Compound Renewal Processes (Encyclopedia of Mathematics and its Applications, Series Number 184)

381 Pages·2022·2.611 MB·English

by  A. A. Borovkov

COMPOUND RENEWAL PROCESSES
Compoundrenewalprocesses(CRPs)areamongthemostubiquitousmodelsused
inapplicationsofprobability.Atthesametime,theyareanaturalgeneralizationof
randomwalks,themostwell-studiedclassicalobjectsinprobabilitytheory.This
monograph,writtenforresearchersandgraduatestudents,presentsthegeneral
asymptotictheoryandgeneralizesmanywell-knownresultsconcerningrandom
walks.ThebookcontainsthekeylimittheoremsforCRPs,functionallimit
theorems,integro-locallimittheorems,largeandmoderatelylargedeviation
principlesforCRPsinthestatespaceandinthespaceoftrajectories,including
largedeviationprinciplesinboundarycrossingproblemsforCRPs,withanexplicit
formoftheratefunctionals,andanextensionoftheinvarianceprincipleforCRPsto
thedomainofmoderatelylargeandsmalldeviations.Applicationsestablishthekey
limitlawsforMarkovadditiveprocesses,includinglimittheoremsinthedomainsof
normalandlargedeviations.
EncyclopediaofMathematicsandItsApplications
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Encyclopedia of Mathematics and Its Applications
Compound Renewal Processes
A. A. BOROVKOV
SobolevInstituteofMathematics,Russia
Translatedby
ALEXEY ALIMOV
SteklovInstituteofMathematics,Moscow
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Informationonthistitle:www.cambridge.org/9781009098441
DOI:10.1017/9781009093965
©A.A.Borovkov2022
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Firstpublished2022
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Names:Borovkov,A.A.(AleksandrAlekseevich),1931-author.|
Alimov,Alexey,translator.
Title:Compoundrenewalprocesses/A.A.Borovkov;translatedbyAlexeyAlimov.
Description:Cambridge;NewYork:CambridgeUniversityPress,2022.|
Series:Encyclopediaofmathematicsanditsapplications|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2021054195(print)|LCCN2021054196(ebook)|
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Contents
Introduction pagexi
1 MainLimitLawsintheNormalDeviationZone 1
1.1 PreliminaryResults 1
1.1.1 ConvergenceofDistributionsandMomentsofSome
FunctionalsofCPRs 1
1.1.2 CRPswithStationaryIncrements 5
1.1.3 StrongLawofLargeNumbersforaSimpleRenewal
Processη(t) 7
1.1.4 Almost Sure Convergence of Some Functionals of
CRPs 7
1.2 FirstMomentsoftheProcesses Z(t) andY(t).StrongLaws
ofLargeNumbers 10
1.2.1 AsymptoticsforFirst-andSecond-OrderMoments
of Z(t)andY(t) 10
1.2.2 StrongLawsofLargeNumbers 13
1.3 CentralLimitTheoremandtheLawoftheIteratedLogarithm 13
1.3.1 Anscombe’sTheorem 13
1.3.2 CentralLimitTheorem 14
1.3.3 LawoftheIteratedLogarithm 16
1.4 Convergence to a Stable Law. Analog of the Law of the
IteratedLogarithm 17
1.4.1 ConvergencetoaStableLaw 17
1.4.2 AnalogoftheLawoftheIteratedLogarithm 19
1.5 InvariancePrinciple 21
1.5.1 Introduction 21
1.5.2 Analog of Anscombe’s Theorem in the Case of
ConvergencetoaContinuousProcess 22
1.5.3 InvariancePrincipleforCompoundRenewalProcesses 24
vi Contents
1.6 Convergence ofNormalizedCompoundRenewalProcesses
toStableProcessesintheCasewhenξ hasInfiniteVariance 28
1.6.1 S-ConvergencetoStableProcesses 28
1.6.2 AbsenceofS-ConvergencewithoutCondition(1.6.5) 32
1.6.3 D-ConvergencetoStableProcesses 33
1.7 Limit Theorems for the First Passage Time of an Arbitrary
BoundarybyaCompoundRenewalProcess 38
1.7.1 Introduction 38
1.7.2 CaseofFiniteVariance 39
1.7.3 CaseofInfiniteVariance 44
1.8 MainLimitLawsforMarkovAdditiveProcesses(forSums
ofRandomVariablesDefinedonStatesofaMarkovChain) 49
1.8.1 ErgodicTheoremsforHarrisMarkovChains 49
1.8.2 MarkovAdditiveProcess 51
1.8.3 MainLimitLawsforMarkovAdditiveProcesses 53
2 Integro-LocalLimitTheoremsintheNormalDeviationZone 57
2.1 Integro-LocalLimitTheoremsintheCaseofIndependentor
LinearlyDependentτandζ 57
2.1.1 Integro-LocalTheoremforRandomWalks 58
2.1.2 Integro-LocalTheoremsforHomogeneousCRPsin
theCaseofIndependentorLinearlyDependentτandζ 60
2.2 Refinement of Stone’s Integro-Local Theorem for Random
Walks 67
2.3 Integro-Local Theorems for Compound Renewal Processes
intheGeneralCase 74
2.4 ExtensionofResultstotheInhomogeneousCase 84
2.5 Integro-LocalTheoremsforMarkovAdditiveProcesses 87
3 LargeDeviationPrinciplesforCompoundRenewalProcesses 90
3.1 Introduction 90
3.2 RelationshipbetweenCompoundRenewalProcessesandthe
RenewalMeasure.DeviationFunctionfortheRenewalMeasure 93
3.2.1 RenewalMeasuresandCRPs 93
3.2.2 Asymptotics for the Renewal Measure and the
CorrespondingDeviationFunction 95
3.2.3 PreliminaryVersionoftheLocalLDPforCRPs 98
3.3 Deviation Functions for the Renewal Measure and for
CompoundRenewalProcesses 99
3.3.1 Properties of the Function D(t,α) and of the
DeviationFunctionsforCRPs 99
3.4 LargeDeviationPrinciplesfor Z(T) 105
3.4.1 TheGeneralCase 105
Contents vii
3.4.2 HomogeneousProcessesandProcesseswithStation-
aryIncrements 109
(cid:2) (cid:3)
3.4.3 LDPfortheProcess Z(t),γ(t) andItsConsequences 111
3.5 FundamentalFunctionsandTheirProperties.FurtherProper-
(cid:2)
tiesoftheDeviationFunctionD(α),D(α).OntheCondition
λ+ < D(0) 112
3.5.1 FundamentalFunctionsandTheirProperties.Further
PropertiesoftheFunctionD(α) 112
3.5.2 PropertiesoftheFunctions μ(α)and μ(α) 124
3.5.3 PropertiesoftheDeviationFunctioninItsGeneral
FormandoftheCorrespondingFundamentalFunction 130
3.5.4 Condition λ+ < D(α) and Strong Dependence
betweenτandζ intheLargeDeviationZone 135
3.5.5 Examples 137
3.6 On Large Deviation Principles for the ProcessY(t) and for
MarkovAdditiveProcesses 141
3.6.1 LDP for the ProcessY(t) on the Narrowing of the
(cid:4) (cid:5)
Set Y(T) ∈TΔ[α) 142
3.6.2 LDPforY(t)whenτandζ AreIndependent 144
3.6.3 OnLargeDeviationPrinciplesforMarkovAdditive
Processes 145
3.7 Rough Asymptotics for the Laplace Transform of the
DistributionofaCompoundRenewalProcess 147
4 LargeDeviationPrinciplesforTrajectoriesofCompoundRenewal
Processes 154
4.1 ConditionsfortheFulfillmentoftheLDPfortheIncrements
ofaProcessandforFinite-DimensionalDistributions 154
4.1.1 LDPforIncrementsofaCRP 154
4.1.2 ProofofLemma4.1.5 161
4.2 First Partial Local Large Deviation Principles for the
TrajectoriesofaCompoundRenewalProcess 163
4.2.1 MainAssertionandItsProof 164
4.2.2 ProofsofLemmas4.2.2and4.2.4 170
4.3 SecondPartialLocalLargeDeviationPrinciple 176
4.3.1 MainResults 176
4.3.2 OntheMostProbableTrajectories 179
4.3.3 AuxiliaryAssertions 181
4.3.4 ProofofTheorem4.3.1 183
4.4 CompleteLocalLargeDeviationPrinciple 186
4.5 IntegralLargeDeviationPrincipleforTrajectoriesofaCom-
poundRenewalProcess 189
4.5.1 MainResultandItsProof 189
viii Contents
4.5.2 OntheRelaxationoftheConditionsofTheorem4.5.1 196
4.6 LargeDeviationPrinciplesfortheFirstBoundaryCrossing
Problem 197
4.6.1 LevelLines 198
4.6.2 Inequalities for the Distribution of the Maximum
ValueofaCRP 201
4.6.3 Large Deviation Principles for the First Boundary
CrossingProblem 204
4.7 LargeDeviationPrinciplesfortheSecondBoundaryCross-
ingProblem 208
4.7.1 MostProbable(Shortest)Trajectories 208
4.7.2 TheSecondBoundaryCrossingProblem 211
4.8 Moderately Large Deviation Principles for Trajectories of
CompoundRenewalProcesses 215
4.8.1 MainResults 215
4.8.2 Proofs 218
4.8.3 Rough(Logarithmic)InvariancePrincipleforCRPs
intheModeratelyLargeDeviationZone 224
5 Integro-LocalLimitTheoremsundertheCramérMoment
Condition 225
5.1 Introduction 225
5.2 MainResults 226
5.2.1 Integro-LocalTheoremfortheProcess Z(t) 226
5.2.2 Integro-LocalTheoremfortheProcessY(t) 231
5.2.3 Integro-Local Theorem for Finite-Dimensional
DistributionsoftheProcess Z(t) 233
5.2.4 NormalandModeratelyLargeDeviations 235
5.3 Integro-LocalTheoremsfortheRenewalMeasure 237
5.4 ProofofTheorem5.2.1andItsGeneralization 249
5.4.1 ProofofTheorem5.2.1 249
5.4.2 ExtensionofResultstotheCasewhentheDistribu-
tionof(τ,ζ )DependsonaParameter 254
1 1
5.5 ProofsofTheorems5.2.10–5.2.14 256
5.5.1 ProofofTheorem5.2.10 256
5.5.2 Proof of Theorem 5.2.13 on Finite-Dimensional
Distributions 262
5.5.3 ProofofTheorem5.2.14 263
5.6 Exact Asymptotics of the Laplace Transform of the Dis-
tribution of a Compound Renewal Process and Related
Problems 264
5.6.1 MainResult 264
Contents ix
5.6.2 RefinementoftheInequalitiesofTheorem4.6.3for
theDistributionof Z(T) 267
5.6.3 ExactAsymptoticsoftheMomentsofaCRP 269
5.7 Integro-LocalTheoremsforMarkovAdditiveProcessesunder
theCramérConditions 272
6 ExactAsymptoticsinBoundaryCrossingProblemsforCompound
RenewalProcesses 275
6.1 Asymptotics of Distributions of the Maximal Value of
a Compound Renewal Process with Linear Drift. First
PassageTimeofaHighLevel 275
6.1.1 Preliminaries 275
6.1.2 DistributionoftheMaximalValueofaCRPwithDrift 278
6.1.3 DistributionoftheFirstPassageTimeofaHighLevel 281
6.2 LimitTheoremsundertheCramérConditionfortheCondi-
tionalDistributionofJumpswhentheTrajectoryHasaFixed
End 287
6.2.1 LimitConditionalDistributionofJumps 287
6.2.2 OntheDistributionoftheVectorαξ 290
6.3 Integro-LocalTheoremsfortheFirstPassageTimeofaHigh
LevelbytheTrajectoryofaCompoundRenewalProcess 290
6.4 IntegralTheoremsfortheDistributionof Z(T) =maxt≤T Z(t) 295
6.4.1 TheCasea <0,α >0 295
6.4.2 TheCaseα > a ≥ 0 302
6.4.3 TheCasea >0,α ∼aasT →∞ 304
6.4.4 AsymptoticsoftheProbabilitythattheTrajectoryof
aCRPDoesNotCrossaHighLevel xforα = x < a 306
T
6.5 Integro-LocalTheoremsinBoundaryCrossingProblemsfor
CompoundRenewalProcesses 308
6.5.1 Integro-Local Theorems for the First Boundary
CrossingProblem 308
6.5.2 Integro-Local Theorems for the Ruin Probability
Problem 314
6.6 IntegralTheoremsinBoundaryCrossingProblems 315
6.6.1 Integral Theorems in the First Boundary Crossing
Problem 315
6.6.2 OntheSecondBoundaryCrossingProblem 317
6.7 ApplicationstotheRuinProbabilityProblemforInsurance
Companies 319
7 Extension of the Invariance Principle to the Zones of Moderately
LargeandSmallDeviations 325
7.1 StrongApproximationofaCRPbyaWienerProcess 325