Random Recursive Equations and Their Distributional Fixed Points

297 Pages·2012·1.36 MB·English

Gerold Alsmeyer
Random Recursive Equations
and Their Distributional Fixed
Points
January23,2012
Springer

Preface
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Acknowledgements
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vii

Contents
1 Introduction................................................... 1
1.1 Atrueclassic:thecentrallimitproblem........................ 1
1.2 Aprominentqueuingexample:theLindleyequation ............. 4
1.3 Arichpoolofexamples:branchingprocesses................... 7
1.4 ThesortingalgorithmQuicksort ........................... 10
1.5 Randomdifferenceequationsandperpetuities................... 16
1.6 Anonlineartimeseriesmodel ................................ 20
1.7 Anoisyvotermodelonadirectedtree ......................... 22
1.8 Anexcursiontohydrology:theHorton-Strahlernumber .......... 24
2 Renewaltheory ................................................ 29
2.1 Anintroductionandfirstresults............................... 29
2.2 Animportantspecialcase:exponentiallifetimes................. 33
2.3 Lattice-type ............................................... 35
2.4 Uniformlocalboundednessandstationarydelaydistribution ...... 37
2.4.1 Uniformlocalboundedness............................ 37
2.4.2 Finitemeancase:thestationarydelaydistribution......... 38
2.4.3 Infinitemeancase:restrictingtofinitehorizons ........... 40
2.5 Blackwell’srenewaltheorem................................. 42
2.5.1 Firststepoftheproof:shakingofftechnicalities.......... 42
2.5.2 Settingupthestage:thecouplingmodel ................. 44
2.5.3 Gettingtothepoint:thecouplingprocess ................ 45
2.5.4 Thefinaltouch ...................................... 46
2.6 Thekeyrenewaltheorem .................................... 47
2.6.1 DirectRiemannintegrability........................... 48
2.6.2 Thekeyrenewaltheorem:statementandproof............ 52
2.7 Therenewalequation ....................................... 55
2.7.1 Gettingstarted....................................... 56
2.7.2 Existenceanduniquenessofalocallyboundedsolution .... 57
2.7.3 Asymptotics ........................................ 58
2.8 Renewalfunctionandfirstpassagetimes ....................... 63
ix
x Contents
2.9 Anintermezzo:randomwalks,stoppingtimesandladdervariables . 65
2.10 Two-sidedrenewaltheory:ashortpathtoextensions ............. 72
2.10.1 Thekeytool:cyclicdecomposition ..................... 72
2.10.2 Uniformlocalboundednessandstationarydelaydistribution 74
2.10.3 ExtensionsofBlackwell’sandthekeyrenewaltheorem .... 75
2.10.4 Anapplication:Tailbehaviorofsup S inthenegative
n 0 n
driftcase .......................≥.................... 77
3 Iteratedrandomfunctions ...................................... 81
3.1 Themodel,definitions,somebasicobservationsandexamples..... 81
3.1.1 Definitionofaniteratedfunctionsystemanditscanonical
model.............................................. 82
3.1.2 Lipschitzconstants,contractionpropertiesandthetop
Liapunovexponent................................... 85
3.1.3 Forwardversusbackwarditerations..................... 86
3.1.4 Examples........................................... 87
3.2 GeometricergodicityofstronglycontractiveIFS ................ 90
3.3 ErgodictheoremformeancontractiveIFS...................... 96
3.4 AcentrallimittheoremforstronglymeancontractiveIFS.........102
4 Powerlawbehaviorofstochasticfixedpointsandimplicitrenewal
theory ........................................................107
4.1 Goldie’simplicitrenewaltheorem.............................107
4.2 Makingexplicittheimplicit..................................111
4.3 Proofoftheimplicitrenewaltheorem..........................113
4.3.1 ThecasewhenM 0a.s. .............................116
≥
4.3.2 ThecasewhenP(M>0) P(M<0)>0 ...............116
∧
4.3.3 ThecasewhenM 0a.s. .............................120
≤
4.4 Applications...............................................122
4.4.1 Randomdifferenceequationsandperpetuities ............122
4.4.2 Lindley’sequationandarelatedmax-typeequation........129
d
4.4.3 Letac’smax-typeequationX =M(N X)+Q............131
∨
4.4.4 TheAR(1)-modelwithARCH(1)errors .................135
5 The smoothing transform: a stochastic linear recursion with
branching
PartI:Contractionproperties ...................................143
5.1 Settingupthestage:theweightedbranchingmodel ..............145
5.2 Adigression:weightedbranchingandrandomfractals............151
5.3 TheminimalLp-metric......................................155
5.4 ConditionsforS tobeaself-mapofPp(R) ...................163
5.5 ContractionconditionsforS ................................168
5.5.1 Convergenceofiteratedmeanvalues....................168
5.5.2 Contractionconditionsif0<p 1.....................170
≤
5.5.3 Contractionconditionsif p>1 ........................171
Contents xi
5.5.4 Contractionconditionsif p>2and∑i 1 Ti Lp.........179
5.5.5 Aglobalcontractionconditionif p 1≥..|..|.∈.............183
≥
5.5.6 Existenceofexponentialmoments......................184
5.6 Anapplication:Quicksortasymptotics......................188
5.7 TheHausdorffdimensionofrandomCantorsets:completingthe
proofofTheorem5.12 ......................................194
6 Thecontractionmethodforaclassofdistributionalrecursions ......205
6.1 Thesetup:adistributionalrecursionofgeneraladditivetype ......206
6.2 TheZolotarevmetric........................................207
6.3 Asymptoticnormality:ZolotarevversusminimalLp-metrics ......215
6.4 Backtorecurrenceequation(6.2):ageneralconvergencetheorem..217
6.5 Thedegeneratecase:recursion(6.2)withtautologicallimitequation223
6.6 Applications...............................................233
6.6.1 Thetotalpathlengthinarandomrecursivetree...........233
6.6.2 Thenumberofleavesofarandomrecursivetree ..........240
6.6.3 Sizeandtotalpathlengthofarandomm-arysearchtree....243
7 The smoothing transform: a stochastic linear recursion with
branching
PartII:Fixedpoints............................................251
7.1 Thesmoothingtransformwithdeterministicweights .............251
References.....................................................251
A Aquicklookatsomeergodictheoryandtheorems .................257
A.1 Measure-preservingtransformationsandergodicity ..............257
A.2 Birkhoff’sergodictheorem ..................................259
A.3 Kingman’ssubadditiveergodictheorem........................260
B Convexfunctioninequalitiesformartingalesandtheirmaxima......263
C Banach’sfixedpointtheorem....................................267
D Hausdorffmeasuresanddimension ..............................271
Index .............................................................281