Large deviation estimates for exceedance times of perpetuity sequences and their dual processes

52 Pages·2016·0.46 MB·English

TheAnnalsofProbability
2016,Vol.44,No.6,3688–3739
DOI:10.1214/15-AOP1059
©InstituteofMathematicalStatistics,2016
LARGE DEVIATION ESTIMATES FOR EXCEEDANCE TIMES OF
PERPETUITY SEQUENCES AND THEIR DUAL PROCESSES1
BY DARIUSZ BURACZEWSKI∗, JEFFREY F. COLLAMORE†,
∗ ∗
EWA DAMEK AND JACEK ZIENKIEWICZ
UniversityofWrocław∗ andUniversityofCopenhagen†
In a variety of problems in pure and applied probability, it is rele-
vant to study the large exceedance probabilities of the perpetuity sequence
Yn:=B1+A1B2+···+(A1···An−1)Bn,where(Ai,Bi)⊂(0,∞)×R.
Estimatesforthestationarytaildistributionof{Yn}havebeendevelopedin
theseminalpapersofKesten[ActaMath.131(1973)207–248]andGoldie
[Ann. Appl. Probab. 1 (1991) 126–166]. Specifically, it is well known that
ifM:=supnYn,thenP{M>u}∼CMu−ξ asu→∞.Whilemuchatten-
tionhasbeenfocusedonextendingsuchestimatestomoregeneralsettings,
littleworkhasbeendevotedtounderstandingthepathbehaviorofthesepro-
cesses.Inthispaper,wederivesharpasymptoticestimatesforthenormalized
firstpassagetimeTu:=(logu)−1inf{n:Yn>u}.Webeginbyshowingthat,
conditionalon{Tu<∞},Tu→ρ asu→∞foracertainpositiveconstant
ρ. We then provide a conditional central limit theorem for {Tu}, and study
P{Tu ∈G} as u→∞ for sets G⊂[0,∞). If G⊂[0,ρ), then we show
thatP{Tu∈G}uI(G)→C(G)asu→∞foracertain largedeviation rate
function I and constant C(G). On the other hand, if G⊂(ρ,∞), then we
showthatthetailbehaviorisactuallyquitecomplexanddifferentasymptotic
regimesarepossible.Weconcludebyextendingourresultstothecorrespond-
ing forward process, understood in the sense of Letac [In Random Matri-
cesandTheirApplications(Brunswick,Maine,1984)(1986)263–273Amer.
Math.Soc.],namelytothereflectedprocessMn∗:=max{AnMn∗−1+Bn,0},
n∈Z+.UsingSiegmundduality,werelatethefirstpassagetimesof{Yn}to
thefinite-timeexceedanceprobabilitiesof{M∗},yieldinganewresultcon-
n
cerningtheconvergenceof{M∗}toitsstationarydistribution.
n
CONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3689
2. Statementofresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3692
2.1. Aclassofstochasticrecursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3692
2.2. Mainresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3694
2.3. Acomparisonwithpreviousestimatesforclassicalrandomwalks . . . . . . . . . . .3698
ReceivedNovember2014;revisedAugust2015.
1SupportedbytheNCNGrantUMO-2011/01/M/ST1/04604.
MSC2010subjectclassifications.Primary 60H25; secondary 60K05, 60F10, 60J10, 60G70,
60K25,60K35.
Keywordsandphrases.Randomrecurrenceequations,stochasticfixedpointequations,firstpas-
sagetimes,Siegmundduality,asymptoticbehavior,ruinprobabilities.
3688
LARGEDEVIATIONESTIMATESFOREXCEEDANCETIMES 3689
2.4. Exceedanceprobabilitiesforforwardrecursivesequences . . . . . . . . . . . . . . . .3701
3. ProofofTheorem2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3702
4. ProofofTheorem2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3711
4.1. Preliminaryconsiderations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3711
4.2. Theconditionalcentrallimittheoremforthecriticalcase . . . . . . . . . . . . . . . .3714
5. ProofofTheorems2.3and2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3722
5.1. ProofofTheorem2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3722
5.2. ProofofTheorem2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3731
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3737
1. Introduction. Since the pioneering work of Kesten (1973) and Vervaat
(1979),therehasbeenacontinuedinterestintheprobabilisticstudyofperpetuity
sequences.Muchofthisinteresthasbeendrivenbyawidevarietyofapplications.
Perpetuity sequences arise naturally in connection with the ARCH and GARCH
financial time series models, the Asian option in discrete and continuous time
andininsurancemathematics.[Foradetaileddescriptionofthesediverseapplica-
tions,see,forexample,Engle(1982),Bollerslev(1986),Mikosch(2003),Geman
andYor(1993),Carmona,PetitandYor(2001),Paulsen(2002),Klüppelbergand
Kostadinova (2008), and Collamore (2009).] From a theoretical perspective, per-
petuity sequences also appear in connection with the weighted branching process
andbranchingrandomwalk.Indeed,utilizinganargumentinGuivarc’h(1990)and
Liu(2000),itispossibletorelatethetailbehaviorofaperpetuitysequencetothat
of an associated nonhomogeneous recursion, leading to further applications, for
example, to the Quicksort algorithm in computer science and to Mandelbrot cas-
cades.[SeeAlsmeyerandIksanov(2009),Buraczewski(2009),Guivarc’h(1990),
Liu(2000),Buraczewskietal.(2014),andreferencestherein.]
Acentralissuearisinginalloftheseproblemsisthecharacterizationofthetail
behavioroftheperpetuitysequence.Namely,letting{(Ai,Bi):i∈Z+}beani.i.d.
sequenceofrandomvariablestakingvaluesin(0,∞)×R,andletting
(1.1) Yn=B1+A1B2+···+(A1···An−1)Bn, n=1,2,...,
thenitisofinteresttoconsider
(1.2) P{V >u} asu→∞,
where,typically,
V := lim Y or V :=supY .
n→∞ n n n
Ineithercase,itiswellknownthatundermildregularityconditions,
(1.3) P{V >u}∼Cu−ξ asu→∞
forcertainpositiveconstantsC andξ [cf.Kesten(1973),Goldie(1991)].
Much recent work has been devoted to showing that the estimate in (1.3) ex-
tends well beyond the setting of perpetuity sequences. Following Letac (1986), it
3690 BURACZEWSKI,COLLAMORE,DAMEKANDZIENKIEWICZ
ishelpfultofirstobservethat{Y }canbeidentifiedasthebackwardprocessgener-
n
D D
atedbytheaffinemap(cid:4)(x)=Ax+B,where(A,B)=(A ,B )(and=denotes
1 1
equalityindistribution).Moreprecisely,letting(cid:4)i(x)=Aix+Bi fori∈Z+,then
(1.4) Y =(cid:4) ◦···◦(cid:4) (0), n=1,2,....
n 1 n
Thelimitingbehaviorofthissequenceis,ofcourse,thesameasthatofthecorre-
spondingforwardprocess,namely
(1.5) Y∗:=(cid:4) ◦···◦(cid:4) (0), n=1,2,....
n n 1
Then it is natural to consider more general random functions, including Markov-
dependent sequences and random matrices. Extensions of this type can be also
found,forexample,inrecentworkofAlsmeyer(2003),AlsmeyerandMentemeier
(2012), Brofferio and Buraczewski (2015), Buraczewski et al. (2009), Collamore
(2009), Collamore and Vidyashankar (2013a, 2013b), Guivarc’h and Le Page
(2013a),KlüppelbergandPergamenchtchikov (2004),Enriquez,SabotandZindy
(2009),Mirek(2011)andRoitershtein(2007).Wenotethat,fortheprocess(1.5),
recursionsgeneratedbyrandommatriceswerealsoconsideredinKesten’s(1973)
original work. Moreover, some refined large deviation asymptotics for related re-
cursive structures can be found in Buraczewski et al. (2013) and Buraczewski,
DamekandZienkiewicz(2015).
Incontrast,verylittleisknownconcerningthepathpropertiesofperpetuityse-
quences.Twonaturalquestions,wellmotivatedbythetheoryofrandomwalks,are
thecharacterizationofthedistributionofthefirstpassagetimeofthesequencein
(1.4)andtheconvergenceofthesequencein(1.5)toitsstationarydistribution.In-
deed,thesetwoquestionsareverymuchthesame,sinceitisknownbyextensions
ofclassicaldualityforrandomwalksthat
(cid:2) (cid:3)
(1.6) P{Y >u, somek≤n}=P M∗>u ,
k n
where {M∗} is defined as in (1.5), but with (cid:4)(x) replaced with (cid:4)(cid:4)(x):=(Ax +
n
B)+, and Y = 0 = M∗. [Cf. Siegmund (1976), Asmussen and Sigman (1996),
0 0
andthediscussioninSection2below.]Thus,thefinite-timeexceedancesof{M∗}
n
canbeanalyzedthroughthefirstpassagetimesof{Y },andviceversa.
n
The primary objective of this article is to study the asymptotic distribution of
thescaledfirstpassagetime
1
T := inf{n:Y >u} asu→∞.
u n
logu
Motivatedbythelargedeviationtheoryforrandomwalks,developedintheclassic
papers of Donsker and Varadhan [cf. Varadhan (1984)], we study the asymptotic
behaviorof
P{T ∈G} asu→∞, whereG⊂[0,∞).
u
LARGEDEVIATIONESTIMATESFOREXCEEDANCETIMES 3691
Webeginbyshowingthat,conditionalon{T <∞},
u
T →ρ inprobability
u
for some positive constant ρ, thus describing the “most likely” first passage time
into the set (u,∞). We then characterize the asymptotic distribution of {T } on
u
the respective time intervals [0,τ], where either τ <ρ, τ =ρ, or τ >ρ, and the
analysisonthesevariousregionsturnsouttobequitedifferent.Onthefirstofthese
regions, we show that there exists a certain “rate function” I :[0,∞)→[0,∞)
suchthatasu→∞,
C(τ)(λ(α))−(cid:8)(u)
(1.7) P{T ≤τ}∼ √ u−I(τ) forallτ <ρ,
u
logu
where C(τ) is a positive constant which we explicitly identify and (cid:8)(u) :=
τlogu−(cid:11)τlogu(cid:12)∈{0,1}. [For logarithmic large deviation asymptotics describ-
ingthedecayparameterI(τ)forthiscase,seealsoNyrhinen(2001).]
Next, we examine the behavior of {T } around its central tendency, that is,
u
around the constant ρ which describes the most likely exceedance time. To this
end,weestablishaconditionalcentrallimittheorem;namely,weshowthatcondi-
tionalon{T <∞},
u
(cid:5)
(1.8) a logu(T −ρ)(cid:13)⇒Z,
u
where Z has the standard Normal distribution function (denoted (cid:4)) and a is a
certainconstant.Thus,inparticular,P{T ∈[0,ρ]}∼(cid:4)(0)P{T <∞},andhence
u u
asadirectanalogof(1.7),weobtainthat
(cid:2) (cid:3) C
(1.9) P T ∈[0,ρ] ∼ u−ξ asu→∞,
u
2
where (C,ξ) isgivenasin(1.3).Both(1.7)and(1.8)arecomparabletoclassical
estimatesforrandomwalks,asweexplaininSection2.3below.
Finally, we turn to a description of {T } for large times, when τ >ρ. Here we
u
findthatthebehaviorisquitecomplex,requiringnewmathematicaltechniques.As
wedemonstrate,theseasymptoticscanbequitedistinctfromthoseexpectedfrom
the large deviation theory of random walks, which, based on Arfwedson (1955),
Asmussen[(2000),Chapter4],andCollamore(1998),wouldsuggestthat
C(τ)
(1.10) P{τ ≤T <∞}∼ √ u−I(τ) asu→∞.
u
logu
As we show, under certain conditions, the previous formula fails to hold and we
obtain very different asymptotic behavior, not only for P{τ ≤T <∞}, but also
u
for logP{τ ≤T <∞}; thus, even the polynomial decay rate predicted by (1.10)
u
need not hold, in general. Indeed, in Theorems 2.3 and 2.4 below, we provide
asymptoticestimatesshowingthat,undercertainconditions,
(1.11) limsuplogP{τ ≤T <∞}≤−I(τ),
u
u→∞
3692 BURACZEWSKI,COLLAMORE,DAMEKANDZIENKIEWICZ
whileunderotherconditions,
(1.12) liminflogP{τ ≤T <∞}>−I(τ).
u→∞ u
Inthisway,weexhibitaninterestingasymmetrybetweenthelarge-timebehavior
andthesmall-timebehaviorof{T }.Thelatterresultsarequitetechnicalandshow
u
thatforτ >ρ,thestoryisveryinteresting,challengingandnotfullyunderstood.
Wenowturntoamoreprecisestatementofourresults.Intheprocess,wealso
connecttheconvergencein(1.7)tothatofthedualprocessof{Y }.Theproofsare
n
deferredtoSections3–5,whereweestablishourmainresultsforthethreeregimes
(τ <ρ,τ =ρ,τ >ρ)whichwehavejustdescribed.
2. Statementofresults.
2.1. Aclassofstochasticrecursions. Beforestatingourmainresults,wefirst
introducesomenotationrelatedtoourstochasticrecursionsandformulateafewof
their basic properties. Let {(A ,B ):i =1,2,...} be a sequence of i.i.d. random
i i
variablestakingvaluesin(0,∞)×R.Throughoutthepaper,weassume:
• E[logA]∈(−∞, 0)andE[log+|B|]<∞.
• For every x ∈R, P{Ax+B =x}<1, which implies, in particular, that P{B =
0}<1.
Wewillbeinterestedinthefollowingtwoprocesses:theperpetuitysequence
(cid:6)n
(2.1) Yn:=B1+ A1···Ak−1Bk, n=1,2,..., Y0=0,
k=2
and,particularly,theprocessofpartialmaximumsofthissequence,namely,
(2.2) M := max Y , n=0,1,....
n k
0≤k≤n
Thesesequencesrepresentthebackwardprocessesgeneratedbytherandommap-
pings (cid:4) (x) = A x +B and (cid:4) (x) =(A x +B ,0)+, respectively. The corre-
i i i i i i
sponding forward processes [defined in (1.5)] are Markov chains satisfying the
respectiveequations
Y∗=A Y∗ +B ,
n n n−1 n
(2.3) (cid:7) (cid:8)
M∗= A M∗ +B +.
n n n−1 n
If E[logA]<0 andE [log+|B|]<∞, then it is well known that {Y } converges
n
pointwiseto
(cid:6)∞
Y =B1+ A1···Ak−1Bk,
k=2
LARGEDEVIATIONESTIMATESFOREXCEEDANCETIMES 3693
while{M }convergesa.s.to
n
M =supY ,
n
n≥0
whereY andM arefinitea.s.ThenY andM arecalledstationarysolutions,since
theysatisfythestochasticfixedpointequations
D
(2.4) Y =AY +B, Y independentof(A,B);
(2.5) M =D (AM +B)+, M independentof(A,B).
In this paper, our objective will be to describe the path behavior of {Y } and
n
{M },and,inthisconnection,itwillbeofinteresttocomparethelimitingquanti-
n
ties we obtain to the tail behavior of Y and M. To this end, define the generating
functions
(cid:9) (cid:10)
λ(α)=E Aα , (cid:9)(α)=logλ(α), α∈R;
(cid:9) (cid:10)
λ (α)=E |B|α , (cid:9) (α)=logλ (α), α∈R.
B B B
Notebytheconvexityof(cid:9)and(cid:9) that,if(cid:9)(α)<∞and(cid:9) (α)<∞forsome
B B
α>0,then(cid:9)(β)and(cid:9) (β)arefiniteforeveryβ ∈(0,α).Moreover,thesefunc-
B
tionsareinfinitelydifferentiableontheinteriorsoftheirrespectivedomains.
Wewillusesomefundamentalpropertiesofthesolutionstothestochasticequa-
tions (2.4). First, if (cid:9)(α)< 0 and (cid:9) (α) <∞ for some α >0, then their αth
B
momentsmustbefinite,namely,
(cid:9) (cid:10) (cid:9) (cid:10)
(2.6) E |Y|α <∞ and E Mα <∞;
see Vervaat (1979). Next, to describe the tail behavior of Y and M, we focus on
thenonzerosolution,ξ,totheequation(cid:9)(ξ)=0.Moreprecisely,assumethatfor
someξ >0,
(cid:9)(ξ)=0, (cid:9)(cid:16)(ξ)<∞ and (cid:9) (ξ)<∞.
B
Then,iftherandomvariablelogAisnonarithmetic,itiswellknownthatthetails
ofY andM areregularlyvaryingwithindexξ;thatis,
P{Y >u}∼C u−ξ asu→∞;
Y
(2.7)
P{M >u}∼C u−ξ asu→∞;
M
see Goldie (1991). Various explicit expressions for the constants C and C are
Y M
alsoavailable;seeRemark2.2below.
3694 BURACZEWSKI,COLLAMORE,DAMEKANDZIENKIEWICZ
2.2. Main results. Let {Y } denote the perpetuity sequence defined in (2.1),
n
andlet
1
(2.8) T := inf{n:Y >u}
u n
logu
denotethescaledfirstpassagetimeof {Y } intotheset (u,∞).Thenourprimary
n
objective is to study the asymptotic decay, as u→∞, of P{T ∈G}, where G⊂
u
[0,∞).Wewillshowthatthisprobabilitydecaysatapolynomialrate,andprovide
sharpasymptoticestimatesdescribingthisrateofdecay.
Set
(cid:11)
(2.9) μ(α)=(cid:9)(cid:16)(α) and σ(α)= (cid:9)(cid:16)(cid:16)(α).
To characterize the behavior of {T } as u→∞, it is helpful to first observe that,
u
conditionalontheeventofruin,therandomvariableT convergesinprobabilityto
u
ρ=(μ(ξ))−1,whereξ isgivenasin(2.7).Thisconstantρ willplayanimportant
roleinthesequel.
LEMMA 2.1. Assumethereexistsavalueξ >0suchthat(cid:9)(ξ)=0,andsup-
pose that (cid:9) and (cid:9) are finite in a neighborhood of ξ and the law of logA is
B
non-lattice.Setρ=(μ(ξ))−1.Thenforanyε>0,
(cid:2) (cid:3)
(2.10) P T ∈/(ρ−ε,ρ+ε)|T <∞ →0 asu→∞.
u u
Lemma2.1willfollowasadirectconsequenceofastrongerresult,Lemma4.3,
whichwillbeprovedinSection4.
Turning now to our main results, we first introduce the rate function which we
willusetodescribethepolynomialratesofdecay.Recallthattheconvexconjugate
(orFenchel–Legendretransform)ofthefunction(cid:9)isdefinedby
(cid:2) (cid:3)
(cid:9)∗(x)=sup αx−(cid:9)(α) , x∈R.
α∈R
Next,define
(cid:12) (cid:13)
1
(2.11) I(τ)=τ(cid:9)∗ , τ >0, I(0)=∞.
τ
This rate function appears in the large deviation study for random walks, and it
is closely related to the “support function” in convex analysis, whose properties
are well known [see Rockafellar (1970), Chapter 13]. Various convexity proper-
ties of the function I(·) itself (developed primarily for random walk in higher
dimensions) can be found in Collamore (1998), Section 3. Note that if we set
τ =(μ(α))−1 forsomeα∈dom(μ)(thedomainofμ),thenitfollowsthat
(cid:9)(α)
(2.12) I(τ)=α− ;
μ(α)
LARGEDEVIATIONESTIMATESFOREXCEEDANCETIMES 3695
cf.DemboandZeitouni(1993),page28.
We turn now to the characterization of P{T ∈[0,τ]} when τ <ρ. Recall that
u
the function (cid:9) is differentiable on the interior of its domain. Moreover, if (cid:9) is
also essentially smooth (namely, if we further assume that |(cid:9)(cid:16)(α )|↑∞ for any
i
{α } ⊂ int(dom(cid:9)) whose limit lies on the boundary of dom(cid:9)), then it is well
i
known that (cid:9)(cid:16) maps R onto the entire real line. Thus, in this case, there always
existsapointα(τ)satisfyingtheequation
(cid:7) (cid:8) 1
(2.13) μ α(τ) = .
τ
More generally, if τ−1 lies in the interior of the domain of (cid:9)∗, then a solution
α(τ)existsin(2.13);cf.Ellis(1984),TheoremVI.5.7;Rockafellar(1970),Theo-
rem23.5.
Thus, the assumption of a solution to (2.13) is a very weak condition, which
also appears to be necessary. In particular, when there fails to be a solution, one
usuallyexpectstoobtainonlylogarithmiclargedeviationasymptoticsratherthan
thesharpasymptoticswhicharethefocusofthispaper.
Themostimportantsolutionto(2.13)appears,forourpurposes,whenwetake
τ =ρ, where ρ is given as in the previous lemma. Then by definition of ρ, we
haveα(ρ)=ξ.Thenτ ∈(0,ρ)ifandonlyifα(τ)>ξ,whichisthesettingofour
firstmainresult.
THEOREM 2.1. Let τ ∈ (0,ρ) and suppose that there exists a point α ≡
α(τ) ∈ R such that (2.13) holds. Assume that (cid:9) and (cid:9) are finite in a neigh-
B
borhoodofα andthelawoflogAisnon-lattice.Then
C(τ)(λ(α))−(cid:8)(u) (cid:7) (cid:8)
(2.14) P{T ≤τ}= √ u−I(τ) 1+o(1) asu→∞,
u
logu
where(cid:8)(u):=τlogu−(cid:11)τlogu(cid:12).Moreover,
(cid:12) (cid:13)
(cid:2) (cid:3) u−I(τ)
(2.15) P T ≤τ −L (u) =o √ asu→∞,
u τ
logu
whereL (u)={clog(logu)}/loguandc≥{2(α+1)}/(cid:9)(α).TheconstantC(τ)
τ
isgivenby
1 1 (cid:9) (cid:10)
(2.16) C(τ)= √ lim E Mα ∈[0,∞).
ασ(α) 2πτ n→∞λn(α) n
Moreover,ifP{A>1,B >0}>0thenC(τ)>0.
Notethat(2.15)shows,heuristically,thatthecriticalevent{Y >u}occursnear
n
theendofthetimeinterval[0,τlogu].
Next, we turn to the behavior around the critical case, where we would take
τ =ρ intheprevioustheorem.Inthefollowingresult,weestablishamoreprecise
estimate, which yields a conditional central limit theorem for the normalized first
passagetime.
3696 BURACZEWSKI,COLLAMORE,DAMEKANDZIENKIEWICZ
THEOREM 2.2. Suppose that there exists a value ξ >0 such that (cid:9)(ξ)=0.
Also,assumethat(cid:9)and(cid:9) arefiniteinaneighborhoodofξ andthelawoflogA
B
isnon-lattice.Thenforanyy∈R,
(cid:2) (cid:3) (cid:7) (cid:8)
(2.17) P T ≤ρ+a(logu)−1/2y =C u−ξ(cid:4)(y) 1+o(1) asu→∞,
u M
where (cid:4) is the standard Normal distribution function, a =ρ3/2σ(ξ), and where
C ∈[0,∞)isgivenasin(2.7).Moreover,ifP{A>1,B >0}>0,thenC >0.
M M
REMARK 2.1. Itfollowsfrom(2.17)and(2.7)that
(cid:2) (cid:3) (cid:7) (cid:8)
(2.18) P T ≤ρ+a(logu)−1/2y|T <∞ =(cid:4)(y) 1+o(1) ;
u u
√
consequently, conditional on {T <∞}, we have that a logu(T −ρ)(cid:13)⇒Z,
u u
whereZ ∼Normal(0,1).
REMARK 2.2. Using Goldie’s (1991) original characterization, the constant
C inTheorem2.2canbeexpressedas
M
1 (cid:9)(cid:7) (cid:8) (cid:10)
(2.19) C = E (AM +B)+ ξ −(AM)ξ .
M
ξμ(ξ)
Recently, certain more explicit representation formulas have been derived for
C and C in (2.7); see Enriquez, Sabot and Zindy (2009) and Collamore
M Y
and Vidyashankar (2013b). The main representation formula in Collamore and
Vidyashankar(2013b)statesthat,underaweakcontinuityassumptiononlogA,
(cid:14)(cid:12) (cid:13) (cid:15)
1 B B ξ
(2.20) CM = ξμ(ξ)E[τ]Eξ V0+ A1 + A A2 +··· 1{τ=∞} ,
1 1 2
whereE [·]denotesexpectationintheξ-shiftedmeasure(definedformallyinSec-
ξ
tion 3 below), τ −1 is the first regeneration time of the forward process {M∗} in
n
(2.3), and M∗ is chosen such that M∗ =D M∗. Specifically, if P{B <0}>0, then
0 0 τ
τ−1canbetakentobethereturntimeof{M∗}totheorigin.Inparticular,thepos-
n
itivityofC followsreadilyundertheseconditionsfrom(2.20).Moreover,under
M
the weaker requirements of Collamore and Vidyashankar (2013b), Theorem 2.2,
together with the additional assumption that {M∗} is ψ-irreducible (which is im-
n
plicitly assumed in Section 9 of that article), one obtains (2.20) for the k-chain
{M∗ :n=1,2,...},aswellasthealternativerepresentation
kn
1 1 (cid:9) (cid:10)
(2.21) C = lim E Mξ ,
M ξμ(ξ)n→∞n n
whichisreadilyseentohaveacloselyrelatedformto(2.16).
Finally, we turn to the case where τ >ρ. Interestingly, in this case, we do not
obtain a complete analog of Theorem 2.1. Indeed, counterexamples can be con-
structed where the asymptotics differ from those one might expect from the large
LARGEDEVIATIONESTIMATESFOREXCEEDANCETIMES 3697
deviationtheoryforrandomwalks,asdescribedin(1.10).Forτ >ρ,thecondition
that appears to lead to these counterexamples is that E[logA]>(cid:9)(α(τ)). In this
case, the true probability may decay at a slower polynomial rate than I(τ). More
precisely,withinaratherflexibleclassofprocesseswithE[logA]>(cid:9)(α(τ)),we
havethat
(2.22) P{τ ≤T <∞}≥D u−I(τ)+δ forsufficientlylargeu.
u 0
On the other hand, under different hypotheses which, in particular, imply
E[logA]<(cid:9)(α(τ)),wehavethat
D
(2.23) P{τ ≤T <∞}≤ √ 1 u−I(τ) forsufficientlylargeu.
u
logu
Thus,onecannotexpectadirectanalogofTheorems2.1and2.2here,andournext
theoremprovides,ineffect,asourceofcounterexamplestothenaturalconjecture
suggestedby(1.10).
THEOREM 2.3. Let τ ∈ (ρ,∞), and suppose that there exists a point α ≡
α(τ)∈int(dom(cid:9))suchthat(2.13)holdsand
(2.24) μ(0)=E[logA]>(cid:9)(α).
Moreover, assume that B =1 a.s. and the law of A has a strictly positive con-
tinuous density on R. Then there exist positive constants D and δ such that, for
0
sufficientlylargeu,
(2.25) P{Ynu−1≤uandYnu >u}≥D0u−I(τ)+δ, nu=(cid:11)τlogu(cid:12).
REMARK 2.3. Since the construction in the theorem is quite involved, we
have restricted our attention to the case B =1; however, the theorem can also be
establishedundertheweakerassumptionthat B >0 a.s.Formoredetails,seethe
discussioninSection5.1followingtheproofofthetheorem.
While the previous theorem leads essentially to a negative conclusion, we also
havethefollowingcomplementaryresult.
THEOREM 2.4. Let τ ∈ (ρ,∞), and suppose that there exists a point α ≡
α(τ)∈int(dom(cid:9))suchthat(2.13)holdsand
(2.26) (cid:9)(β)<(cid:9)(α) forsomeβ <min{1,α}.
AssumethatB >0a.s.andλ (−α)<∞,andassumethatthelawof(A,B)has
B
compactsupportandAhasaboundeddensity.ThenthereexistfiniteconstantsD
andU suchthat,forallu≥U,
D(cid:16)k
(2.27) P{Ynu+k−1≤uandYnu+k >u}≤ √loguu−I(τ), nu=(cid:11)τlogu(cid:12),