Renewal Theory for Perturbed Random Walks and Similar Processes

260 Pages·2016·1.611 MB·English

by  Alexander Iksanov

Probability and Its Applications
Alexander Iksanov
Renewal Theory
for Perturbed
Random Walks
and Similar
Processes

Probability and Its Applications
Serieseditors
SteffenDereich
DavarKhoshnevisan
AndreasKyprianou
SidneyI.Resnick
Moreinformationaboutthisseriesathttp://www.springer.com/series/4893
Alexander Iksanov
Renewal Theory
for Perturbed Random Walks
and Similar Processes
AlexanderIksanov
FacultyofComputerScience
andCybernetics
TarasShevchenkoNational
UniversityofKyiv
Kyiv,Ukraine
ISSN2297-0371 ISSN2297-0398 (electronic)
ProbabilityandItsApplications
ISBN978-3-319-49111-0 ISBN978-3-319-49113-4 (eBook)
DOI10.1007/978-3-319-49113-4
LibraryofCongressControlNumber:2016961210
MathematicsSubjectClassification(2010):60-02,60G,60K
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To myfamily
Preface
The present book offers a detailed treatment of perturbed random walks, per-
petuities, and random processes with immigration. These objects are of major
importanceinmodernprobabilitytheory,boththeoreticalandapplied.Furthermore,
these havebeenused tomodelvariousphenomena.Areasofpossibleapplications
includemostofthenaturalsciencesaswellasinsuranceandfinance.Recentyears
haveseenanexplosionofactivityaroundperturbedrandomwalks,perpetuities,and
randomprocesseswithimmigration.Overthelastdecade,severalniceresultshave
beenproved,andsomeefficienttechniquesandmethodshavebeenworkedout.This
bookisaresultofagrowingauthor’sconvictionthatthetimehascometopresent
inabookformatmaindevelopmentsintheareaaccumulatedtodate.Summarizing,
thefirstpurposeofthisbookistoprovideathoroughdiscussionofthestateofthe
artintheareawithaspecialemphasisonthemethodsemployed.Althoughmostof
the resultsaregivenina finalformasultimate criteria,thereare stilla numberof
openquestions.Someofthesearestatedinthetext.
Formally,themainobjectsarerelatedbecauseeachoftheseisaderivedprocess
ofi.i.d.pairs.X1;(cid:2)1/,.X2;(cid:2)2/;:::.Here(cid:2)1,(cid:2)2;:::arereal-valuedrandomvariables,
whereasX1,X2;:::arereal-valuedrandomvariablesinthecaseofperturbedrandom
walksandperpetuities(withnonnegativeentries)andX1,X2;:::areDŒ0;1/-valued
random processes in the case of random processes with immigration. As far as
perturbedrandomwalks.Tn/n2N definedby
Tn WD(cid:2)1C:::C(cid:2)n(cid:2)1CXn; n2N
are concerned,the main motivation behind our interest is to what extent classical
results(someofthesearegiveninSection6.3)forordinaryrandomwalks.(cid:2)1C:::
C(cid:2)n/n2N must be adjusted in the presence of a perturbating sequence. A similar
motivation is also our driving force in studying weak convergence of random
processeswithimmigration.X.t//t(cid:3)0definedby
X
X.t/WD XkC1.t(cid:2)(cid:2)1(cid:2):::(cid:2)(cid:2)k/1f(cid:2)1C:::C(cid:2)k(cid:4)tg; t(cid:3)0:
k(cid:3)0
vii
viii Preface
IfX .t/(cid:4)1and(cid:2) (cid:3)0forallk 2N,thenX.t/isnothingelsebutthefirsttimethe
k k
ordinaryrandomwalkexitstheinPterval.(cid:2)1;t(cid:3).Thisisaclassicalobjectofrenewal
theory,anditiswellknownthat. k(cid:3)01f(cid:2)1C:::C(cid:2)k(cid:4)utg/u(cid:3)0satisfiesafunctionallimit
theoremast ! 1.IfX .t/ isnotidenticallyone,theasymptoticbehaviorofX.t/
k
is affected by both the first-passage time process as above and fluctuations of the
“perturbating”sequence.Xk/k2N.Withthispointofview,thesubjectmatterofthe
bookisonegeneralizationofrenewaltheory.Thus,thesecondpurposeofthebook
istoworkouttheoreticalgroundsofsuchageneralization.Actually,theconnections
betweenthemainobjectsextendfarbeyondtheformaldefinition.Thethirdpurpose
ofthebookistoexhibittheselinksinfull.Asawarm-up,wenowgivetwoexamples
in whichperturbedrandomwalksare linkedto perpetuitiesandrandomprocesses
withimmigration,respectively.
(a) To avoid at this point introducingadditionalnotation, we only discuss perpe-
tuitieswith nPonnegativeentriesthatarealmostsurelyconvergentseriesofthe
formY WD exp.T /.Itturnsoutthatwheneverthetailofthedistribution
1 n(cid:3)1 n
ofY issufficientlyheavy,theasymptoticbehaviorofPfY > xgasx ! 1
1 1
is completelydeterminedbythatofPfsup T > logxg.In particular,if the
n(cid:3)1 n
powerorlogarithmicmomentsofsup exp.T /arefinite,soarethoseofY ;
n(cid:3)1 n 1
see Sections 1.P3.1and 2.1.4.A similar relation also exists between the finite-
n-perpetuities nkD1exp.Tk/ and maxima max1(cid:4)k(cid:4)nexp.Tk/, thoughthis time
withrespecttoweakconvergence;seeSection2.2.
(b) The number of visits to .(cid:2)1;t(cid:3) of the perturbed random walk is a certain
randomprocesswithimmigrationevaluatedatpointt.Themomentresultsfor
generalrandomprocesseswithimmigrationderivedinSection3.4areakeyin
the analysis of the moments of the numbers of visits (see Section 1.4 for the
latter).
As has already been mentioned, the random processes treated here allow for
numerousapplications. The fourth purpose of the book is to add two less known
examplestothe listofpossibleapplications.InSection4we showthatacriterion
forthefinitenessofperpetuitiescanbeusedtoproveanultimateversionofBiggins’
martingaleconvergencetheoremwhichisconcernedwiththeintrinsicmartingales
insupercriticalbranchingrandomwalks.Fortheproof,wedescribeandexploitan
interestingconnectionbetweentheseatfirstglanceunrelatedmodelswhichemerges
when studying the weighted random tree associated with the branching random
walk under the so-called size-biased measure. In Section 5 we investigate weak
convergenceofthenumberofemptyboxesintheBernoullisievewhichisarandom
allocationschemegeneratedbyamultiplicativerandomwalkandauniformsample
onŒ0;1(cid:3).Wedemonstratethattheproblemamountstostudyingweakconvergence
of a particular random process with immigration which is actually an operator
definedona particularperturbedrandomwalk. We emphasizethatthe connection
betweentheBernoullisieveandcertainrandomprocesseswithimmigrationremains
veiled unless we consider the Bernoulli sieve as the occupancy scheme in a
randomenvironment,andfunctionalsinquestionareanalyzedconditionallyonthe
environment.
Preface ix
Iclosethisprefacewiththanksandacknowledgments.Ithankmyfamilyforall
theirloveandcreatinganiceworkingatmosphere,bothathomeanddacha,where
most of the research and writing of this monograph was done. The other portion
oftheresearchunderlyingthepresentdocumentwasmostlyundertakenduringmy
frequentvisitstoMünsterunderthegeneroussupportoftheUniversityofMünster
andDFGSFB878“Geometry,GroupsandActions.”IthankGeroldAlsmeyerfor
makingthese visits to Münster possible and always being ready to help. Matthias
MeinershelpedinarrangingmyvisitstoMünster,too,whichishighlyappreciated.I
thankOlegZakusylo,myformersupervisor,forall-roundsupportatvariousstages
of my scientific career. I thank my colleagues and friends (in alphabetical order)
Gerold Alsmeyer, Darek Buraczewski, Sasha Gnedin, Zakhar Kabluchko, Sasha
Marynych,Matthias Meiners, Andrey Pilipenko, Uwe Rösler, Zhora Shevchenko,
and Vladimir Vatutin, in collaboration with whom many of the results presented
inthisbookwereoriginallyobtained.SashaMarynychscrutinizedtheentirebook
and found many typos, inconsistencies, and other blunders of mine. Apart from
this, Iowe specialthanksto Sasha forhisability tobe helpfulalmostatanytime.
I am grateful to Darek Buraczewski, Matthias Meiners, Andrey Pilipenko, and
IgorSamoilenkowhoreadsomechaptersof the bookandgavemeusefuladvices
concerningthepresentationanddetectedseveralerrors.
Kyiv,Ukraine AlexanderIksanov
Contents
1 PerturbedRandomWalks .................................................. 1
1.1 DefinitionandRelationtoOtherModels .............................. 1
1.2 GlobalBehavior......................................................... 4
1.3 SupremumofthePerturbedRandomWalk............................ 6
1.3.1 DistributionalProperties........................................ 6
1.3.2 ProofsforSection1.3.1 ........................................ 8
1.3.3 WeakConvergence ............................................. 20
1.3.4 ProofsforSection1.3.3 ........................................ 22
1.4 First-Passage Time and Related Quantities
forthePerturbedRandomWalk........................................ 29
1.5 ProofsforSection1.4................................................... 32
1.6 BibliographicComments............................................... 41
2 Perpetuities ................................................................... 43
2.1 ConvergentPerpetuities................................................. 44
2.1.1 CriterionofFiniteness.......................................... 44
2.1.2 ExamplesofPerpetuities....................................... 45
2.1.3 ContinuityofPerpetuities...................................... 52
2.1.4 MomentsofPerpetuities........................................ 57
2.1.5 ProofsforSection2.1.4 ........................................ 59
2.2 WeakConvergenceofDivergentPerpetuities ......................... 65
2.3 ProofsforSection2.2................................................... 68
2.4 BibliographicComments............................................... 83
3 RandomProcesseswithImmigration...................................... 87
3.1 Definition................................................................ 87
3.2 LimitTheoremsWithoutScaling....................................... 89
3.2.1 StationaryRandomProcesseswithImmigration.............. 90
3.2.2 WeakConvergence ............................................. 91
3.2.3 ApplicationsofTheorem3.2.1................................. 93
3.2.4 ProofsforSection3.2.2 ........................................ 96
xi