Cramer's theorem for nonnegative multivariate point

We consider a continuous time version of Cramer's theorem with nonnegative summands $ S_t=\frac{1}{t}\sum_{i:\tau_i\le t}\xi_i, t \to\infty, $ where $(\tau_i,\xi_i)_{i\ge 1}$ is a sequence of random variables such that $tS_t$ is a random process with indep

6

002 tcO 32 ]RP.tham[ 2v8527050/tham:viXraCRAMER’STHEOREMFORNONNEGATIVEMULTIVARIATEPOINTPROCESSESWITH

INDEPENDENTINCREMENTS

KLEBANERF.ANDR.LIPTSER

Abstract.WeconsideracontinuoustimeversionofCramer’stheoremwithnonnegativesummandsSt=1

n

obeystheLargeDeviationprinciple nξi,n→∞

i=1

(LDP)inthemetricspace(R+, )(fortheEuclideanmetric )withtherate1

t

i:

ξi,

τi≤t

We consider a continuous time version of Cramer's theorem with nonnegative summands $ S_t=\frac{1}{t}\sum_{i:\tau_i\le t}\xi_i, t \to\infty, $ where $(\tau_i,\xi_i)_{i\ge 1}$ is a sequence of random variables such that $tS_t$ is a random process with indep

2KLEBANERF.ANDR.LIPTSER

where(ξi,τi)i≥1isasequenceofrandompairs,whereξi’sandτi’sarerandomvariables:

ξide nedonaprobability≥0andspaceτ0=0<τ1<τ2<...τi<...

( ,F,P).Let(Gn)n≥0bethe ltrationwithG0=( , )andGn:=σ{(τi,ξi)i≤nbeidenticallydistributedandindependent}.RandomofvariablesGi 1withξiareassumedtothedistributionfunction

G(x)=P(ξ1Theconditional≤x),x≥0.

Pwhereris distributionτi≤t|Gi 1 of= τigivenGi 1is1 e r(t τi 1) exponential:

,t≥τi 1,PThefollowing apositivenumber.ξi≤x,theoremτi≤t|Gi 1isan Moreover,=G(x)analogue weassumethat1 e r(t τi 1)ofTheorem ,i≥1.

1.

Theorem2.Thefamily

S1

t=

antherate

function

t

I(u)=

sup

λu rWegivetwo λ∈( ∞,Λ)

r[1 G(0+)] ,

∞λz

0(e 1)dG(z) ,

u>0u=0.

examplesillustratingcompatibilitywithTheorems1and2.Forbothdiscreteandcontinuoustimecases,let

P(ξ1sothat,ξ≤x)=1 e x,x≥0,

1hastheLaplacetransformwithΛ=1.Thelogmomentgeneratingfunctionis

g(λ)= log(1 λ),λ<1.

ForG(x)=1 e x ,x≥0.

∞

r(eλzrλ

1)dG(z)=

r

√

We consider a continuous time version of Cramer's theorem with nonnegative summands $ S_t=\frac{1}{t}\sum_{i:\tau_i\le t}\xi_i, t \to\infty, $ where $(\tau_i,\xi_i)_{i\ge 1}$ is a sequence of random variables such that $tS_t$ is a random process with indep

CRAMERTHEOREM3

8

6

4

2

Figure1.TheratefunctionId(u)

0.80.60.40.20

Figure2.TheratefunctionIc(u)forr=1

Remark1.RelatedtopicstoTheorem2canbefounde.g.inGeorgiiandZessin,[3],servingaclassofmarkedpointrandom elds.Probably,theproofofTheorem2canbeadaptedwithargumentsfromproofsin[3]providedthatmanydetailsnotrelatedtooursettinghavetobeomittedandotheronesconcerningtotheboundarye ecthavetobeadded.

WeprefertogiveacompleteanddirectproofofTheorem2.

2.Countingrandommeasure,itscompensator,

Laplacetransform

Weconsider(ξi,τi)i≥1asamultivariate(marked)pointprocess(see,e.g.[4],[5])withthecountingmeasure

µ(dt,dy)=I{τi<∞}δ{τi,ξi}(t,y)dtdy,

i≥1

whereδ{τi,ξi}istheDiracdelta-functiononR+×R+.Parallelto

(Gn)n≥0,weintroduceonemore ltration(Gt)t≥0relatedto(ξi,τi)i≥1:

Gt:=σ(µ([0,t′]×Γ):t′≤t,Γ∈B(R+)),

We consider a continuous time version of Cramer's theorem with nonnegative summands $ S_t=\frac{1}{t}\sum_{i:\tau_i\le t}\xi_i, t \to\infty, $ where $(\tau_i,\xi_i)_{i\ge 1}$ is a sequence of random variables such that $tS_t$ is a random process with indep

4KLEBANERF.ANDR.LIPTSER

Then,theLevymeasureν(ds,dx),relatedtoµ,isexplicitlycomputed(see,e.g.TheoremIII.1.33,[5])

whereB(R+)istheBorelσ-algebraonR+,andassumethatG0isaugmentedbyP-zerosetsfromF(noticethat,then,(Gt)t≥0satis esthegeneralconditions).Withthehelpofthecountingmeasureµ,onecanpresenttStinaformofastochasticintegralwithrespecttoµ:

t tSt=xµ(ds,dx).(2.1)

x>0

dx)= IdG(x)de r(s τi 1)

ν(ds,]]τi 1,τi]](t)

i≥1

We consider a continuous time version of Cramer's theorem with nonnegative summands $ S_t=\frac{1}{t}\sum_{i:\tau_i\le t}\xi_i, t \to\infty, $ where $(\tau_i,\xi_i)_{i\ge 1}$ is a sequence of random variables such that $tS_t$ is a random process with indep

CRAMERTHEOREM5

Thiscanbewritteninanequivalentformofdi erentialequation

d(EUt)

tlim

1

→∞

t

logP3.1.Theexponentialtightness. |St u|≤δBychoosing

.= I(u).

Kj={x∈R+:x∈[0,j]}

andapplyingCherno ’sinequalitywithparameter0.5Λ,we ndthat

P(S0.5ΛtSt

t>j)≤e 0.5Λj+logEe

.

ByLemma1,

Ee

0.5ΛtSt

=e

rt

and,therefore,

x>0

[e0.5Λx 1]dG(z)

1

lim

δ→0

tlogP(St≤δ)≥ r[1 G(0+)]

tlim

1

→∞

We consider a continuous time version of Cramer's theorem with nonnegative summands $ S_t=\frac{1}{t}\sum_{i:\tau_i\le t}\xi_i, t \to\infty, $ where $(\tau_i,\xi_i)_{i\ge 1}$ is a sequence of random variables such that $tS_t$ is a random process with indep

6KLEBANERF.ANDR.LIPTSER

Itisacountingprocesswiththecompensatorν(0,t],{x>0}=r[1 G(0+)]t.Therefore,bytheWatanabetheorem,[8],πtisaPois-sonprocesswithparameterr[1 G(0+)].Hence,duetowellknownpropertyofthePoissonprocess

P(πt=0)=e tr[1 G(0+)].

We ndthat

1

logP(πt=0)= r[1 G(+)]t

implyingthelowerbondfrom(3.1).

Theupperboundfrom(3.1)isderivedwiththehelpofLaplace’stransformwith0<λ<Λ.Tothisend,weuseidentity

λx

1=EexpλtSt tr[e 1]dG(x)

x>0

implyingtheinequality

beingequivalentto

1

1≥EI{St≤δ}exptλδ r

[e

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