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1、Chapter2RenewalandRegenerativeProcessesRenewalandregenerativeprocessesaremodelsofstochasticphenomenainwhichanevent(orcombinationofevents)occursrepeatedlyovertime,andthetimesbetweenoccurrencesarei.i.d.Modelsofsuchphenomenatypicallyfocusondetermininglimitingaveragesforcostsorothersy
2、stemparameters,orestablishingwhethercertainprobabilitiesorexpectedvaluesforasystemconvergeovertime,andevaluatingtheirlimits.Thechapterbeginswithelementarypropertiesofrenewalprocesses,in-cludingseveralstronglawsoflargenumbersforrenewalandrelatedstochasticprocesses.Thenextpartofthec
3、haptercoversBlackwell’srenewaltheorem,andanequivalentkeyrenewaltheorem.Theseresultsareimportanttoolsforcharacterizingthelimitingbehaviorofprobabilitiesandexpectationsofstochasticprocesses.WepresentstronglawsoflargenumbersandcentrallimittheoremsforMarkovchainsandregenerativeprocess
4、esintermsofaprocesswithregenerativeincrements(whichisessentiallyarandomwalkwithauxiliarypaths).Therestofthechapterisdevotedtostudyingregenera-tiveprocesses(includingergodicMarkovchains),processeswithregenerativeincrements,terminatingrenewalprocesses,andstationaryrenewalprocesses.2
5、.1RenewalProcessesThissectionintroducesrenewalprocessesandpresentsseveralexamples.ThediscussioncoversPoissonprocessesandrenewalprocessesthatare“embed-ded”instochasticprocesses.Webeginwithnotationandterminologyforpointprocessesthatweuseinlaterchaptersaswell.Suppose0≤T1≤T2≤...arefini
6、terandomtimesatwhichacertaineventoccurs.ThenumberofthetimesTnintheinterval(0,t]is∞N(t)=1(Tn≤t),t≥0.n=1R.Serfozo,BasicsofAppliedStochasticProcesses,99ProbabilityanditsApplications.cSpringer-VerlagBerlinHeidelberg20091002RenewalandRegenerativeProcessesWeassumethiscountingprocessis
7、finitevaluedforeacht,whichisequivalenttoTn→∞a.s.asn→∞.Moregenerally,wewillconsiderTnaspoints(orlocations)inR+(e.g.,intime,oraphysicalorvirtualspace)withacertainproperty,andN(t)isthenumberofpointsin[0,t].Theprocess{N(t):t≥0},denotedbyN(t),isapointprocessonR+.TheTnareitsoccurrencetim
8、es(orpointlocations).Thepointproc
