Lumpability and Absorbing Markov Chains
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1 umbility nd Absorbing rov Chins By Ahmed A.El-Sheih Dertment of Alied Sttistics & Econometrics Institute of Sttisticl Studies & Reserch (I.S.S.R Ciro University Abstrct We consider n bsorbing rov Chin Y (t tht result of n ggregtion finite rov Chin X (t of higher dimension with resect to the rtition A nd unnown trnsition robbility mtrix (t..m P the question of whether X (t is(not lso bsorbing will be study for the lumed nd the wely lumed bsorbing rov Chin for ll (certin rtitions within isomorhic with resect to the cses ( + nd (. Secil cses will be introduced for ech cse for ech rtition. ey Words Absorbing rov Chins Aggregtion of Sttes Identifibility Problem umbility Isomorhic.. Introduction G.A. Wtterson (96 gve the theoreticl formul for the robbility distribution its generting function nd moments of the time ten to first rech n bsorbing stte of finite rov Chin (discrete time discrete sttes. Also these formuls hve been lied to n exmle ten from genetics. A.B.Volidis (985 gve the sufficient conditions for l( P A P where P is the trnsition robbility mtrix (t..m of n bsorbing rov Chin X (t nd P is the t..m of the lumed rocess Y (t which is lso n bsorbing chin. He exressed
2 the fundmentl mtrix of Y (t in terms of the fundmentl mtrix of the originl chin X (t..a.novotny (999 hs demonstrted severl dynmiclly fithful lgorithms to study homogeneous nucletion nd growth in discrete sin models which include the Constrined Trnsfer trix(ct method the Projective Dynmics(PD method nd the onte Crlo with Absorbing rov Chins (CAC method which is generliztion of the n-fold wy lgorithm. edoux nd eguesdron ( exlored how n initil distribution with resect to the lumed chin my differ from seudosttionry initil distribution. They lso illustrted the stochstic equivlence nd the cone invrince roches by chrcterizing the set of ll bsortion time distributions of finite stte bsorbing rov Chin which re geometric. E.V.Denrfo nd et l (7 discussed bsorbing rov Decision Processes (DPs which re generliztions of discounted DPs. They resented two results on slitting of rndomized sttionry olicy into mixture of nonrndomized sttionry olicies for bsorbing DPs with Borel stte nd ction sets. E.A.vn Doorn (7 considered rov Chins in continuous time with n bsorbing coffin stte nd finite set S of trnsient sttes. The limiting distribution of the chin s t conditionl on survivl u to time t chin when is nown to equl the qusi-sttionry distribution of the S is irreducible. He ddressed the roblem of generlizing this result to setting in which S my be reducible nd resented comlete solution when the eigenvlue with mximl rel rt of the genertor of the (sub rov Chin on S hs multilicity one. A rov Chin is clled n bsorbing chin if (i It hs t lest one bsorbing stte ( stte i is n bsorbing stte if nd only if P nd P for ii i j nd (ii For every stte in the chin the robbility of reching n bsorbing stte in finite number of stes is nonzero.
3 Assume tht Y (t is n bsorbing rov Chins tht result of n ggregtion finite rov Chin X (t of higher dimension under given rtition { A A Am A nd unnown trnsition robbility mtrix (t..m P with t..m P. In this er two cses will be considered: - Y (t hs one bsorbing stte nd - Y (t hs two bsorbing sttes. l( P A P So in the next sections the form of ll t..m P such tht (umed nd l( P A S P (wely lumed where S is non emty subclss of robbility vectors of the clss V of ll initil robbility vectors for ll (certin rtitions within isomorhic will be given.. The Generl Cse m n+ form: In this cse we hve just one rtition within isomorhic te the A {{{ { + A- The umed Sitution-One bsorbing stte The t..m P of the rov chin Y (t will be j P i (. The form of ll trnsition mtrices P of the rov Chin X (t of order ( + ( + such tht l( P A P will be I O P (. R T I is the identity mtrix of order ( O is the zero mtrix of order (
4 R T ( ( + ( + ( + ( + ( + ( + ( + j i +. The Secil Cse m n5 The form of the t..m P such tht l( P A P is: j P i So the form of ll t..m P will be: P j i B- The umed Sitution-Two bsorbing sttes The t..m of the lumed chin Y (t will be j P i ( (. ( The form of ll trnsition mtrices P of the rov Chin X (t of order ( + ( + such tht l( P A P will be
5 I O P (. R T I is the identity mtrix of order ( O is the zero mtrix of order ( ( R ( T ( + ( + ( + ( + ( + ( + ( ( + j i +. The Secil Cse m n5 The form of the t..m P P such tht l( P A P is: j i So the form of ll t..m P will be: P j i C- The Wely umed Sitution-One bsorbing stte In this section we generlize the solution of the identifibility roblem nd introduce the form of ll trnsition mtrices P of the rov Chin X (t of order ( + ( + such tht l( P A S P. 5
6 et + j P ( + ( + ( + ( + ( + ( + i + The t..m P of the rov chin (.5 Y (t will te the sme form s in (.. So the form of ll trnsition mtrices P of the rov Chin X (t will be I is the identity mtrix of order ( O is the zero mtrix of order ( R ( + ( + ( I O P (.6 R T T ( + ( + ( + ( ( + ( +. The Secil Cse m n The form of the t..m P such tht ( nd + ( + ( + ( + & ( ( + ( + l( P A S P is: j ( + j P So the form of ll t..m P will be: 6
7 P ( nd D- The Wely umed Sitution-Two bsorbing sttes The form of ll trnsition mtrices P of the rov Chin X (t of order ( + ( + where l( P A S P will be the sme s in (.5. Also j j the t..m P of the rov chin will te the sme form s in (.. Y (t So the form of ll trnsition mtrices P of the rov Chin X (t will be I O P (.7 R T I is the identity mtrix of order ( O is the zero mtrix of order ( ( R ( + ( + ( ( + ( + ( nd + & T ( + ( + ( + ( + ( + ( + ( ( ( + ( ( + ( + ( + j ( + j 7
8 . The Secil Cse m n The form of the t..m P such tht l( P A S P is: P j i So the form of ll t..m P will be: P. The Generl Cse m n ( nd In this section the solution of the identifibility roblem will be generlized nd the forms of ll t..m P where l( P A P nd when l( P A S P for certin rtitions within isomorhic will be introduced. The rtitions re: The t..m - { {... A { { -... A P of the rov chin Y (t will be: j j P b (. b & b b A- The umed Sitution-One bsorbing stte The t..m of the rov chin Y (t will be P b b (. 8
9 b b The rtition {{{... A The form of ll trnsition mtrices P of the rov Chin X (t of order ( where l( P A P will be I O P (. R T I is the identity mtrix of order ( O is the zero mtrix of order ( b b R b ( T b b b b b b b b b ( ( i j. The Secil Cse m n The form of ll t..m P will be: b P b b b b b b b b b b b j i The rtition { {... A The form of ll trnsition mtrices P of the rov Chin X (t of order ( where l( P A P will be Α Α O P (. Β C j i 9
10 O is the zero mtrix of order ( b b b b Β b b ( j i C b b b b b b b b b ( ( j i. The Secil Cse m n The form of ll t..m P will be: j P b b i & b b b b b b j i & j i B- The umed Sitution-Two bsorbing sttes The t..m of the rov chin The rtition {{{... A Y (t will be P (.5 The form of ll trnsition mtrices P of the rov Chin X (t of order ( where l( P A P will be
11 I O P (.6 t O T I is the identity mtrix of order ( O is the zero mtrix of order ( T ( ( j i. The Secil Cse m n The form of ll t..m P will be: P j i The rtition { {... A The form of ll trnsition mtrices P of the rov Chin X (t of order ( where l( P A P will be Α O P (.7 t O C Α j i O is the zero mtrix of order (
12 C ( ( j i. The Secil Cse m n5 The form of ll t..m P P will be: j i & j i C- The Wely umed Sitution-One bsorbing stte In this section the solution of the identifibility roblem nd the form of ll trnsition mtrices P of the rov Chin X (t of order ( where l( P A S P cn be considered s secil cse of A.A.El-Sheih (7. The t..m P of the rov chin Y (t will te the sme form s in (.. The rtition {{{... A The form of ll trnsition mtrices P of the rov Chin X (t of order ( where l( P A S P will be I O P (.8 R T I is the identity mtrix of order ( O is the zero mtrix of order (
13 R T ( ( ( ( ( ( ( b ( b i ( i+ i ( i+ i i ( b i ( i+ i j j (.5 The Secil Cse m n The form of ll t..m P will be: P b i i ( i+ ( b ( b j ( j The rtition { {... A The form of ll trnsition mtrices P of the rov Chin X (t of order ( where l( P A S P will be Α O P (.9 Β C Α + nd + O is the zero mtrix of order (
14 Β C ( ( ( ( ( ( ( ( ( b i ( i+ i nd b ( b i ( i+ i ( i+ i i j j & i ( i.6 The Secil Cse m n The form of ll t..m P will be: P nd will be the sme s bove nd b b ( b j j & i ( i D- The Wely umed Sitution-Two bsorbing sttes The solution of the identifibility roblem nd the form of ll trnsition mtrices P of the rov Chin X (t of order ( where l( P A S P The t..m is considered s secil cse of A.A. El-Sheih (7. P of the rov chin Y (t will te the sme form s in (.5. The rtition {{{... A The form of ll trnsition mtrices P of the rov Chin X (t of order ( where l( P A S P will be
15 I O P (. t O T I is the identity mtrix of order ( O is the zero mtrix of order ( T ( ( ( ( ( ( i ( i+ i ( i i ( i+ j ( j.7 The Secil Cse m n The form of ll t..m ( ( P will be: P ( The rtition { {... A nd j j The form of ll trnsition mtrices P of the rov Chin X (t of order ( where l( P A S P will be Α Α O P (. t O C + nd + 5
16 O is the zero mtrix of order ( C ( ( ( ( ( ( nd ( i ( i+ i j j & i ( i.8 The Secil Cse m n The form of ll t..m P will be: P nd will be the sme s bove nd ( nd & i ( i. Summry In this er we considered the question of whether the originl finite rov Chin X (t will be bsorbing chin if the lumed or the wely lumed rov Chin Y (t ws bsorbing in cse of hving one (two bsorbing sttes for ll (certin rtitions within isomorhic with resect to the cses ( + nd (. First for the generl cse ( + there is only one rtition within isomorhic so we reresented the trnsition robbility mtrix P of X (t s mtrix of rtition mtrices with different dimensions ccording to the number of bsorbing sttes. It cn be shown tht the originl rov Chin X (t will be lso bsorbing with the sme number 6
17 of bsorbing sttes whether the bsorbing rov Chin Y (t lumed or wely lumed. Second for the cse ( the roblem hs been studied for certin rtitions within isomorhic nd gin the t..m P of rov Chin X (t hs been reresented s mtrix of rtitioned mtrices ccording to the rtition within isomorhic number of bsorbing sttes nd whether Y (t lumed or wely lumed bsorbing rov Chin. The t..m P of the lumed (wely lumed Y (t will te the form of the identity mtrix I ( if there exist two bsorbing sttes nd s result the cse of two bsorbing sttes cn be considered s secil cse of one bsorbing stte. With resect to the rtition A the rov Chin X (t will be bsorbing including just one bsorbing stte whether the number of bsorbing sttes of the lumed (wely lumed rov Chin one or two sttes nd this result cn be exlined by the nture of the rtition A {{{... where there is one stte in the first clss nd the remining sttes in the second clss. X (t According to the rtition A { {... Y (t the rov Chin will not be bsorbing whtever the number of bsorbing sttes of the lumed (wely lumed rov Chin Y (t where there is two sttes in the first clss nd the remining sttes in the second clss. Finlly it cn be seen tht if Y (t bsorbing chin (lumed or wely lumed the originl chin X (t my or my not be lso bsorbing deending on mny resons such s the cse under study the number of bsorbing sttes nd the rtition within isomorhic. A ossible future wor ln is to discuss the generl cse if we hve bsorbing sttes of the rov Chin Y (t nd see under wht conditions the originl rov chin X (t will be lso bsorbing? 7
18 References - Ahmed A. A. E. (7 On Wely umed Discrete Stochstic Process Proceeding of the nd Annul Conference on Sttistics Comuter Sciences nd Oertion Reserches I.S.S.R Ciro University Egyt. - E.A.vn Doorn (7 Survivl in qusi-deth rocess th INFORS Alied Probbility Conference University of Technology Eindhoven. - E.V.Denrdo E.A.Feinberg nd U.G. Rothblum (7 Slitting of rndomized sttionry olicies in bsorbing rov Decision Processes th INFORS Alied Probbility Conference University of Technology Eindhoven. - G.A. Wtterson (96 rov Chins with Absorbing Sttes: A Genetic Exmle Ann.th.Sttist. ( edouxj. nd eguesdronp. ( We umbility nd Pseudo- Sttionrity of Finite rov Chins Commun.Sttist.- Stochstic odels 6( A.Novotny (999 Advnced Dynmic Algorithms for the Decy of etstble Phses in Discrete Sin odels: Bridging Disrte Time Scles Int. J. od.phys.c ( VolidisA.B.(985 On the Proerties of Functions of rov Chins. An.Sc.Thesis t Institute of Sttisticl Studies nd Reserch Ciro University. 8
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