Limit theorems for continuous-state branching processes with immigration 2022 Clément Foucart.pdf

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Limittheoremsforcontinuous-statebranching
processeswithimmigration
Cl´ementFoucart∗&ChunhuaMa†&LinglongYuan‡
July22,2021
Abstract
Acontinuous-statebranchingprocesswithimmigration,whosebranchingmecha-
nismisΨandimmigrationmechanismΦ,CBI(Ψ,Φ)forshort,mayhavetwodifferent
asymptoticregimesdependingonwhetherΦ(u)du<∞orΦ(u)du=∞.When
∫0SΨ(u)S∫0SΨ(u)S
Φ(u)du<∞,CBIseitherhavealimitdistributionoragrowthratedictatedbythe
∫0SΨ(u)S
(u)du=∞,immigrationoverwhelmsbranchingdynam-
∫0SΨ(u)S
-lineartime-dependentrenormaliza-

-

statebranchingprocesseswithimmigration[(1972)].
-statebranchingprocesses,Immigration,Grey’smartingale,Limit
distribution,Non-linearrenormalization,Regularlyvaryingfunctions.
MSC(2020):primary60J80;secondary60F05;60F15
1Introduction
Continuous-statebranchingprocesseswithimmigration(CBI)havebeendeflnedbyKawazu
arXiv:[]21Jul2021
andWatanabe[23].TheyarescalinglimitsofGalton-WatsonMarkovchainswithimmi-
gration,.[23,].Recentyearshaveseenrenewedinterestinthisclassof
-
tialequationswithjumps,seeDawsonandLi[9],andfromamoreappliedpointofview,
formanimportantsubclassoftheso-calledaffineprocesses,whichareknownintheflnan-
cialmathematicssettingformodellinginterestrates,see[11].Wementionforinstancethe
worksinthisdirectionofJiaoetal.[21]andBarczyetal.[2]wherecertainCBIprocesses
arestudiedfromastatisticalpointofview.
TheasymptoticbehaviorsofGalton-Watsonprocesseswithimmigrationhavebeenex-
[8],Heathcote[19],
∗LAGAUniversit´eSorbonneParisNordE–mail:******@-
†SchoolofMathematicalSciencesandLPMC,NankaiUniversityE–mail:******@
‡DepartmentofMathematicalSciences,UniversityofLiverpool;DepartmentofMathematicalSciences,
Xi’anJiaotong-LiverpoolUniversity;E–mail:******@
1:.
Heyde[20],Pakes[31]andSeneta[34]
CBIshavebeencharacterizedbyDuhaldeetal.[10].Finepropertiesofthestationarydis-
tributionsofCBIprocesses,whentheyexist,havealsobeenrecentlyestablishedinChazal
etal.[7]andKeller-ResselandMijatovic[24].Inthecasewherenostationarydistribution
exists,
notbesurprisingthattheresultsfoundintheseventiesforGalton-Watsonprocesseswith
immigration,havecounterpartsinthecontinuous-stateandcontinuous-
yearafterKawazuandWatanabe’sfoundingwork,Pinskythuspublishedashortnote[32]
withoutproof,
andresumeinthisarticlethestudyoflimittheoremsforCBIsinitiatedbyPinsky.
Westartbyprovinganalmost-sureconvergenceforCBIprocessesbyadaptingGrey’s
approach[17]totheframeworkwithimmigration(Theorem2).Wethenprovideageneral
non-linearrenormalizationinlaw(Theorem6).Tothebestofourknowledgethislatter
renormalizationdoesnotappearintheliteratureaboutGalton-Watsonprocesseswithim-

andimmigrationmechanisms,
caseofsupercriticalbranching,weshowtheexistenceoftwodistinctalmost-sureasymp-
Φ(u)
toticregimesaccordingtotheconvergence/divergenceoftheintegral∫0SΨ(u)
integralconverges,(u)du<∞,thebranchingdynamicstakesprecedenceoverimmi-
∫0SΨ(u)S
,underthe
classicalLlnLmomentassumption(alsocalledKesten-Stigumcondition)overthebranch-
ingL´evymeasure,theCBIprocessgrowsatthesameexponentialrateasthepurebranching
,whenitdiverges,(u)du=∞,immigrationissosubstan-
∫0SΨ(u)S
tialthatthebranching,althoughsupercritical,,typicallythe
processgrowsfasterthanthepurebranchingprocessontheeventofitsnon-extinction.
Asimilardichotomyoccursmoregenerallyfornon-criticalCBIprocesseswhenwecon-
sidertheirlong--linear
time-dependentrenormalizationinlawofanon-criticalCBI(Ψ,Φ)-process(Yt,t≥0)satis-
Φ(u)
fying∫0SΨ(u)Sdu=∞.Weshallflnd,seeTheorem6,adeterministicfunctionλ↦rt(λ)only
dependingonΨandΦsuchthat
rt(1~Yt)Ð→e1inlaw()
t→∞

equivalenttothefollowingproperty,(Ψ,Φ)
˜Φ(u)
processes(Yt,t≥0)and(Yt,t≥0)suchthat∫0SΨ(u)Sdu=∞,onehas
Yt
Ð→Λinlaw()
Y˜tt→∞
whereP(Λ=0)=P(Λ=∞)=(),weshallseethatnofunction
2
(η(t),t≥0)existssuchthatη(t)Ytconvergesinlawtowardsanondegeneraterandom
variable({0,∞}).ThishasbeenshownbyCohn[8]in
()willbemade
moreexplicitbyintroducingfurtherassumptionsontherateofdivergenceoftheintegral
Φ(u)Φ(u)
∫0SΨ(u)Sdu,namelyonthespeedatwhich∫εSΨ(u)Sdugoesto∞
veinasPakes,see[31],wedesignthreeregimesofdivergence:Slow(S),Log(L)andFast
(F).
theintegraldiverges,
,
2:.
inthefastcase(F)’s
result[32,Theorem2]whichcorrespondstothesubcriticalcaseundercondition(S)has
nowaproof,seeRemark14–ii),andamisprintinhisstatementiscorrected.
dp
Notation:WedenoterespectivelybyÐ→andÐ→theconvergenceinlawandthe
∼whentheratioofthetwoterms
onthetwosidesofitconvergesto1(ifanyofthetwotermsisrandom,theconvergence
holdsalmostsurely).TheprobabilitymeasureanditsexpectationaredenotedbyPandE.
Foranyx≥0,
∞
functionfinaneighbourhoodof0isdenotedby∫0f(x)dx<∞(similarly∫f(x)dx<∞
denotestheintegrabilityoffinaneighbourhoodof∞).Last,wedenotefunctions,either
deterministicorrandom,vanishinginthelimitbyo(1).


-sureconvergenceresultsinthesupercritical
,-critical
Φ(u)
casewhen∫0SΨ(u)Sdu=∞.Wedeflnethethreeregimes(S),(L),(F)
lastsectiontreatssomecriticalbranchingmechanismsfulflllingcertainregularvariation
properties.
2Preliminaries
WerecallhereafterthedeflnitionofaCBIprocessandsomeofitsmostfundamentalprop-
’sbook[29]andChapter12ofKyprianou’s
book[26].Wesaythatarandomvariableisnondegenerateifitssupportisnotcontained
in{0,∞}andproperifitisflnitealmostsurely.
Writeπandνfortwoσ-flnitenonnegativemeasureson(0,∞)satisfyingrespectively
∞2∞
∫0(z∧z)π(dz)<∞and∫0(1∧z)ν(dz)<∞.Consideratriple(σ,b,β)suchthat
σ≥0,b∈Randβ≥´evyprocess
withflnitemean(hereweassumeSΨ′(0+)S<∞,sothatinparticulartheCBIprocessdoes
notexplode)andwhosecharacteristictripleis(b,σ,π).LetΦbetheLaplaceexponentofa
subordinatorwithdriftβandL´´evy-Khinchine
formula
122∞−qu
Ψ(q)=bq+σq+S(e−1+qu)π(du),q≥0,
20
SoΨisconvex(.,Ψ′′(q)≥0,∀q≥0)withΨ(0)=,
∞−qu
Φ(q)=βq+S(1−e)ν(du),q≥0,
0
SoΦisaconcavecontinuous,strictlyincreasingfunctionwithΦ(0)=0.
ACBIprocesswithbranchingandimmigrationmechanismsΨandΦ,isastrongMarkov
process(Yt,t≥0)takingvaluesin[0,∞)whosetransitionkernelsarecharacterizedbytheir
≥0andx∈R+,
−λYt
E[et]=exp‹−xv(λ)−Φ(v(λ))ds,()
xtSs
0
wherethemapt↦vt(λ)isthesolutiontothedifferentialequation
∂
vt(λ)=−Ψ(vt(λ)),v0(λ)=λ.()
∂t
3:.
Notethatvt+s(λ)=vt(vs(λ))fromtheMarkovproperty.
ExistenceandunicityofCBIprocesseshavebeenestablishedin[23,].Re-
centlyDawsonandLi[9],seealso[22],haveshownthatanyCBIisthestrongsolutionofa
certainstochasticdifferentialequation(SDE)withjumps.
Supposethat(Ω,Ft,P)isafllteredprobabilityspacesatisfyingtheusualhypotheses.
Let{Bt}t≥0bean(Ft)-(ds,dz,du)andN1(ds,du)denotetwo
(F)-Poissonrandommeasureson(0,∞)3and(0,∞)2withintensitiesdsπ(dz)duand
t
dsν(dz).WeassumethattheBrownianmotionandthePoissonrandommeasuresare
˜0(ds,dz,du)bethecorrespondingcompensatedmeasure
ofN0,namelyN˜0(ds,dz,du)∶=N0(ds,dz,du)−dsπ(dz)
t»
Yt=Y0+σSYsdBs
0
t∞Ys−tt∞()
+SSSzN˜0(ds,dz,du)+S(β−bYs)ds+SSzN1(ds,dz).
000000
admitsauniquestrongsolutionwhoselawisthatofaCBIwithbranchingmechanismΨ
,thatistosayΦ≡0,thedrift
βandthePoissonrandommeasureN1vanishandtheprocess(Yt,t≥0)solutionto()
isacontinuous-statebranchingprocess(CBprocessforshort)withbranchingmechanism
≡0,onlytheimmigrationpartremainsand(Yt,t≥0)isasubordinatorwith
,werecallthata(sub)-criticalCB(Ψ)processconditionedon
thenon-extinctionisaCBI(Ψ,Φ)processwithΦ=Ψ′−Ψ′(0+),seeLambert[27],Li[28,
]andFittipaldiandFontbona[14].Fromnowon,unlessexplicitelymentioned,
weconsiderprocesseswithimmigration,namelyΦ(q)>0forallq>0.
RecalltheformofthefunctionΨandnoticethatb=Ψ′(0+).ACBIprocessissaidto
becritical,subcriticalorsupercriticalaccordingasb=0,b>0orb<

ρ=inf{z>0,Ψ(z)≥0},inf∅=∞.
Weseethatρ=0ifb≥0andρ>0ifb<=∞ifandonlyif−Ψisthe
(),if0<λ<ρ(>ρ),thenvt(λ)∈[λ,ρ]
isincreasing((λ)∈[ρ,λ]isdecreasing)()implies
λdz
S=t,∀t∈[0,∞),∀λ∈(0,∞)~{ρ},()
vt(λ)Ψ(z)
Recall¯vt∶=lim↑vt(λ)∈[0,∞]andset¯v∶=lim↓v¯t∈[0,∞].Greyshowsin[17]that
λ→∞t→∞
∞dq
v¯t<∞forallt>0ifandonlyifS<∞(Grey’scondition).()
Ψ(q)
Notethatρ≤v¯,andif¯v<∞then¯v=ρ.
Recall().Deflnethemapr(λ)∶=tΦ(v(λ))
t∫0s
⎧⎪⎧⎪λΦ(u)
⎪∫vt(λ)Ψ(u)duifΨ≡~0
rt(λ)=⎨()
⎪⎪⎪⎪tΦ(λ)ifΨ≡0.
⎩
Then()canalsobewrittenas
E[e−λYt]=exp(−xv(λ)−r(λ)).()
xtt
4:.
Notealsothatforanyt≥0andanyn∈N,Y=Y1+⋯+Yninlawwhere((Yi),1≤i≤n)
ttttt≥0
(Ψ,1Φ),Yhasaninflnitedivisiblelaw
nt
onR+andλ↦rt(λ)istheLaplaceexponentofasubordinator(withnokillingterm).For
∞Φ(u)
anyt≥0,wesetrt(∞)=∫v¯tΨ(u)du∈(0,∞],where¯vt∶=lim↑vt(λ),withtheconvention
λ→∞
∞Φ(u)
thatif∫Ψ(u)du=∞thenrt(∞)=∞forallt>(),weeasilycheckthat
r(∞)<∞assoonas∞Φ(u)du<∞.Lettingλtendto∞in()readilyentailsthat
t∫Ψ(u)
rt(∞)<∞ifandonlyifPx(Yt=0)>-setof
CBIsto[16].
[29]forproofsofthefollowingtechnicalstatements;seealso
[17].Wegatherinthenextlemmaanalyticalresultsonthemapλ↦vt(λ)anditsinverse
(wheneveritexists).
↦vt(λ)isstrictlyincreasingon[0,∞).Foranyt≥0,letλ↦v−t(λ)
betheinversemapofλ↦vt(λ).Thisisastrictlyincreasingfunction,well-deflnedon[0,v¯t)
whichsatisflesforalls,t≥0and0≤λ<v¯s+t
v−(s+t)(λ)=v−s(v−t(λ)).
For0≤λ<v¯t,suchthatΨ(λ)≠0,by()onehas
v−t(λ)dzv−t(λ)dz
S=S=t.()
λΨ(z)vt(v−t(λ))Ψ(z)
Inparticular,inthesupercriticalcase,∈(−∞,0),forλ∈(0,ρ)
∂v−t(λ)
=Ψ(v−t(λ)),v0(λ)=λ.()
∂t
Themapt↦v−t(λ)isdecreasingandbylettingt→∞in()weseethatv−t(λ)Ð→0.
t→∞
Moreover,v(λ)~v(λ)Ð→ebuforanyu≥0.
−(t+u)−tt→∞
Thefollowingthe