Birth Times, Death Times and Time Substitutions in Markov Chains

Transcription

1 Marti Jacbse Birth imes, Death imes ad ime Substitutis i Markv Chais Preprit Octber Istitute f Mathematical Statistics ~U_iy_erS_iW_OfC_p_eh_ag_e ~ ~~~~~ JI

2 Harti Jacbse * ** BIRH IMES, DEAH IHES AND IHE SUBSIUIONS IN MARKOV CHAINS Preprit 1983 N~ 7 INSIUE OF MAHEHAICAL SAISICS UNIVERSIY OF COPENHAGEN Octber 1983

3 SUMMARY Give a Markv chai (X ) ;;:O ' radm times are studied which are birth times r death times I the sese that the pst- ad pre- prcesses are Idepedet give the preset ex I'X) - at time ad the cditial pst- prcess (birth times) r the cditial pre- prcess (death times) is agai Markvia. he mai result fr birth times characterizes all time substitutis thrugh hmgeeus radm sets with the prperty that all pits i the set are birth times. he mai result fr death times is the dual f this ad appears as the birth time therem with the directi f time reversed.

4 INRODUCION AND NOAION A earlier paper, [7], by Jim Pitma ad the authr, hereafter referred t as BDC, ctaied a study f certai classes f birth times ad death times fr Markv chais i discrete time with statiary trasiti prbabilities. Much f the mtivati fr that paper came frm David Williams' [14] fudametal results path decmpsitis f diffusis, ~ particular the e-dimesial Brwia mti BM(l). he types f e.g. birth times csidered i BDC were radm times determied by the evluti f the Markv chai, with the prperty that the pst- prcess is agai Markv with a trasiti fucti pssibly differet frm that f the give chai, ad furthermre shuld have a cditial idepedece prperty similar t that i the strg Markv prperty, with past ad future idepedet give the preset at time. Hwever, t all discrete time aalgues f Williams' path decmpsitis are cvered by BDC. Fr istace Williams shwed that fr BM(l) made trasiet by absrpti at a high level, the time f the ultimate m~~mum f the path is a birth time ad a death time i the sese that give the value x f the path at time, the pre- ad pst- prcesses are bth Markvia ad cditially idepedet. But sice the trasitis fr the tw fragmets bviusly deped x, results f this type are t icluded i BDC. Oe mtivati behid the preset paper has bee the desire t fill this gap. A first discussi f the larger classes f birth ad death times eeded t accmplish this, appeared i the preprit Jacbse [5], ad sme f the fudametals there are repeated here. But the mai results t be give belw deal with a particular class f birth times (ad its dual class f death times) which apart frm pssessig several ice prperties is relevat t the thery f time substitutis i Markv prcesses, cf. the papers by Pitteger [11]

5 - 2 - ad Glver [3]. he mai results f BDC prvide characterizatis f classes f birth times ad death times frmulated as equivaleces betwee bjects defied relative t the prbabilistic structure ad bjects defied i terms f path-algebra. his iterplay betwee prbability thery ad path-algebra is als the essece f the philsphy behid the preset paper. What it amuts t techically will be discussed at the ed f this itrducti. Oe bvius pe prblem left by BDC was hw the results culd be carried ver t prcesses i ctiuus time. his has w bee de by Pitteger [la] (herem 3.9 f BDC birth times), Sharpe [13] (herem 5.2 death times) ad, mst fittigly ad mst recetly, by Pitteger ad Sharpe [12] (herem 6.2 times that are birth as well as death times). Ather imprtat recet referece is the paper [2] by Getr ad Sharpe where they discuss what types f cditial idepedece are relevat i ctiuus time, ad prvide ecessary ad sufficiet cditis fr the differet types t be valid. he basic referece fr this paper is BDC ad it may be useful fr the reader t have a cpy available. he tati t be used here is that f BDC with sme mir chages: give a cutable state space J, let fl dete the space f all sequeces w = (wo,w l '...) l J idexed by the -egative itegers N, let (X, EN) be the crdiate prcess, 1. e. X (w) = W, ad dete by (Y, EN.) the sequece f trasitis Y = (X I,X ) - defied fr EN+ = {l,2,... }. Writig F fr the (ucmpleted) cr-algebra spaed by all X, a prbability p (,F) is said t be Markv r Markv (p) if P makes (X) a Markv chai with statiary trasitis p. If ]J is

6 - 3 - the P-Iaw f may be writte istead f P ad, as is the custm, ]1 ~s degeerate at x. he fllwig cveti ~s adapted thrughut: the same letter is used t dete a Markv prbability (capital letter) ad its trasiti fucti (small letter), Adjiig a state /j, t J, write J = JU {M /j, ad let [2/j, be the space f all sequeces ~ J/j, that remai i /j, ce they get there. he lifetime f a sequece we [2/j, ~s l;(w) = if{en: X (w) =/j,}. he space [2/j, ad the subspace [20 = (l; <00) f paths with fiite lifetime will be used maily i Secti 4 death times. Fr bjects pertaiig t [2/j,' the same tati will be used as fr the crrespdig bjects [2. Fr EN, the killig peratr K: [2 +[2/j, are defied by ad shift peratr 8 :[2+[2 8 (wo' w 1 '.. ) = (w,w I".. ) + Fr = 1, 8 ~s writte istead f 8 1, A radm time is a measurable mappig frm [2 t the exteded time set N = N U {}. Give a radm time idetificati, e.g. X =X, X, Y, K,8 are defied by lcal (=). Als, X, K, 8 are defied ly the set ( <00) ad Y (0 < < 00) As a csequece, fr i stace (Y = (a,b» will be the tati fr the subset {w: 0 < (w) <00, Y (w) = (a,b)} f [2. Fr a fixed EN (X,...,X ). he atms the pre- a-algebra F ~s the a-algebra spaed by A are the sets f the frm A = (X = x'...,x = x). Fr a radm time, the pre- a-algebra F csists f the sets which

7 - 4 - are cutable u~s f sets f the frm (i) (A, = ) where E N, A EA r Cii) e-pit sets {w} where (w) = 00. A radm time splits the prcess (x ) it tw parts, the pre- prcess, cveietly idetified with ad therefre labelled K give as (x 0 K, E N) = (X O ' Xl,!J.,1'1, ), - ad the pst- prcess 8 give as (x 0 8, E N) = (X,X 1"" ) + As discussed earlier, the ma~ theme i BDC ad here is t prvide equivaleces betwee prbabilistic ad path-algebraic bjects. hus fr istace tw differet types f defiitis f radm times will be used: (i) peratial defiitis ad (ii) algebraic defiitis. Defiitis f type (i) give the prperties f a radm time relative t a Markv prbability, while thse f type (ii) are ccered exclusively with the prperties f a radm time as a fucti rl. he latter may be implicit r explici t ~ ature, fr example a descripti f a radm time ivlvig a cllecti f parameters is explicit if the parameters may be chse idepedetly f each ther ad implicit if they are iterrelated. Fr a example, csider stppig (ptial) times. Give a Markv prbability P, ~s a peratially defied stppig time fr P if cditially F withi ( <00), 8 is Markv with the same trasitis p as P. O the ther had, ~s a algebraically defied stppig time if the fllwig three equivalet cditis are satisfied: Ca) ( = ) E F fr E N; Cb) (<)EF fr EN; = (c) (w) = i { E N:w E F} fr sme sequece (F,EN) f sets F EF. Here e wuld call (a), (b) implicit ad (c) a explicit defiiti, because i Ca) the sets ( = ) must be mutually disjit,

8 - 5-1.U (b) the sets ( ~) must icrease with,while ~ (c) the F EF are arbitrary. he characterizati therems t be give here, as thse preseted i BDC (r Jacbse [6]),prvide prbabilistic equivaleces betwee peratially ad algebraically defied bjects. Fr istace, ad a easy csequece f the results i Secti 3 f BDC, fr stppig times the fllwig is true: a radm time ~s a peratially defied stppig time fr the Markv prbability P, iff it is P-equivalet t a algebraically defied stppig time.

9 CONDIIONAL INDEPENDENCE IMES I BDC tw slightly differet tis f cditial idepedece were used 1 the study f birth times ad death times respectively: fr the birth times cditial idepedece f the pre- ad pst- prcesses give X was demaded, while fr the death times it tured ut that the relevat cditial idepedece ccurs whe cditiig Xl' - I this paper we shall use the same cditial idepedece ccept fr birth times ad death times as described i the fllwig defiiti which replaces Defiiti 3.11 i BDC. (2.1). Defiiti A radm time 1S called a cditial idepedece time fr the Markv prbability P, if uder P the pre- ad pst- prcesses are cditially idepedet give y hus 1S a cditial idepedece time iff there 1S a cditial distributi f 8 give (XO,...,X ) withi (O<<OO) r equivaletly f K give (x l'x",,) withi (O<<OO), whichisafuctifthetras- - iti Y ale. It shuld be ticed that cditiig (XO,...,X ) is equivalet t cditiig F ad ivlves 1 particular the cditiig the value f. By ctrast, cditiig (x l'x,,,,) - f the exact value f, wherefre i particular, as 1S essetial, the cditial pre- prcess has a radm lifetime. des t imply kwledge It seems mst atural t have a uified ccept fr cditial idepedece applyig t the birth time as well as the death time thery. A secd reas fr usig Defiiti 2.1 is the fllwig: csider fr a real-valued prcess 1 ctiuus time with, say, right-ctiuus, left-limit paths, the time where the prcess attais its ultimate miimum. With jumps pssible,

10 - 7 - the trasiti fucti f fr istace the cditial pst- prcess give the past will i geeral deped the trasiti (X,X) - rather tha X ale. raslatig this it the discrete time situati makes it atural t study K ad e give the trasiti Y he fllwig result prvides a useful characterizati f cditial idepedece times. he prf prceeds exactly as that f Lemma 3.12 i BDC ad is therefre mitted. (2.2). Lemma A radm time ~s a cditial idepedece time fr the Markv prbability P iff fr every E N+ ad every (a, b) E J2 there exists F E F ' Gab E F respectively such that (2.3) ( =, Y = (a, b» = (F,Y = (a, b), e E Gb) a P - a.s. r equivaletly iff fr every E N+ ad every 2 (a,b)ej there exists FIb EFl' G E F -,a - such tha t (2.4) (=,Y =(a,b»=(f 1 b'y =(a,b),e leg) -,a - P - a.s. Remark Cditial idepedece times satisfyig (2.3) r (2.4) exactly are t splittig times as rigially defied by Williams, see [4J, equati (3.3) r [15J, Secti It appears mst atural t geeralize the defiiti there ad call a splittig time if (2.5) ( = ) = (F,e 1 E G ) - fr sme F E F, G E F. If (2.5) hlds with G = G t depedig, is a statiary splittig time, cf. [llj. he defiiti i (2.5) is implicitly algebraic. Lemma 2.2 may w be refrmulated as statig that cditial idepedece times are the peratially defied versis f statiary splittig times (see als the remark fllwig

11 - 8 - Lemma 3.12 i BDC ad Lemma 2.8 l [11]), he peratial defiiti f geeral splittig times demads that K ad 8 be idepedet give y ad

12 Cl-BIRH INES AND HE CLASS BR he purpse f this secti is t study varius classes f birth times which are cditial idepedece times. (3.1). Defiiti A radm time is a birth time with cditial idepedece (i shrt a Cl-birth time) fr the Markv prbability P if it is a cditial idepedece time fr P ad if cditially Y withi (0 < <00), the pst- prcess is Markv with a statiary trasiti fucti (depedig pssibly Y ) As defied i BDC a radm time is a regular birth time fr P if there is a trasiti fucti q such that cditially F the pst- prcess is Markv (q). hus clearly a regular birth time is a Cl-birth time. (Ntice that if ls a regular birth time fr P, the pst- prcess ls itself Markv withut cditiig the past. his is f curse t true l geeral fr a Cl-birth time). Suppse P ls Markv. Recall that by herem 2.3 f BDC, if DEF with P(D) > 0, the the cditial prbability D = (X E H,C) P - a.s. where HcJ ad C fr sme VCJ 2 with CV=(YEV,EN+) P =P( ID) D ls Markv iff ls a ctermia1 evet, i.e. ad C E F ivariat fr 00 8 C =C C V his result cditiig evets was used i BDC, herem 3.9, t give the fllwig explicitly algebraic characterizati f regular birth times: defiig B t be the class f radm times f the frm (3.2) where C is a arbitrary ctermia1 evet, ls the assciated ctermia1 time

13 ad p ~s a stppig time fr the family (F, E N) C+ f cr-algebras, it was shw that is a regular birth time fr P iff is P-equivalet t a radm time i B. he prf f the therem als shwed that is a regular birth time fr P iff there exists F E F,C cterrial such that (3.3) ( = ) = (F,e E C) P - a.s. fr all EN, cf. (3.16) f BDC. his bservati amuts t a implicitly algebraic characterizati f regular birth times. give by he trasitis q fr e are the same as thse f p( lc) ad are (3.4) where z g(z)=p(c). It is easy t see that istead f usig (3.2), B may be defied as fllws: E B iff there is a ctermial evet C ad evets F E F, E N such that (3.5) (W) =if{en: wef,e wec} Prpsitis 3.6 ad 3.9 prvide tw simple implicitly algebraic characterizatis f Cl-birth times. We are uable t give a explicit algebraic characterizati. he mai results belw deal with the prperties f the explicitly defied class BR, see Defiiti Prpsiti 3.6 is the aalgue f (3.3), ad is prved exactly like that usig (2.3), the fact that fr a radm time satisfyig (2.3) the cditial distributi f the pst- prcess give F withi (Y = (a, b» b P ( IG ab ), ad the characterizati f cditiig evets quted abve. is

14 (3.6). Prpsiti A radm time is a Cl-birth time fr P if ad ly 2 if fr every E N+ ad every Ca, b) E J there exists F E F ad c termial Cab respectively, such that (3.7) (=,Y =(a,b»=(f,y =(a,b),e EC b) a P-a.s. Ntice that if satisfies (3.7) exactly ( exceptial sets), the ~s a statiary splittig time with the G=G f (2.5) give by (3.8) G= U (Y l = (a,b),e EC ab ) (a,b) With a Cl-birth time fr P we shall write qab fr the trasiti fucti f the pst- prcess give Y = (a, b). hus, if satisfies (3.7), qab is the trasiti fucti fr the Markv prbability Qb =pb(.ic ). ab ab he secd characterizati f Cl-birth times ~s a bservati due t J.W. Pitma (private cmmuicati). It fllws immediately frm the defiitis f regular birth times ad Cl-birth times. (3.9). Prpsiti A radm time ~s a Cl-birth time fr the Markv prbability P fied by if ad ly if, fr every (Y = (a, b ) ) therwise ~s a regular birth time fr P. 2 (a,b) E J the radm time de Of curse herem 3.9 f BDC prvides a explicitly algebraic characterizati f each ab' But t btai frm this a explicitly algebraic characterizati f all Cl-birth times requires that the ab be chse simultaeusly i such a way that the sets ( b <00) = (Y = (a,b» be disjit, a ad it is t at all clear hw this shuld be de.

15 Csider a Cl-birth time fr P satisfyig (3.7). Sice, whe igrig sme ull sets, the sets the right are mutually disjit fr, (a,b) varyig, it is clear that fr P-almst all w (3.10) (w) = ih EN: we F,e we Cy ( )}, + w cf. (3.5). he mai difficulty arisig whe attemptig t characterize Cl-birth times i a explicit algebraic fashi rests the fact that the cverse is t true: give a arbitrary cllecti (Cab) f ctermial evets ad. F EF, if ~s defied by (3.10), it is t i geeral true that (3.7) hlds (exactly rather tha a.s.) matter what is the chice f the F, C a b appearig there. We shall w discuss systems f ctermial evets fr which the implicati (3.10) t (3.7) hlds (fr all chices f F ) Fr the tw defiitis belw, let C = (Cab) (a, b)ej2 be a cllecti f ctermial evets. he iclusi ~ ~s strict, allwig fr equality. (3.11). Defiiti C ~s a trasiti reprducig cllecti f ctermial evets if (a,b)pr(c,d) implies that either Cab~ (X =d,c cd ) r (X = d, Cab Cc d) = f/j. Here (a, b) pr (c, d) meas that there exists we (X = b, Cab) ad E N+ such that Y (w) = (c,d). [J Let p be a stchastic trasiti fucti J. (3.12). Defiiti C is trasiti reprducig fr p if (a,b)pr(p) (c,d) implies that either r d P - a.s. Here (a,b)pr(p) (c,d) meas that = b 2: P (C b' Y = (c, d» > 0 =l a [J Give a Markv prbability P we shall call C trasiti reprducig fr p if C is trasiti reprducig fr the trasiti fucti p f P.

16 It seems plausible that if C satisfies the peratial defiiti (3.12), the each Cab may be replaced by a ew ctermial evet C: b such that b C a b = C*b a P - a. s. ad C* = (C* ) ab but this fact is t verified here. satisfies the algebraic defiiti (3.11), Defiiti 3.11 puts sme implicitly g~ve cstraits the Cab' here des t appear t be ay explicit receipt fr describig all trasiti reprducig C. Of curse C ~s trasiti reprducig if all Cab = C. A mre subtle example is the fllwig (3.13). Example Let (C' ) ab Ca, b)ej2 be a arbitrary cllecti f ctermial evets ad defie where C b = (Y E V*b' E N C i b) a a + a v* = {(x y) : Cl:::> C' }. ab ' ab xy he we claim that C = (Cab) ~s a trasiti reprducig cllecti f ctermial evets. see this, suppse (a,b)pr(c,d), ad fid w, with we (X = b,cab)' Y (w) = (c,d). We shall shw that Cab:::> (X = d,ccd). Firstly, by the defiiti f Cab ad because we Cab' (c,d) E V: b ad C~b:::> C ~d' But the t shw that w' E (X = d,c cd) implies w' E Cab' it is eugh t see that w' E (X = d,c cd ) implies Y (w')e V* k ab fr all k E N+ ' i.e. implies C~b::::>C~ (w')' Sice by assumpti Yk(w') EV~d' l.e. k C~d:::>C~k(w')' ad sice C~b:::>C~d' the implicati is evidet. We shall w shw that trasiti reprducig cllectis f ctermial evets lead t a uiversal class f Cl-birth times. (3.14). Defiiti Let BR dete the class f radm times f the frm (3.15) (w) = i { E N : we F ewe C y ( )} + w

17 where fr every, F EF ad where LS a trasiti reprducig cllecti f ctermial evets. (3.16). Prpsiti Suppse belgs t the class BR. he is a CIbirth time fr ay Markv prbability P. Prf We shall shw that satisfies (3.7) exactly (with the F there t the F frm the defiiti f ). Clearly (w) = ad Y (w) = (a, b) iff Y (w) = (a, b), Sw E C b ad fr all k, 1:5_ k < e f (i), (ii) hlds: a (i) (ii) defie 00 Represet each C xy as (Y E V,mE N+,C ) with m xy xy V = {(u, v) (x, y)pr(u, v)}, xy 00 C xy ivariat. he recallig the meaig f pr frm Defiiti I particular vo cv xy xy We claim that subject t (w) =, Y (w) = (a, b), fr every k, 1 ~ k <, it is true that (i) r (ii) hlds iff e f (i)', (ii)', (iii), hlds: (i), Cii), Y,e.(w) ~V~k(W) fr sme,e., k<,e.~ (iii) I Y,e.(w) EV ykcw ) fr all,e., k <. < ad Sice these three cditis ivlve ly w09 ""w, the prpsiti will the be established. Nw fix w with (W) =, Y(w) = (a,b) ad k with l~k<. Suppse

18 that ID E F k. Write (x,y) = Y k (ID). he (i) r (ii) ls equivalet t e f (i) I, (ii)' r (iii)" where (iii) 11 his is clear if i (ii)' ad (iii)" v is writte istead f V But the defiiti f v O esures that ( c) ( Y rf vo f XO=y'Cxy :::> XO=y, m~ xy r sme ad this tgether with v O cv xy xy legitimate. shws that the use f istead f V ls he prf is cmpleted by shwig that (with the assumptis abut ID, k made abve), (iii)'<=:>(iii)". Here => is evidet because 8 IDE (XO=b,C ab ). Cversely, if (iii)" hlds, we have (x,y)pr(a,b) by the defiiti f hece sice C ls trasiti reprducig, either (X O = b, C Cb) = (/J r xy a C :::> (X O = b,c b). Were the last pti pssible we wuld have 8 k ID E C ~ a ~ because 8 ID E (X O == b,c b) c C a xy ad Y(w)EV cv -t.. xy xy fr k <I ~, ad slce we Fk by assumpti this wuld frce (w) = k ctradictig the assumpti (w) =. hus ecessarily (X O = b,c C b) = (/J, ad we are back xy a t (iii) I Remark akig F = ~ fr all, it fllws i particular frm the prf that if C is trasiti reprducig, the if{: 8 E Cy } is a radm time satisfyig (3.7) exactly. hus there exists F E F such that (3.17) Itrducig G as i (3.8), we have 8 m w E Cy (ID) iff 8 m - l we G, s m (3.17) may be writte

19 Als =if{en :e leg} + - s (with a mir mdificati), is e f the peetrati splittig times characterized by Pitteger [11], herem 4.4. Cl (3.18). Example he class f radm times f the frm (3.15) with C as ~ Example 3.13, ~s the class BO first itrduced i [5]. Specializig further e fids that with 2 f : J ~ R sme fucti, the times ad give as the firs t, respectively the last time that the sequece (f (Y ), E N) attais its ultimate miimum, bth belg t BO. It was shw by Millar [9], that ~, f Markv prcesses i ctiuus time. are Cl-birth times fr a wide class Cl It is easy t give examples f Cl-birth times fr a Markv prbability P such that is t P - a.s. equal t a member f BR. But apart frm the explicit descripti (3.15), the times i BR pssess a umber f ice prperties, as we shall w see. Suppse that O>O,>O satisfy (3.7) s that fr ctermial, the idetities F,G EF (3.19) (0 =, Y = (a, b» = (F,Y = (a, b), e E Cb), a ( =, Y = (c, d» = (G,Y = (c, d),e E D d) c hld exactly fr E N+, (a,b), (c,d) E J 2. Nw csider the radm time p = 0 + Oe. he (p=,y = (a,b),y =(c,d»=(h d'y =(c,d),e ECbD_ d) p,c a c fr suitable H be F, ad csequetly fr P Markv,a (3.20) p(e E IF,Y = Ca,b),Y = (c,d» =pdc lc bd d). p per p ac hus p will t be a Cl-birth time fr P uless fr all (a,b),(c,d)

20 with P(Ya = (a,b),yp = (c,d» >0, the iclusi Cab:::>Dcd hlds d P -a.s. Hwever with the prf f Prpsiti 3.16 ad Defiiti 3.14 i mid, the fllwig result is t surprisig ad easily prved. (3.21). Prpsiti If a> 0, > 0 belg: t BR ad bth satisfy (3.15) with the same family C f ctermial evets, the als pe BR ad satisfies (3.15) with the cllecti C, where p=+08. his result prvides e explaati fr the termilgy 'trasiti reprducig' i Defiiti 3.l2:1etP be Markv ad let qab be the cmm trasiti fucti fr P(8 E IY = (a,b»where K = O, r p. he t ly are the K K the trasiti fuctis fr 8 p give Y, but they p als arlse by the tw stage prcedure csistig i firstly csiderig the pst-a prcess give Y a ad its distributi, ad the secdly, fr this ew prcess, evaluatig the trasitis fo the pst- prcess. he mai result f this secti gives a characterizati f hmgeeus radm sets with all pits Cl-birth times, I terms f cllectis f times i BR. By a stadard defiiti a prcess (Uri)EN defied (rl,f) ls hm + geeus N+ if (3.22) U k = U 0 8 k + (Usually (3.22) is required t hld ly utside a exceptial set, but here we shall assume that it is a idetity all f rl fr all k,. Nte that we require hmgeeity ly fr k,~l) A radm subset M f N+ ls hmgeeus if (U ) where U=l(EM)' I terms f M, (3.22) becmes is hmgeeus, (M - k) N+ = M 0 8 k

21 writig A-k= {l-k: lea} fr A a subset f N. (hat M is a radm + subset f N+ meas that fr every w there 1S defied a subset M(w) f N+ such that w ~ 1 (EM) (w) is F-measurab le fr every E N +) I discrete time the structure f hmgeeus prcesses is f curse very simple: let U=U 1 ad take =l i (3.22) t fid U =U e 1 - Cversely, give U measurable, (U) defied this way 1S hmgeeus. Similarly, if M is a hmgeeus radm set, the (3.23) M={EN : 8 leg} + - where G= (lem) Cversely, glve GEF, M defied by (3.23) 1S hmgeeus. Give a radm subset M f N+, let l, 2,... dete the pits f M 1 icreasig rder f magitude. hus 1 = if M, 2 = ih > 1 : E M} etc. I particular ( 1 = 1) = (1 E M) I rder that. (w) 1 be defied fr all w, we adpt the cveti that if the cardiality /M(w) / f M(w) 1S < i, the.(w)=. l(w)=... = hus ( 1 = 00) = (M = C/J), ad. < (. <00) 1 I this secti, whe writig M = hi' 2, }, we shall always assume the. t have bee defied i this maer. 1 frm It 1S easy t see that M= h l, 2, } 1S hmgeeus iff. 1 1S f the

22 (3.24) (i ~ 2) with satisfyig (3.25) = ih EN: e 1 E G} + - fr sme GEF. Fr arbitrary M, itrduce (Z.). l' l l> the prcess f trasitis made at the timepits i M, i.e. Z.=Y he Z. ls defied ly (. <00) l. l l l ad the prcess (2.) may have fiite lifetime. Als dete by G. the l l smalles t a-algebra ctaiig F..,'",F (hus G. -::J F,but i geeral l i l i the iclusi may be strict, als if M is hmgeeus. If fr every j < i,. J ls F -measurable, the. l G. =F ) l. l We shall shw (3.26). herem (a) Let P be Markv ad let M = h l' 2,.. } be a hmgeeus radm set. Suppse each. l t G. l such that the trasitis fr e. l i.e. fr all i, (a,b), ls a Cl-birth time fr P give G. l with respect d t deped p(e E IG.) = Qbb. l a l (. <00, Z. = (a,b», where l l ls Markv with trasitis ad iitial state b. he there exists a cllecti C = (Cab) f ctermial evets, trasiti reprducig fr the trasitis p f P, such that M={EN:8 + -l EG'} P-a.s. where G' = U (Y l = (a,b), 8 E Cab). Als ecessarily, fr every i (a,b) F =G. P-a.s.. l l (b) Let C = (Cab) be a trasiti reprducig cllecti f ctermial evets

23 ad let M=hl' 2,... } dete the hmgeeus radm set {EN+: 8 - l EG} where G= U (Y l =(a,b),8ec ab ). he. E BR ad F = G. fr all 1. ad (a,b) 1. with respect t ay Markv prbability P, each. is a Cl-birth time fr P 1. with P(S E IF )=pb(.ic b ).. a (. < =, Z. = (a, b», i particular the trasitis fr S t deped 1.. give F. 1. d Prf (a) Sice F eg., each. 1.S a rdiary Cl-birth time fr P, hece by Prpsiti 3.6 we ca fid F(i) E F ad ctermial such that (3.27) _ (i) _ (i) (.-,Z.-(a,b»-(F,Y -(a,b),8 EC b ) a P - a.s. fr every i,,(a,b) But the (Z. = (a,b» 1. ad S1.ce by assumpti the right had side must t deped i we deduce that fr every (a,b), C(b i ) =C(j) pb-a.s. fr all 1.,J with P(Z. = (a,b» >0, a ~ 1. P(Zj = (a,b» > 0. It fllws that (3.27) may be assumed t hld with c~~)= Cab ctermial t depedig 1. prvided (a, b) ER: = {(x,y): I: P(Z. = (x,y» > O} ad fr (a, b) (/. R we may ad shall take Cab = (/J With C = (Cab) as just fud, defie G' as 1. the statemet f the therem ad put M' = { EN: 8 1 E G'}. he defiiti f M ad (3.27) (with + - replaced by Cab) shw that HeM' P - a.s. Fr the ppsite iclusi, write M = { EN: S 1 E G}, let + - (a,b)er ad fid i, ad a atm A f F,Ae (Y = (a,b», such that PCA,. =) >0. With (1.' =)A the secti f 1. (. = ) 1. CA,. = ) 1. beyd A (cf. BDC, Defiiti 2.7), we fid that the F -atm 1.

24 P(8 E jf )=pbc jc.=»,.. 1. A while by (3.27) this als equals pb(.jc ab ). Csequetly ( i =)A =C ab ad sice by the defiiti f b M, ( i = ) A c Gab p - a. s. it fllws that b p where - a.s. G = ab ad b p -a. s. P(EM"M) =P(8 ECy,8 1 (/. G) - I: P(Y =(a,b),8 EC b,8 l(/.g) Ca,b)ER a - I: P(Y = (a,b»pb(c b'g b) (a,b)er a a = 0 S we have shw M=M' P-a.s. Writig MI ={ 1, 2,... }, it 1.S the clear that! =.,F =F I,G! =G. P-a.s. fr every We cmplete the prf by shwig that C is trasiti reprducig fr p, ad that F =G , P - a.s. But if C 1.8 trasiti. reprducig it is easy t see that i - 1.S l measurable with respect t the P-cmpleti f F I r F, (a argumet fr a similar asserti 1.S prvided i the prf f Cb) belw), hece s is ad F = G. P - a. s. f llws he argumet that C 1.S trasiti reprducig is mre legthy ad uses that. 1. is Cl-birth with respect t G. 1. rather tha just F. 1. Let i<j ad (a,b), (c,d) be give such that P(Z. = (a,b),z. = (c,d» >0. 1. J he fid k, E N+, a ad a F -atm Bc (XO =b'y = (c,d» with PD> 0, where D= (A,. =k,8 k EB,. =k+). Sice 1. J. 1.S G.-measurable, DEG. ad 1. J J (3.28) P ( 8 E j D) = Q dd. c J because. J Py (. j C ). xy 1.S Cl-birth relative t G.. Here as belw we write QY = J xy he cditial prbability may hwever als be fud usig that

25 S Cl-birth. Usig the hmgeeity f M, which implies 1 particular that 1. = a J 1 J-1 i (. < =), we fid 1 p(a EB,,=k+,e E.jA,.=k). J, 1 1 J = p (a E B,.. 0 a =, a 0 e E j A,. = k). J J-1 1 b = Q b (B,.. =, a E.) a J-1 J-1 ad frm this it fllws directly that p ( e E ID) = Q bb ( e (. I B,.. =: ). a J-1 J Usig the Markv prperty f at time shws the right had side t L equal Qdb( j(.. =)B), ad cmparig with (3.28) we therefre fid a J-1 d d I Q d = Q b (. (.. = ) B) c a J-1 Nw t ly is Qb pb( Ic ) Qd =pd(.jc ), but we als have ab =: ab' cd cd Summarizig we have shw that if i < j, the (3.29) P(Z. = (a,b),z. = (c,d» >0 => C d CC b 1 J C a d p - a.s. We must shw (see Defiiti 3.12) that (a,b)pr(p) (c,d) implies CcdcC ab pd _ a. s. r pd(c ab C cd) = O. By assumpti may be fud with (3.30) b.. b b p (C b'y = (c,d» =p (C b)q bey = (c,d» >0, a a a 1particular CablC)'J ad (a,b)er. If (c,d)~r,ccd=c)'j ad there is thig t prve, s we may als assume (c,d) ER.

26 Nw fid i with P(Z. == (a,b» > 0, ad csider the tw pssible cases: 1 (i) 1 > 0, (H) 1 = 0, where (3.31) 1=P(Z.=(a,b),Y + =(c,d),e + ECcd) I case (i), the idetificati betwee M ad M' shws that P-a.s. ad the 1 > 0 (Z.=(a,b),Y =(c,d),e EC d)c U (z.=(a,b),zj.=(c,d», c.. 1 i 1 J>l ad (3.29) gives d C cd C Cab P - a. s as desired. I case (ii), because. 1 1S Cl-birth we have b O=p(Z. = (a,b»q bey = (c,d),e EC d) 1 a c he first factr is > 0 by assumpti, the secd is > 0 by (3.30), d hece Q d ab (C cd ) = 0, 1.e. P d (C cd I Cab) = 0 r P (CabCcd)=0 ad the prf f (a) is cmplete. (b) Sice satisfies (3.15) with F = [2, IE BR Nw. = e, s usig Prpsiti 3.21 ad prceedig by iducti it 1S 1- i - l clear that each i EBR ad that (3.15) hlds with (Cab) the give cllecti (!. hat each. is Cl-birth fr ay P as stated, fllws the frm 1 Prpsiti shw that F = G. we must check that. 1S F -measurable fr i 1 J i j<i. But the F -atm A= (X ==x O '".,X =x,. =), ly thse time pits k< ca belg t H fr which C :::> (X O = x,c ), ad the xk-l~ x_lx amg these exactly thse succeed fr which (x_ l,x)e V fr k < l <. -t.. -t.. ~-lxk hus M fl {l,..., - l} ad hece als. is cs tat A. J he startig pit 1 herem S a hmgeeus radm set M, i.e. a

27 set with a specific algebraic structure. It is the shw essetially that the pits i M are Cl-birth times iff the set G characterizig M (cf. (3.23» has a special frm which is described explicitly algebraically. Alteratively e might have take a arbitrary radm set M=hl' Z ""}' assumig as i herem 3.26 (a) that each is Cl-birth fr P. his hwever wuld allw fr the pssibility M = {} 1 with a arbitrary Cl-birth time fr P, ad as maitaied abve, explicit algebraic descripti f such appears pssible. [ 11]. As we shall presetly see, herem 3.26 has cectis t results 1 [3], Suppse that the cditis 1 herem 3.26 (b) are satisfied ad let P be Markv. Because ( i <=), it 1S easy t see that the prcess Z = (Zi) ien (which may have + fiite lifetime) 1S Markv with trasiti fucti b q((a,b),(c,d» =P (Zl = (c,d)ic ab ). hus the Markv chai Z 1S btaied frm a time chage i the rigial chai P ime chages f this type were discussed by Pitteger [11] - if abve Cab:::: Cb depeds b ly, the (X )'EN becmes a Markv chai ad we have a example fittig exactly it Pitteger's thery. Mre geerally Pitteger csidered time chages M={ 1, 2,... } (here we take Z rather tha (X ) t be the time chaged chai) such that Z 1S. 1 Markv ad each. 1S a cditial idepedece time fr P (but t 1 ecessarily. Cl-birth). Oe essetial algebraic cditi such M 1S

28 (3.32) see [11], (3.5b). his leads f curse t much mre geeral time chages tha thse treated i herem 3.26, eve if all. l are required t be Cl-birth. Fr a example, let C be trasiti reprducig, let (J be give by (3.15), put 1 = (J, ad defie by (3.32). he the satisfy (J.5a,b,c) f [J ad therefre e.g. Z is Markv with respect t ay P. (he set r i [llj is the set G frm (3.8». Hwever, because f the F.. appearigi(3l5) M={"" }..,.. 1'"2' " will t I geeral be hmgeeus. he time chage crrespds t takig ly a subset f the full hmgeeus set {~l: 8 - l EG}, G as i (3.8). We shall als discuss the relati f the precedig t the time chage results f Glver [3]. raslated it discrete time ad the setup used here, his herem 1.5 states the fllwig: let A= (A)EN be a raw additive fuctial (RAF) defied with A =0 t:.a : = A - A is either 0 r 1. Defie fr i E N+ -l ad suppse that fr all = i f{ EN: A = i} l + ' ad suppse fially (cditi (a) f herem 1.5, [3]) that fr every l, (3.33) A 1 (<.) = l ls F -measurable fr. l P, each. l Markv chai. E N+. he with respect t ay Markv prbability ls a cditial idepedece time ad Z = (Z.), Z. = Y, l l. l is a By defiiti, A is a RAF if ~+ = ~ + A 0 8 k exactly fr all k, EN. If als M = 0 r 1, ecessarily -l A = L k=o lg 0 8k ' where G = (A = 1) = (t:.a = 1). But the 1 1 (t:.a = 1) = (8 1 E G),ad it ls clear -

29 that the time chage determied by A ls the same as that iduced by the hmgeeus radm set M={:8 leg} - ad that all time chages frm hmgeeus M arise frm RAF's I this maer. he extra cditi (3.33) is critical whe establishig the Markv prperty fr Z. A little path algebra shws it t be equivalet t the fllwig cditi, expressed i terms f G: fr every E N+ such that there exists F EF (G,8. 1 E G) = (F,8 1 E G), - - which except fr a small mdificati is cditi (b) f herem 4.4, [11]. S the time chages i [3] appear as special cases f thse i [11]. Summarizig the abve discussi, we have ecutered three types f time chages: thse i herem 3.26 ivlve hmgeeus radm sets with all. l Cl-birth times; thse i [3] are iduced by hmgeeus radm sets with all. cditial idepedece times; ad fially thse I [11] arise frm cerl tai subsets f hmgeeus radm sets with all. l times. cditial idepedece hrughut this secti we have discussed Cl-birth times which by defiiti bey a strg Markv prperty ivlvig cditial idepedece f the pre- ad pst- prcesses give Y Other authrs have studied birth times where this cditial idepedece ccurs whe cditiig t ly Y but als sme auxiliary F -measurable variable. hus, i [8] Millar has itrduced (fr prcesses i ctiuus time) radmized ctermial times ad shw that if is such a time, the cditially F pedig X the pst- prcess ls }furkv with a trasiti fucti dead a F -measurable variable a fucti f (X,X), will t be a Cl-birth time. - U. herefre, uless U ls

30 Millar's defiiti f radmized ctermial times ~s quite cmplicated. Simpler examples f birth times with a cditial idepedece prperty ~vlvig a extra variable ca be fud i Getr [1]. Frm (3.19) it fllws that with 0, give by (3.19), the radm time p = will be a birth time with the kid f cditial idepedece discussed by Millar ad Getr, prvided y ~s F -measurable. Fially it p may be remarked that Millar pits ut that the class f radmized ctermial times ~s clsed uder the additi (0,)-?0+O 8.

31 Cl-DEAH IMES AND HE CLASS DR his secti ctais the defiitis ad results which are the death time aalgues f thse i Secti 3. (4.1). Defiiti A radm time ~s a death time with cditial idepedece (i shrt a Cl-death time) fr the Harkv prbability P if it is a cditial idepedece time fr P ad if cditially y withi (0 < <=), the pre- prcess ~s Markv with a statiary trasiti fucti (depedig pssibly y ). Accrdig t a defiiti i Secti 5 f BDC, a radm time is a regular death time fr a Markv prbability P if the pre- prcess is Markv (q) fr sme (substchastic) trasiti fucti q with the pre- ad pst- prcesses cditially idepedet give < < 00 ad Xl' - Ay regular death time fr P ~s a Cl-death time. his statemet is t as trasparet as the similar e fr birth times, s we shall prduce a argumet ad at the same time itrduce sme tati. Recall that if Q ~s a Markv prbability [1,1::, with iitial measure \! ad trasiti fucti q, such that (X ) with psitive prbability, has fiite lifetime, the the prcess reversed frm the lifetime defied by the reversal trasfrmati R: [I, I::, ~ [I, I::, g~ ve by if <r;<= therwise ~s agai Markv with a substchastic trasiti fucti q J. Here (4.2) A -1 q(x,y) =~(y)q(y,x)~ (x) fr x,yf{o<~<} with ~(z) L Q(X =z) =O 00

32 herefre, if ls a regular death time fr the Markv prbability P ~, the reversed pre- prcess RK is Mark'V with statiary trasitis, hece s ls R 0 K give <00 ad fr this latter prcess cditiig X -1 simply amuts t freezig the iitial state, s that R 0 K, ad there fre als R 0 R 0 K = K, cditially X is Harkv with statiary -1 trasitis. By the cditial idepedece prperty shared by all regular death times it w fllws that ay regular death time is a Cl-death time. he mai result, herem 5.2, i Secti 5 f BDC states that a radm time is a regular death time fr the Markv prbability px iff ls equivalet t a radm time i the class D. A remark i BDC shws hw this result may be geeralized t Markv prbabilities with a -degeerate start. Sice we shall wrk with this geeralizati here, we shall redefie D ad restate the regular death time therem. If 2 HcJ, VcJ, let dete the mdified termial time if X EH therwise he class D ls w defied t cmprise all radm times f the frm (4.3) = sup{: 1 ~~ HV ' 8-1 E F} fr sme 2 HcJ, VcJ, FE F. (By the usual cveti = 0 if the set I brackets is empty; I particular = 0 (X E H)). he the fllwig ls true: ls a regular death time fr the Markv prbability P iff is P-equiva1et t a radm time i D. As pited ut i BDC, the results regular birth times ad the crrespdig results death times are duals. his duality is prevalet als i the thery f Cl-birth times ad Cl-death times, s the death time results

33 will be preseted i the same rder as their aalgues i Secti 3. Als. t keep dw the legth f the paper. we shall t give prfs. I the death time thery the cuterpart f a ctermial evet ls a se quece = (,E N+) f termial evets (4.4) where = (XO E H, Y k E V, 1 < k < ) 2 HcJ, VcJ are sets t depedig. Of curse the ivariat part f the ctermial evet is matched by the iitial part (X O E H) f each. Ntice that he tati frm (4.4) will be used belw with subscripts ab where (a,b) E J 2. Ntice that there is really a switch i tati frm (4.3) t (4.4): (4.3) frbids trasitis I V prir t while (4.4) demads that all pre- trasitis belg t V. Of curse (4.3) is mdelled up the defiiti f D frm BDC, but i the remaider f the secti we shall use the tati (4.4). he first tw results are the duals f Prpsitis 3.6 ad 3.9. (4.5). Prpsiti A radm time ls a Cl-death time fr P if ad ly if there exists FE F ad fr every (a, b) E J2 subsets Hab C J, Vab c J2 such that (4.6) (=,Y a =(a,b»=(b,y =(a,b),e lef) - P - a.s. fr (4.7). Prpsiti A radm time ls a Cl-death time fr the Markv prbability P by if ad ly if fr every 2 (a,b) E J the radm time ab defied

34 (4.8) ab { (Y = (a, b) ) therwise ~s a regular death time fr P. [J Suppse that ~s a Cl-death time fr P s that (4.6) hlds. Csider (a,b) with P(Y = (a, b))> 0, itrduce 00 J b ={xej: 2: P(X =x,<<oo,y = (a,b)) >O}, a =O the state space fr K that K give Y = (a,b) give Y = (a, b). Straightfrward calculatis shw ~s Markv with trasitis where 0:> z a a = - =l g b (z) = 2: P (Y k E Vb' 1 < k <, X 1 = a) Hwever, the symmetry betwee the birth time ad death time theries is brught ut mre clearly by csiderig K reversed, as was de i herem 2 f Jacbse [6], ad leads i a atural way t the death time aalgue f herem Itrduce i; (z) = 2: P (X = z) ad the trasiti fucti ~ atural =O P-duality t p (cf. [6]), A -1 p(x,y) = i;(y)p(y,x)!; (x) Withut lss f geerality we may assume i; > 0, ad the p(x,y) ~s defied if i;(x),i;(y) <00. Fr cveiece we w assume i; <00 (s the P-chai ~s trasiet), althugh it is eugh that i; <00 the state space U Jab fr Crrespdig t ( ) there ~s a atural dual ctermial evet Cab ab EN +

35 which is a subset f the space Q = (l;;<00) cq!:, amely f paths with fiite lifetime, (4.9) C b = (Yk E Vb' 1 < k < l;; < 00, X 1 E H b) a "a = l;;- a where f curse A 2 Vab ={(x,y) EJ : (y,x) EVab } (Abut tati ~ the sequel: symbls with a A refer t bjects pertaiig t the path space QO)' A A (4.10). Prpsiti With P,F,Cab as abve ad satisfyig (4.6) (4.11) P (R 0 K E / Y = (a, b» = P a (. / Cb) a Remark his result states that the distributi f K reversed give Y (a,b) ~s the same as that f ea give YA = (b, a) fr a prcess P i atural duality t P ad with a Cl-birth time fr P satisfyig (3.7) P - a. s. with the Cab there replaced by the dual Cab f (~b). Prf Bth chais uder csiderati start i a, s t prve (4.11) it remais t idetify their (substchastic) trasiti fuctis which by (4.2) ad (3.4) are -1 e(y)qab(y,x)e (x) ad la (x y)pa(x y)g(y) V ' 'A ab g(x) respectively, where 00 e(z) = 2: P(X = z,< </Y = (a,b», =O

36 It is essetial t te, ad fairly bvius t verify, that the tw chais have Jab as state space. Isertig the expressis fr qab ad p abve, it is see that we eed ly shw that the fuctis -1 eg ad ~g are prprtial Jab' (We write g=gab)' Nw ex> e(z) 0:: I:P(X =z,<<ex>,y =(a,b» =O ex> I: I: P(X = Z, =k'y k = (a,b» =O k> ad usig the Markv prperty at times ad k - 1 (recall that this reduces t ex> a +l g(z)p (Xl=b,F) I:P(X =z, b ) =O a Csequetly -1 e(z)g (z) ~s prprtial t (4.12) ex> + 1 I: P (X = z, b). a =O O the ther had A Z P (s <ex» = 1 ad By duality ad Hce with fl the iitial distributi fr P = pfl, ~(u);(u,l'i) = fl(u) it fllws that 00 ~(z)g(z) = I: P(X l=z, b) - a =l which is (4.12) exactly, s the prf f (4.11) is cmplete. D

37 With the precedig discussi as mtivati, we shall withut further cmmets state the aalgues f the defiitis ad results frm Defiiti 3.11 wards. Csider a cllecti evets ad defie Cab 2 =(b,(a,b)ej,en, a + as i (4.9) abve. f sequeces f termial (4.13). Defiiti ~s a trasiti reprducig cllecti f sequeces f termial evets if (a,b)pr(c,d) implies that either Cab::> (X = c,c cd ) r (X = c,cabc cd ) = (/J. Here (a,b)pr(c,d) meas that there exists w E(X O = a,c ab ) ad E N+ such that Y (w) = (d,c). Let p be a substchastic trasiti fucti J such that fr all x E J. (4.14). Defiiti is trasiti reprducig fr p if (a,b)pr(p) (c,d) implies that either AC P - a.s. r AC P - a.s. Here (a,b)pr(p) (c,d) meas that 00 A a A I: P (C b'y = (d,c» >0 =l a I (4.15). Example Let (a~) be a arbitrary cllecti f sequeces f termial evets ad defie where * A' A, Vab = {(x,y) : Cab::> Cxy}. he = (: b ) ~s a trasiti reprducig cllecti f sequeces f termial evets. (4.16) Defiiti Let DR dete the class f radm times f the frm

38 (4.17) where FE F (W) =sup{en+: we~ (w),8_l wef}, 2 ad = (ab,(a,b) EJ,EN+) 1S a trasiti reprducig cllecti f sequeces f termial evets. (4.18). Prpsiti Suppse belgs t the class DR. he is a CIdeath time fr ay Markv prbability P. If 1S give by (4.17) ad P 1S trasiet, the P(RK E ly =(a,b»=pa('lc b ), a where p 1S the trasiti fucti 1 atural P-duality t p. (4.19). Example he class f radm times f the frm (4.17) with as i Example 4.15, is the class DO first itrduced i [5]. he times ad frm Example 3.18 bth belg t DO. Let w G be a measurable subset f ~O ad csider the radm subset f N give by (4.20) L={EN:RK leg} + Sets f this frm appear as the duals f hmgeeus radm sets. Of special iterest t us is the situati where is a cllecti f sequeces f termial evets, Cab is as i (4.9) ad (4.21) G = U (Y 1 = (b, a),8 E C b) (a,b)ej2 a he E L (w) iff w E ~ (w) With G give by (4.21), Gc (r; > 1) ad frm w, whe csiderig L f the frm (4.20), we shall always assume the G there t be a subset f (r; > 1). he 0 EL is impssible.

39 With this assumpti ~ frce, give 1 f the frm (4.20), defie radm times (cr., i EN) ~ + ~ by lettig cr. = 00 ~ (11i =00), cr. =0 ~ (111 <i) ad by writig 1={cr-t,.. "cr1} (11i =-t) with cr-t<... <cr1, Als defie W. = (X,X 1) A.: = (0 < cr. <00) ad let H. dete the cr-a1gebra f ~ cr. cr.- ~ ~ ~ ~ ~ subsets f A. geerated by 8 ad cr.-cr. fr j < i ~ cri-1 J ~ As usual, i the therem belw, p detes the trasiti fucti ~ atural P-dua1ity t p. (4.22) herem (a) 1et P be Markv ad trasiet ad let 1 be a radm set f the frm (4.20) with Gc (r; > 1). Suppse each cr. ~ fr P with respect t H. ~ t deped ~, Le. fr all i, (a,b), such tha t the trasi tis fr R 0 K cr. ~s a Cl-death time ~ gi ve H. d ~ E I A.,H.) = Qa b per 0 K cr. ~ ~ a ~ (A., W. = (b, a», where ~ ~ ~s Markv ~ 0 with trasitis t depedig ~ ad iitial state a. he there exists a cllecti = (: b ) f sequeces f termial evets, trasiti reprducig with respect t p such that 1={EN :RK EG'} + +1 A P -a. s. Cl 1 I < 00), where G' U (Y1 = (b,a),8 E Cab) (a,b) (b) 1et = (: b ) be a trasiti reprducig cllecti f sequeces f termial evets ad let 1 dete the radm set give by (4.20) with G as i (4.21). he cr. E DR ~ fr all ~, ~ particular cr. ~ is a Cl-death time fr ay Markv prbability P, ad if P is trasiet, 1 Aa,A P (R 0 K E A., 8 1) = P (. I C b) cr. ~ cr.- a ~ ~ (A.,W. = (b, a» ~ ~

40 Suppse the cditis i (b) are satisfied. Rather tha csiderig the full set L (i which case thig iterestig is said whe wrkig OLI =00» e may csider the part L* f L precedig a give cptial time. Defiig (cr'!', i EN) 1. + cclusis i (b) remai valid with cr. 1. frm L* as the cr. were defied frm L, the 1. replaced by cr'!' 1. As a fial remark, te that uder the assumptis 1. (b), (Wi,i E N+) 1.S a Markv chai with respect t ay Markv prbability P ad if P 1.S trasiet the trasitis are give by P(W. 1 = (d,c)iw. = (b,a» =pa(y = (d,c) le b) A a '[ where '[ is defied ~O by (4.23)

41 PAH DECOMPOSIIONS WIH HARKOVIAN EXCURSIONS We shall briefly discuss what kid f path decmpsitis btai whe the results frm Sectis 3 ad 4 may be cmbied. Csider a hmgeeus radm set H={EN : 8 leg} with G give by + - (3.8) ad C = (Cab) a trasiti reprducig cllecti f ctermial evets.. ~ With ~1={l,2""} as ~ Secti 3, we saw ~ herem 3.26 (b) that each H a Cl-birth time fr ay Harkv prbability P. Writig the pst -prcess splits it the Harkv chai (z.) = (Y ) (e.,i EN) f skew excursis where ~ + ~. ~ ad the sequece e.. =(X,X +l""'x 1) ~ i i i +l - here may be fiitely r ifiitely may excursis accrdig as (z.) has ~ fiite lifetime r t. All excursis have fiite legths except the last e ~ the case where there are ly fiitely may excursis. It fllws immediately frm the results i Secti 3, that give (z.,i EN), (which icludes cditiig the lifetime f the Z. -chai) ~ + ~ the excursis are, with respect t ay Markvia P, cditially idepedet with the distributi f e. ~ t depedig i but ly hw e. ~ bb/ cditied t start ad ed. Mre specifically, with Q ab = P (. Cab), ~s P(e. E -/Z. = (a,b),z. 1 = (c,d» = Qbb(K E '/Zl = (c,d», ~ ~ ~+ a l P(e. E '/Z. = (a,b), ~ ~s the lifetime f (Z.» ~ ~ J = Q~b (. / 1 = =). I geeral the cditial excursis will f curse t be Harkv. Hwever, if = l ~s a Cl-death time fr each the certaily all the fiite excursis are Markv, ad if i additi ~s b Q ab - a.s. equal t a ctermial evet, the als the last ifiite excursi will be

42 Harkv. Withut aimig fr cmplete geerality, we shall discuss a simple example ivlvig Harkvia excursis. Suppse g1.ve a trasitive relati >- J2 ad ivariat evets (C*b ' (a, b) E J2) cmpatible with >- i the sese that a,00 (a,b) >-(c,d) =>C*b ~ C*d a,00 c,00 Defiig (5.1) Cab 1.S ctermial ad C= (Cab) is trasiti reprducig. (his setup prvides a alterative descripti f the class BO frm Example 3.18, see Prpsiti 3.31 i [5]). With this chice f C e fids -l (5.2) (=,Y =(a,b»=[ U (Yk*","Y)](y =(a,b),8 EC b) k=l l=k+l ~ a writig (x,y) :I-(u,v) if it 1.S t true that (x,y) >-(u,v) Suppse w 1. additi that the relati *- is als trasitive. We claim that the (5.3) (=,Y =(a,b»= (Yk:l-(a,b),l<k<,Y =(a,b),8 EC b ). = a see this, suppse (w)=, Y (w)=(a,b). Frm (5.2) it fllws (fr k = - 1) that Y - l (w) :I-(a,b). Suppse it has bee shw that Yl(w) ;j.-.(a,b) fr k;;;.t<. he Y k - l (w) :I-(a,b) fllws because by (5.2), Y k - l (w) :}-Yl(w) fr sme l,k;;;l;;;, by assumpti Yl(w) :I-(a,b), hece Y k - l (w) >(a,b) S1.ce :I- is trasitive. S a Iducti argumet yields (5.3) frm (5.2).

43 But cmparig (5.3) with (4.6) it is clear that ~s a Cl-death time fr ay Markv prbability. S by the precedig discussi, with C give by (5.1), >- ad *" trasitive as described abve, we btai a path decmpsiti with the excursis (e. ) ~ beig idepedet ad Markv give (z.). ~ We have here discussed path decmpsitis iduced by certai hmgeeus radm sets. But it is f curse als pssible t btai decmpsitis based the dual hmgeeus sets L csidered i herem 4.22 (b). Fially, there are examples f path decmpsitis which are perfect ~ the fllwig sese: suppse L is as ~ herem 4.22 (b) with the frm (4.23) f the frm (4.6) (relative t ~O), suppse M ~s as ~ herem 3.26 (b) with = 1 f the frm (4.6), ad suppse that the begiig = if M f M equals the ed sup L f L. he because ad are always Cl-death times the give Markv chai P may be decmpsed as fllws: chse a ra-,... dm variable U = Z =W with distributi the P-distributi f Y, ad 1 1,... defie W l by iterchagig the tw cmpets f the trasiti U = W l. Give D, cstruct tw idepedet Markv chais, (z.) ad (W.) with di- ~ ~ stributis matchig thse f the (z. ) ~ f Sectis 3 ad 4. Fially, give D, (Z.) ~ ad (Iv.) ~ ad (W.), establish tw idepedet sequeces f mutual ~ ly idepedet Markvia excursis (e.) ad (f.), where the (e.) t- ~ ~ ~ gether with (z. ) ~ abve, while the prcess K. are t cstitute the pst- prcess 8 as described (f. ) ~ ad (W.) ~ are t yield i a similar maer the pre- We leave it t the reader t check that with = r (Examples 3.18 ad 4.19), examples f such perfect path decmpsitis are btaied.

44 ACKNOWLEDGEMEN I am happy t thak Je Glver fr helpful discussis f time substitutis.

45 REFERENCES 1. Getr, R.K.: Splittig imes ad Shift Fuctia1s. Z. Wahrsch. Verw. Gebiete 47, (1979). 2. Getr, R.K., Sharpe, M.J.: Markv Prperties f Markv Prcesses. Z. Wahrsch. Verw. Gebiete~, (1981). 3. Glver, J.: Raw ime Chages f Markv Prcesses. A. Prbab. ~, (1981). 4. Jacbse, M.: Splittig imes fr Markv Prcesses ad a Geeralized Markv Prperty fr Diffusis. Z. Wahrsch. Verw. Gebiete 30, (1974). 5. Jacbse, M.: Markv Chais: Birth ad Death imes with Cditial Idepedece. Istitute f Mathematical Statistics, Uiversity f Cpehage, Preprit 7 (1979). 6. Jacbse, M.: w Operatial Characterizatis f Cptia1 imes. appear i A. Prbab. 7. Jacbse, M., Pitma, J.W.: Birth, Death ad Cditiig f Markv Chais. A. Prbab. ~, (1977). 8. Mi11ar, P.W.: Radm imes ad Decmpsiti herems. Prbability (Prc. Symps. Pure Math., Vl. XXXI, Uiv. Illiis, Urbaa, Ill., 1976), pp Amer. Math. Sc., Prvidece, R.I. (1977). 9. Mi11ar, P.W.: A Path Decmpsiti fr Markv Prcesses. A. Prbab. ~, (1978). 10. Pitteger, A.O.: Regular Birth imes fr Markv Prcesses. A. Prbab. ~, (1981). 11. Pi t teger, A. 0.: ime Chages f Markv Chais. Stchas tic Prces s. App1. ll, (1982). 12. Pitteger, A.O., Sharpe, M.J.: Regular Birth ad Death imes. Z. Wahrsch. Verw. Gebiete 58, (1981).

46 Sharpe, M.J.: Killig imes fr Markv Prcesses. Z. Wahrsch. Verw. Gebiete 58, (1981). 14. Wi11iams, D.: Path Decmpsiti ad Ctiuity f Lcal ime fr Oe-dimesial Diffusis, Prc. Ld Math. Sc. ~, (1974). 15. Wi11iams, D.: Diffusis, Markv Prcesses, ad Martigales. Vlume 1: Fudatis. Chichester, Wi1ey 1979.

47 PREPRINS 1982 COPIES OF PREPRINS ARE OBAINABLE FROM HE AUHOR OR FROM HE INSIUE OF MAHEMAICAL SAISICS, UNIVERSIESPARKEN 5, 2100 COPENHAGEN 0, DENMARK. N. 1 Hlmgaard, Sim ad Yu, Sg Yu: Gaussia Markv Radm Fields Applied t Image Segmetati. N. 2 Aderss, Stee A., Br~s, Has K. ad Jese, S~re lver: Distributi f Eigevalues i Multivariate Statistical Aalysis. N. 3 jur, ue: Variace Cmpet Mdels i Orthgal Desigs. N. 4 Jacbse, Marti: Maximum-Likelihd Estimati i the Multiplicative Itesity Mdel. N. 5 Leadbetter, M.R.: Extremes ad Lcal Depedece i Statiary Sequeces. N. 6 Heigse, Ige ad Liest~l, Kut: A Mdel f Neurs wi th Pacemaker Behav iur Recievig Strg Syaptic Iput. N. 7 Asmusse, S~re ad Edwards, David: Cllapsibility ad Respse Variables i Ctigecy ables. N. 8 Hald, A. ad Jhase, S.: O de Mivre's Recursi Frmulae fr Durati f Play. N. 9 Asmusse, S~re: Apprximatis fr the Prbability f Rui Withi Fiite ime.

48 PREPRINS 1983 COPIES OF PREPRINS ARE OBAINABLE FROM HE AUHOR OR FROM HE INSIUE OF MAHEMAICAL SAISICS, UNIVERSIESPARKEN 5, 2100 COPENHAGEN ~, DENMARK. N. 1 Jacbse, Marti: imes. w Operatial Characterizatis f Cptia1 N. 2 Ha1d, Aders: Nich1as Beru11i's herem. N. 3 Jese, Erst Lykke ad Rtze, H1ger: Limit herems: Easy Prfs. A Nte De Mivre's N. 4 N. 5 Asmusse, S~re: Cjugate Distributis ad Variace Reducti ~ Rui Prbability Simulati. Rtze, H1ger: Cetral Limit hery fr Martigales via Radm Chage f ime. N. 6 Rtze, H1ger: Extreme Value hery fr Mvig Average Prcesses. N. 7 Jacbse, Marti: Birth imes, Death imes ad ime Substitutis i Markv Chais.