Pseudo-regularly varying functions and generalized renewal processes

496 Pages·2018·2.538 MB·English

by  Buldygin V.V

Probability Theory and Stochastic Modelling 91
Valeriĭ V. Buldygin
Karl-Heinz Indlekofer 
Oleg I. Klesov 
Josef G. Steinebach
Pseudo-Regularly
Varying Functions
and Generalized
Renewal Processes
Probability Theory and Stochastic Modelling
Volume 91
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Valeri˘ı V. Buldygin (cid:129) Karl-Heinz Indlekofer (cid:129)
Oleg I. Klesov (cid:129) Josef G. Steinebach
Pseudo-Regularly Varying
Functions and Generalized
Renewal Processes
123
Valeri˘ıV.Buldygin Karl-HeinzIndlekofer
DepartmentofMathematicalAnalysis DepartmentofMathematics
NationalTechnicalUniversityofUkraine UniversityofPaderborn
Kyiv,Ukraine Paderborn,Germany
OlegI.Klesov JosefG.Steinebach
DepartmentofMathematicalAnalysis MathematicalInstitute
andProbabilityTheory UniversityofCologne
NationalTechnicalUniversityofUkraine Cologne,Germany
Kyiv,Ukraine
ISSN2199-3130 ISSN2199-3149 (electronic)
ProbabilityTheoryandStochasticModelling
ISBN978-3-319-99536-6 ISBN978-3-319-99537-3 (eBook)
https://doi.org/10.1007/978-3-319-99537-3
LibraryofCongressControlNumber:2018956309
MathematicsSubjectClassification(2010):60K05,60F15,26A12,60H10,60F10,60E07
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Preface
Renewal theory is a branch of probabilitytheory rich in fascinating mathematical
problems and also in various important applications. On the other hand, regular
variation of functions is a property that plays a key role in many fields of
mathematics. One of the main aims of this book is to exhibit some fruitful links
betweenthesetwoareasviaageneralizedapproachtobothofthem.
The core of renewal theory is to study (so-called) renewal processes and their
probabilistic and statistical characteristics. One of the most cited examples in
renewaltheoryisthefollowing,whichdealswiththelifespanofalightbulb.
RenewalProcess Assumeonehasalightbulbinaroomandoneturnsiton,keeping
thelightbulbworkinguntilitfails,afterwhichitisreplacedwithanewlightbulb.
Ifξ denotesthelifespanoftheithbulb,thenS =ξ +···+ξ representsthetotal
i n 1 n
lifespanofthefirstnbulbsand
N =max{n:S ≤t} (1)
t n
isthenumberofbulbsneededuntiltimet.{N }iscalledarenewalcountingprocess
t
constructedfromthesequence{S }.
n
Ofcourse,thebulbsinthisexamplecanbeexchangedbyanyotherexpendable
resource,whichmakestheexamplemorerealisticandmoreattractivefromthepoint
ofviewofapplications.Forexample,considerarrivingcustomerswaitinginaqueue
untiloneoftheserversisfreetoservehim/her.Thearrivalcountingprocesstothe
queueisusuallyassumedtobearenewalcountingprocess{N },wherethe{ξ }are
t i
theinter-arrivaltimesbetweenthecustomers.
Nextwegivesomefurtherexamplesofstochasticprocesseswhicharerelatedto
renewalprocessesandwhichoccurbothinpureandappliedmathematics.
Reward Process Let some (random) event occur from time to time. Imagine that
someoneexperiencesarewardateachoccurrenceoftheevent.Letr bethereward
i
earnedatthetimeoftheithoccurrenceoftheevent.DenotingbyN thenumberof
t
occurrencesoftheeventuptotimet,weagainseearenewalprocesswithξ being
i
v
vi Preface
thetimebetweenthe(i−1)thandithoccurrenceoftheevent.Then
(cid:2)Nt
Y = r
t i
i=1
iscalledarewardprocess,whichisaspecialcaseofa(so-called)compoundrenewal
process.
Treating the r ’s as penalties rather than rewards we arrive at several other
i
settings. One of the classical examples here arises in a risk model describing the
evolutionofthecapitalofaninsurancecompanywhichexperiencestwoopposing
cash flows: incoming cash premiums and outgoing claims. Premiums from the
customersmayarriveataconstantratec>0(say)andclaims{r }occuraccording
i
toacountingprocess{N },thatis,N isthenumberofclaimsuptotimet.So,for
t t
aninsurerwhostartswithinitialsurplusx,
(cid:2)Nt
X =x+ct− r , t ≥0,
t i
i=1
representsthecapitalattimet.
General Counting Process There are many situations similar to what has been
describedabove.Let,forexample,ξ bethetimepassedbetweenthe(i−1)thand
i
ithrecordinasportdiscipline.ThenN ,again,denotesthenumberofrecordsupto
t
timet andis,ingeneral,calledacountingprocess.Recordscanalsohaveanegative
meaning,forexamplelossescausedbycatastrophiceventssuchashurricanes.
Such a counting process can also be seen in pure mathematics. Let (say) S
n
be the nth prime number. Then N (usually denoted by π(t) in this case) plays a
t
crucial role not only in number theory, but also in applications like cryptography
whenestimatingthesecurityofprotocolsusedtotransmitdata.Anexamplewhere
arenewalprocessexplicitlyoccursincryptomachinesisdiscussedinGut[159].
Thecentralobjectofalltheabovemodelscanbereducedtoaninvestigationof
thebehaviorofanunderlyingcountingprocess{N }.Inprobabilitytheory,thiscan
t
effectivelybedoneundertheclassicalassumptionsthatthe{ξ }are
i
(a) independent,
(b) identicallydistributed,
(c) positive
random variables. But what about the asymptotic behavior of N in cases where
t
these assumptions fail? Such situations are not rare at all. Imagine, for example,
that a catastrophic event occurs and leads to many claims for damages. Then the
times{ξ }betweentheclaimsbecomedependent,sincemanyofthepolicyholders
i
willreporttoaninsurancecompanyat(roughly)thesametime.Anotherexample,
inwhichAssumption(b)fails,isrelatedtotheso-calledalternatingrenewalprocess
Preface vii
{N }constructedfromasequence
t
S =ξ , S =ξ +η , S =ξ +η +ξ , S =ξ +η +ξ +η ,...,
1 1 2 1 1 3 1 1 2 4 1 1 2 2
wherethe{ξ }areindependentrandomvariableswithdistributionfunctionF and
i ξ
the{η }areindependentrandomvariableswithdistributionfunctionF .Thiskind
i η
of renewalprocessoccursin chromatographicmodelsor in describingthe motion
ofwaterinariver(see,e.g.,Gut[159]forashortdiscussionandfurtherreferences).
Finally, Assumption (c) may fail if, for example, {ξ } describes a money flow
i
including both expenses and incomes, so that ξ could attain both positive and
i
negativevalues.
Formula (1) nevertheless defines a process {N }, called a generalized renewal
t
processinthiscase,evenifoneoftheAssumptions(a)–(c)fails.Moreover,onecan
introduce and study some other important functionals of {S } here. For example,
n
viewing N + 1 defined via (1) as the first time when {S } exceeds the level t,
t n
one can also considerL , the last passage time across the levelt. There are other
t
naturalfunctionals,e.g.,T ,thetotaltimespentby{S }belowthelevelt,etc.The
t n
just mentioned three functionals coincide if Assumption (c) holds, but otherwise
they are different. All these functionals as well as many others are also called
generalizedrenewalprocesses.So,generalizedrenewalprocessesariseeitherifN
t
isconstructedby(1),butoneoftheproperties(a)–(c)fails,orifotherfunctionalsof
{S }(similartoN )areconsidered(weshallspecifylaterwhatkindoffunctionals
n t
wehaveinmindwhenstudyinggeneralizedrenewalprocesses).
One of the basic questions studied in all models including renewal processes
concerns the asymptotic behavior of N as t → ∞ which, in turn, is determined
t
by the asymptotic behavior of S as n → ∞ (we shall clarify what we mean
n
by “asymptotic behavior” later in this book). Therefore one may expect that the
asymptotic properties of the models for (generalized) renewal processes can be
derivedfromtheasymptoticbehaviorof{S }.Inturn,itisalsotobeexpectedthat
n
onecanargueviceversa,i.e.,thattheasymptoticbehaviorof{S }isdeterminedby
n
thatof{N }.Inthiscase,onewouldbeabletoderivestatisticalpropertiesoftheξ ’s
t i
byobservingthecountingprocess{N }.So,{S }and{N }maybeviewedas“dual
t n t
objects”inacertainsense.
Generally,we call two objectsdualif their asymptoticpropertiesare relatedto
each otherasindicatedabove,thatis, if a limit resultforthe first objectimpliesa
correspondingone for the second object, and vice versa. The duality of {N } and
t
{S } has been proved in Gut et al. [160] for the classical setting where all the
n
Assumptions (a)–(c) hold. We would like to mention that Doob [111] predicted
thisdualitypropertyin1948,butuntilveryrecentlythelimittheoremsforrenewal
processesandthoseforsumsofrandomvariablesdevelopedindependently.
Counting processes and their dual (or “renewal”) processes are often observed
in number theory. For example, the total number π(x) of prime numbers up to x
and the n-th prime numberp are dual objects, since π(p ) = n. This duality is
n n
reflectedbytheprimenumbertheoremstatingthatπ(x)∼x/ln(x)asx →∞and
viii Preface
the asymptotic behavior p ∼ nln(n) as n → ∞; note that nln(n) is inverse to
n
x/ln(x)inacertainsense.
Probability theoryprovidesother examplesof dualobjects. One of them is the
number μ of records until time n in a sequence of random variables and the
n
magnitudeτ ofthen-threcord.Obviouslyμ = nandthisdualityallowsoneto
n τn
studytheasymptoticbehaviorofμ andτ simultaneously.Bytheway,thesecond
n n
property of inverse functions fails in both cases since, in general, τ (cid:7)= n and
μn
τ (cid:7)=n.
π(n)
Another example is the duality between the tail F = 1 − F of a distribution
functionF concentratedon the nonnegativehalf-line and its quantile function τ .
q
Here, if F is continuousand increasing, then both propertiesof inverse functions
hold,i.e.,F(τ )=q andτ =x.Ontheotherhand,ifF iseitherdiscontinuous
q F(x)
or non-increasing, then each of these may fail. Nevertheless, one can derive the
asymptoticbehaviorof oneofthese dualobjects,either F(x)as x → ∞ orτ as
q
q →0,fromtheotherone.
AviolationofanyoftheAssumptions(a)–(c)meansingeneralthattheduality
between{N } and{S } disappears.Since dualitypropertiesare importantin many
t n
situations, one main aim of this monograph is to study conditions under which
dualityretains,moreprecisely,underwhichanalmostsureconvergenceofS /a(n)
n
asn → ∞ resultsin an almostsure convergenceofN(t)/a−1(t) ast → ∞, and
viceversa,wherea(·)isasuitablenormalizingfunctionwithinversea−1.
InKlesovetal.[227],forexample,suchdualitieshavebeenprovedinageneral
setting andit becameclearthatthis couldbe donenotonlyforN definedby(1),
t
butalsoforalargeclassofgeneralizedrenewalprocessesincludingthefunctionals
L andT .ThekeyobservationinKlesovetal.[227]wasthat,inordertopreserve
t t
duality,theinversefunctiona−1shouldsatisfyatechnicalcondition,namely
(cid:3) (cid:3)
limlimsup(cid:3)(cid:3)(cid:3)a−1((1±ε)t) −1(cid:3)(cid:3)(cid:3)=0. (2)
ε↓0 t→∞ a−1(t)
Property(2) is satisfied, forexample,if a is a regularlyvarying (RV) function(in
theKaramatasense)withnonzeroindex.
Now, considering (2) as the defining property of a more general class PRV
of functions, called pseudo-regularlyvarying (PRV) functions,and developingits
characteristicsfurther,itturnsoutthatthetheoryofdualobjectscanbeextended.It
willbeshownthatmanypropertiesofPRVfunctionsremainthesameasintheRV
case,including,e.g.,anintegralrepresentationanduniformconvergenceproperties.
Having then developeda theory of PRV functionsone is able to study the duality
of objects in a unified manner, under which the classical setting correspondsto a
particular,stillveryimportantexample.
Thestructureofthebookhascertainlybeeninfluencedbythedirectionwehave
takeninourinvestigationsofPRVfunctions,namely,wefirstobtainedsomebasic
applicationsto classical limit theorems in renewaltheory,then we understoodthe
importance of the PRV property (2) in this field and discovered its central role
for various other limit theorems, and finally we investigated a number of further
Preface ix
applications in different fields. Before we now briefly describe the contents of
the book, we would like to mention that the asymptotics for generalized renewal
processes studied here are essentially based on the fact that the objects under
consideration are inverse to each other (in a certain sense), e.g., S ≈ t and
Nt
N(S )≈n,where“≈”hastobegivenaprecisemeaning,ofcourse.
n
We should mention that we have not aimed to touch all the interesting aspects
and important applications of regular variation theory. Our scope is to add some
new material to the various monographs and texts which are already devoted to
thesetopics(see,forexample,Binghametal.[41]foracomprehensivediscussion
of both the theory and applications of regular variation). For the development
of the theory see also the important contributions of Seneta [324] and Geluk
and de Haan [148]. An excellent presentation of Tauberian theorems which is
heavily related to regularly varying functions has been given in Korevaar [237].
Resnick[300,302],deHaanandFerreira[167],andBorovkovandBorovkov[48]
discussvariousapplicationsoftheconceptofregularvariationinprobabilitytheory.
A role of regularly varying functions for actuarial and finance mathematics is
highlighted in Embrechts et al. [119] and Novak [285]. For recent texts dealing
with applications to statistical problems and long range dependence see, e.g.,
Mikosch[272],Samorodnitsky[316],Solier[340]orPipirasandTaqqu[292](see
also the interesting discussion in de Haan [166], where applications to currency
exchangerates,lifespanestimation,andsealeveldataaregiven).
In Chap. 1, we first assume the classical setting and study equivalences in the
strong law of large numbers and the law of the iterated logarithm for sums of
independent,identicallydistributed(iid)randomvariablesandtheircorresponding
renewal processes. As mentioned above, the proofs are essentially based on the
property that the renewal process N = {N } is the generalized inverse function
t
constructed from the sequence S = {S } of partial sums. Not only equivalence
n
statementscanbe provedinthe case ofsumsofiid randomvariables,butalso the
correspondingmomentconditionscanbederivedfromtheircounterpartsforpartial
sums.Somenontraditionallimitresultsforsumsarealsostudiedinthischapter.
Chapter 2 is a continuation of Chap. 1 in a more general setting, where much
less is knownaboutthe limit propertiesof the underlyingsequences. The random
variablesstudiedinthischapterareneithernecessarilyindependent,noridentically
distributed, nor nonnegative. We propose some new approaches to derive limit
results for generalized renewal processes from their counterparts holding for the
underlyingsequences.Notethatseveraldefinitionsofgeneralizedrenewalprocesses
are introducedin this chapter,where each of them reflects a certain feature of the
classical definitions.Amongthe generalizedrenewalprocessesstudied in Chap. 2
arethefirstexittime,lastexittime,andsojourntime.
Chapters 3–7providethe function-theoreticfoundationsof the book,while the
other chapters are devoted to applications and some necessary complements. The
followingthree classes of functionsplay a key role whenstudyingdualobjectsin
thismonograph,namelyPRV,theclassofpseudoregularlyvaryingfunctions,SQI,
theclassofsufficientlyquicklyincreasingfunctions,andPOV,theclassofpseudo-
O-varying functions.In fact, all three classes appear as naturalgeneralizationsof