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MATH 285: Stochastic Processesmath-old.ucsd.edu/ ynemish/teaching/285Today: Poisson processesBirth and death chainsRecurrence and transience@Homework 5 is due on Friday, March 4, 11:59 PM
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Independentproo-I.co / Xt-.incréments t to.,-,InductionByThmXs tn- tn.-s) for20.7,andofforXt-,; , toIndependent,LPI t.io-,,., ,Xtriin ] t., , tn-( to to.o, -forXs,tnt ,-- . tn.ss.istn ,- tn )fordetermine theanyOtto ---joint *.Independentiunique / yincrémentsofIndependent tn independent of Îtnt tn on stat tn a
BirthanddeathchainsConsider5-We---{ 0,42cà / Iall,.thisµ;allcontinuousa diprocesspureopure ckingdeath -o.mxExamplestateratesbirth andbirthwithin time(E)withdi D
coalescentKingman 'sK ) H Et K ) ( E ) EFFET. (?)Dénotethe timetoCondition edS, timemost recent onspentN atancescommon1-Si ,stateN, sa432lSa --- timeforSN5] ( Et / Si Sat.--Sna ] -i,spent De notecondition etsumon of the LN,TELL ] branch lengths.76.VE [T.ComputewhereatN-I,.
Explosionbe( Xt )etConditionTNThenE on[5, Tn]birthpurea LSaDénote. -- -5N.thetheTTnthe-i?timeto reachand,time,callbydi DénoteWewithprocessexplosionto anwesetcanafter aT?fort T ( minimal )restart fromanother stateIii.,N.
RecurrenceJefandLet21.2transi en ce( ttbe ocontinuous l ed min{t otransientrécurrentpositive iisfor t( transient ) forrécurrentthereembeddedvisitsiif PositiveinfiniteYnrecurrencerefakesiinto i}.Pilli a] if Pif Tico ](Xt )ly manyvisitsif récurrentChainjump:if ispace0. Lif Ei [ Ti ]is arécurrent ( Transient )(Yn )timesinfinite lyaccolentmanynowlongtimesit fakes torevisit i
Recurrenceforbirthanddeathchainsetktlts.iobedi qliiiti ) obirthafori 0,deathandni 2- (i iq-,i)(Xt )isirréductibleoneSimilar / yasstateforhci )FSA{parametersi?I.À.M3'analyseThenfor oi-i-i-i-i-m.siMwithchain: ( allforthedi o,Ni recurrencediscrete-o),itso/transi encetimeMC,isenough( fakedénotetostateo).
birth and death chainsforRecurrenceBythestrongtlarkovpropertyhli )RecallthatG) pli.jt-qli.it/qli )ha )so,(* )becomes'Wecanrewrite thishliti )Applyingthehh'-abovehliti )using-)the identitieshlidifferences) récursive / ygives
ti.atsumshln )hlo )- a ifZpii-- ,o(Xt ) then-andtn i,récurrentisa ifl'Zpi heedwe,findtothe minimal(ThmsolutionowhichThen isha )achieved t-wheneE-o hlo )Iand-ha )( Xe ) istransient.7-o)
Independent incréments Given a stochastic process ( e) tao its incréments are random variables -0 o_0 Suppose that (Xt) is a Count ing process, i.e., (jump times event times # of events that occurred up to time t) Then for set t- s # of events that occurred on Is t] Cor. 20.8 1f (Xt) is a Poisson process with rate d then for any Otto tic - - - tn the incréments tn- tn .
