,Theorem 5andRemark 2]

souslaconditiondusecondmomentpourlaloi

{

^pi

}

i∈N.

Sousdesconditionsunpeuplusfortes,nouspouvonsobtenirledévelopmentasymptotiqued’ordre2 :

Théorème0.3.Onsupposelesconditionsdemoment(1)et(3)pourcertains

λ >

18et

η >

24,etlaconitiondeCramér(4).Alorsp.s., nousavons :pourtoutt

∈

R^,lorsquen

→ ∞

^,

1

mⁿZn

((−∞,

nl

+ √

n

σ

t

]) = (

t

)

W

+ √

¹

nV

(

t

) +

¹

nR

(

t

) +

o

1 n

,

oùV

(

t

)

etR

(

t

)

sontlesvariablesaléatoiresdéfiniespar(11)et(13).

Les résultats présentés dans cette note peuvent être généralisés au cas desmarches aléatoires avec branchement en milieuxaléatoires.Pourlespreuvescomplètesetplusdedétails,nousrenvoyonsleslecteursauxréférences[5,6].

1. Introductionandresults

In thisnote,we shallpresentrecentprogresson first- andsecond-orderasymptoticexpansionsforthe distributionof particlesinabranchingrandomwalkundersuitableconditions.Thereadermayreferto[5,6]forcompleteproofsandmore relatedresults,wheretheresultsareobtainedunderageneralframework,i.e.forabranchingrandomwalkwitharandom environmentintime.

Consider abranching randomwalk onthereallineR^{.Initially,an ancestorparticle}∅ ^{islocatedat S}∅

=

^{0.At time1,} theinitialparticle∅^{isreplacedby N}

=

^N∅newparticles∅ⁱ

=

^{i ofgeneration1,with}displacementL_∅i

=

^Li,sothateach particle∅ⁱ

=

^{i movestoS}∅i

=

^S∅

+

^L∅i,thatis,S_i

=

^Li

,

1

≤

ⁱ

≤

^{N.Ingeneral,attimen}

+

^{1,eachparticleu}

=

^u1u₂

· · ·

^unof generationnisreplacedbyNu newparticlesofgenerationn

+

^1,withdisplacementsLu1

,

Lu2

, · · · ,

LuNu,sothateachparticle uihaspositionSui

=

^Su

+

^Lui,1

≤

ⁱ

≤

^Nu.EachNu isaninteger-valuedrandomvariablewithcommondistribution

{

^pi

}

i∈N, andeach L_u isareal-valuedrandomvariablewithcommondistribution G

(·)

^{;all therandomvariablesN}u

,

L_u,indexedby finitesequencesofintegersu,areindependentofeachother.

Let T ^{be the} genealogical treewith

{

^Nu

}

^{as defining elements. Bydefinition, we have: (a)}∅

∈

T^{; (b) ui}

∈

T^implies u

∈

T^;(c)ifu

∈

T^,thenui

∈

T^ifandonlyif1

≤

ⁱ

≤

^Nu.Let

T

n

= {

^u

∈ T : |

^u

| =

ⁿ

}

bethesetofparticlesofgenerationn,where

|

^u

|

^{denotesthelengthofthesequenceu.}

Denote by Zn

=

u∈Tn

δ

Su the countingmeasure which countsthe numberof particles ofgeneration n situatedin a givenset:forB

⊂

R^,

Zn

(

B

) =

u∈Tn

1B

(

Su

).

Then

{

^Zn

(

R

) }

^formsaGalton–Watsonprocess.Itiswellknownthat

W_n

=

^Zn

(R)

mⁿ with m

=

^∞

i=⁰ ip_i

,

isamartingalewithrespecttothenaturalfiltration.Weshallalwayssupposethat m

>

1 and

∞ i=¹

i

(

lni

)

^1+λp_i

< ∞,

⁽⁵⁾

wherethevalueof

λ >

0 willbespecifiedlater.Under(5),theprocessissupercriticalandthemartingale

{

^Wn

}

^converges a.s. to a non-zero limit W with E^W

=

^{1 by the famous} Kesten–Stigum theorem (c.f.[2]);moreover, the event

{

^W

>

0

}

coincideswith

{

^Zn

→ ∞}

^a.s.

We are interested in the asymptotic behavior of the counting measure Z_n, which describes the configuration of the particlesystemattimen.ThequestionofcentrallimittheoremforZnwasfirstposedbyHarris[8].Underthenear-minimal

conditions(5) forsome

λ >

0 and

x²dG

(

x

) < ∞

^{,Kaplan andAsmussen[11] proved thatforeach fixed t}

∈

R^{,we have} a.s.,

1

mⁿZ_n

((−∞,

^nl

+ √

n

σ

t

]) −−−→

^n→∞

(

t

)

W

,

(6)

where l

=

xdG

(

x

), σ

²

=

(

x

−

^l

)

²dG

(

x

), (

t

) =

t

−∞

φ (

x

)

dx with

φ (

t

) = √

¹ 2

π

^e

−t²/2

.

(7)

Thisis acentral limit theorem,since itrevealsthat a.s. therandom measures Zn withsuitable norming convergetothe standardnormallaw N

(

0

,

1

)

.Indeed,thefactthat(6)holdsa.s.foreachfixed timpliesthata.s.(6)holdsforallrational t;by themonotonicityofthedistributionalfunctionx

→

^Zn

( −∞ ,

x

]

^{andthecontinuityof}

,itfollowsthata.s.(6)holdsforall real t(thatis,theeventthat(6)holdsforallrealt hasprobability1).

Theresult(6)hasbeenextendedinvariousforms(see,e.g.,[3,7,9]).InspiredbytheclassicalEdgeworthexpansiontheory ofcentrallimittheoremsforsumsofindependentrandomvariables(see, e.g.,[12]),wewish toimprove(6)bydescribing itsasymptoticexpansions.

Inthisnote,we aimtogive thefirst- andsecond-order expansionsintheabovecentral limittheoremwhen Cramér’s conditionaboutthecharacteristicfunctionholds:thatis

lim sup

|^t|→∞

e^itxdG

(

x

)

<

1

.

(8) Thisconditionholdsforexampleifthedistribution G hasan absolutelycontinuous component(in Lebesgue’sdecomposi- tion),butdoesnotholdifG isalatticedistribution.

Ourresultswilldependonthefollowingtwomartingales,whichconcernthedeviationsofSufromtheirmeans,oforder 1and2,centeredandnormalized:

N_1,n

=

¹ mⁿ

u∈Tn

(

S_u

−

^nl

)

and N_2,n

=

¹ mⁿ

u∈Tn

(

S_u

−

^nl

)

²

−

ⁿ

σ

²

.

Fortheconvergenceofthesemartingales,wewillneedamomentconditionoftheform

|

x

|

^ηdG

(

x

) < ∞, η >

0

.

(9)

Proposition1.1.Thesequences

{

^N1,n

}

^et

{

^N2,n

}

^aremartingaleswithrespecttothenaturalfiltration

D

0

= {∅ , } , D

n

= σ (

N_u

,

L_ui

:

ⁱ

≥

¹

, |

^u

| <

n

)

for n

≥

¹

.

Thesemartingalesconvergea.s.undersimplemomentconditions:

a) if(5)holdsforsome

λ >

1and(9)holdsforsome

η >

2,thenV1

:=

_n^lim_→∞^N1,nexistsa.s.inR^; b) if(5)holdsforsome

λ >

2and(9)holdsforsome

η >

4,thenV₂

:=

^lim

n→∞N_2,nexistsa.s.inR^.

Ourfirst resultis a first-orderexpansion inthe central limit theorem.As usual, forrandom variables An and Bn,we write An

=

^o

(

Bn

)

a.s.iflimn→∞ ^ABⁿn

=

^{0 a.s.Inthefollowingtheorems,thenumbers8}

,

12

,

18 and24 inthehypothesesare fortechnicalreasons;thebestconstantsarenotyetknown,andseemtobedelicate.

Theorem1.2.Assume(5)forsome

λ >

8,(9)forsome

η >

12,and(8).Thena.s.wehave:forallt

∈

R^,asn

→ ∞

^, 1

mⁿZn

( −∞ ,

nl

+ √

n

σ

t

] = (

t

)

W

+ √

¹

nV

(

t

) +

^o

( √

¹

n

),

(10)

where

V

(

t

) = −

¹

σ ^{φ (}

^t

⁾

^V1

⁺ σ

⁽³⁾

6

σ

³

⁽

¹

⁻

^t

2

)φ (

t

)

W with

σ

⁽³⁾

₌

(

x

−

^l

)

³dG

(

x

).

(11)

Thisresultrevealstheexact convergenceratein(6).Inparticular,itimprovesChen’s Theorem3.1in[4]forbranching Wienerprocess by weakeningthemoment conditionsfortheoffspring lawsanddisplacement lawstherein;moreover, it extendsthattothegeneralcaseincludingnon-Gaussiandisplacementunderweakermomentsconditions.Whenthesecond momentcondition _∞

i=1i²p_i

< ∞

^isassumed,Theorem 1.2wasobtainedbyKabluchko[10,Theorem5andRemark2].

Underslightlystrongermomentsconditions,wecanobtainthesecond-orderasymptoticexpansion.

Theorem1.3.Assume(5)forsome

λ >

18,(9)forsome

η >

24,and(8).Thena.s.,wehave:forallt

∈

R^,asn

→ ∞

^, 1

mⁿZ_n

(−∞,

^nl

+ √

n

σ

t

] = (

t

)

W

+ √

¹

nV

(

t

) +

¹

nR

(

t

) +

^o

1

n

,

(12)

where

R

(

t

) = −

¹

2

σ

^2t

^{φ (}

^t

⁾

^V2

⁻ σ

⁽³⁾

6

σ

^4H3

⁽

^t

^{)φ (}

^t

⁾

^V1

⁻

( σ

⁽³⁾

)

² 72

σ

^{6 H5}

⁽

^t

^{) +}

( σ

⁽⁴⁾

₋

3

σ

⁴

)

24

σ

^{4 H3}

⁽

^t

⁾

φ (

t

)

W

,

(13)

with

σ

^(ν)

=

(

x

−

^l

)

^νdG

(

x

)

for

ν =

³

,

4andH_m

(·)

^istheChebyshev–Hermitepolynomialofdegreem:

H_m

(

t

) =

^m

!

^m₂

k=⁰

(−

¹

)

^kt^m−2k

k

!(

m

−

2k

)!

2^k

(

so that H₃

(

t

) =

^t3

−

^3t

,

H₅

(

t

) =

^t5

−

^10t3

+

^15t

).

From[1,Theorem2],weknowthatforeach

λ >

0,thecondition(5)impliesthat

Wn

−

W

=

o

(

n^−λ

)

a.s.

WiththisweobtainthefollowingvariantsofTheorems 1.2 and1.3:

Corollary1.4.Assume(5)forsome

λ >

8,(9)forsome

η >

12,and(8).Thena.s.on

{

^W

>

0

}

^,wehave:fort

∈

R^,asn

→ ∞

^, 1

Z_n

(R)

^Zn

(−∞,

^nl

+ √

n

σ

t

] = (

t

) + √

¹ n

V

(

t

)

W

+

^o

( √

¹

n

).

Corollary1.5.Assume(5)forsome

λ >

18,(9)forsome

η >

24,and(8).Thena.s.on

{

^W

>

0

}

^{,wehave:forallt}

∈

R^,asn

→ ∞

^,

1 Zn

( R )

^Zn

( −∞ ,

nl

+ √

n

σ

t

] = (

t

) + √

¹ n

V

(

t

)

W

+

¹

n R

(

t

)

W

+

^o

1 n

.

2. Sketchofproofs

Intheproofsofthemainresults,wefollowtheroutein[11,4]:akeydecompositionplays animportantrole.Weneed somenotationtointroducethedecomposition.

LetT

(

u

)

betheshiftedtreeofT^{atuwithdefiningelements}

{

^Nuv

}

satisfying:1)∅

∈

T

(

u

)

,2)vi

∈

T

(

u

) ⇒

^v

∈

T

(

u

)

and 3)ifv

∈

T

(

u

)

,thenvi

∈

T

(

u

)

ifandonlyif1

≤

ⁱ

≤

^Nuv.SetTn

(

u

) = {

^v

∈

T

(

u

) : |

^v

| =

ⁿ

}

^.

Foru

∈ (N

^∗

)

^k

(

k

≥

⁰

)

andn

≥

^1,writeforB

⊂

R^,

Zn

(

u

,

B

) =

v∈Tn(u)

1B

(

Suv

−

^Su

).

Define

tn

=

^nl

+ √

n

σ

t

,

Wn

(

u

,

t

) =

¹

mⁿZn

(

u

, ( −∞ ,

t

] ),

for n

≥

¹

,

t

∈ R .

Let

β

bearealnumberchosensuitably(takemax

{

²_λ

,

³_η

} < β <

¹₄ andmax

{

³_λ

,

_η⁴

} < β <

¹₆ intheproofsofTheorems 1.2 and1.3respectively)andsetkn

=

^nβ

^{,thelargestintegernobiggerthannβ.}

Observefork

≤

^n,

Z_n

(

B

) =

u∈Tk

Z_n−k

(

u

,

B

−

^Su

).

(14)

Weobtainthefollowingkeydecomposition:

1

mⁿZ_n

(( −∞ ,

t_n

] ) =

¹ m^kn

u∈Tkn

W_n−kn

(

u

,

t_n

−

^Su

) − E

W_n−kn

(

u

,

t_n

−

^Su

)

^Su

+

¹ m^kn

u∈Tkn

E

W_n−kn

(

u

,

t_n

−

^Su

)

^Su

=: A

n

+ B

n

.

(15)

Theorems 1.2 and1.3followfromLemmas 2.1 and2.2respectively.

Lemma2.1.UndertheconditionsofTheorem 1.2,foreachfixedt

∈

R^{,thefollowingholda.s.asn}

→ ∞

^: (i)

√

n

A

n

→

⁰

,

(ii)

B

n

= (

t

)

W_kn

+ √

¹

n

σ

⁽³⁾

6

σ

³

⁽

¹

⁻

^t

2

)φ (

t

)

W_kn

−

¹

σ ^{φ (}

^t

⁾

^N1,kn

+

^o

( √

¹ n

),

(iii) W_kn

−

W

=

o

( √

¹

n

).

Lemma2.2.UndertheconditionsofTheorem 1.3,foreachfixedt

∈

R^{,thefollowingholda.s.asn}

→ ∞

^: (i) n

A

n

→

⁰

,

(ii)

B

n

= (

t

)

Wk_n

+ √

¹ n

σ

⁽³⁾

6

σ

³

⁽

¹

⁻

^t

2

)φ (

t

)

Wk_n

−

¹

σ ^{φ (}

^t

⁾

^N1,kn

+

¹ n

−

¹

2

σ

^2t

^{φ (}

^t

⁾

^N2,kn

− σ

⁽³⁾

6

σ

^4H3

⁽

^t

^{)φ (}

^t

⁾

^N1,kn

⁻

( σ

⁽³⁾

)

² 72

σ

^{6 H5}

⁽

^t

^{) +}

( σ

⁽⁴⁾

−

³

σ

⁴

)

24

σ

^{4 H3}

⁽

^t

⁾

φ (

t

)

W_kn

+

^o

(

¹

n

),

(iii) W_kn

−

^W

=

^o

(

¹

n

),

N_1,kn

−

^V1

=

^o

( √

¹ n

).

The maintermsinthe expansionofBn comefromtheuseofthe Edgeworthexpansionsinthecentral limittheorem (see, e.g., [12, P. 159 Theorem 1]). In the proofsof the lemmas, we need to carry out a carefulanalysis by combining thetoolsincludingtheBorel–CantelliLemma,truncatingarguments,momentinequalitiesforsumsofindependentrandom variables,andsoon.Thereadermayreferto[5,6]fordetails.

Acknowledgements

Theauthorsthanktheanonymousrefereeforvaluablecommentsandsuggestions.ThekindhospitalityofLMBA,Univer- sitédeBretagne-Sud,wherepartofthisworkwascarriedout,iswellacknowledged.Theworkhasbeenpartiallysupported by theNationalNaturalScienceFoundation ofChina (GrantsNo. 11101039,No. 11271045,No. 11571052,No.11401590), by the Fundamental Research Funds for the Central Universities (2013YB52), and by the Natural Science Foundation of GuangdongProvince(China),GrantNo. 2015A030313628.

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