A Martingale Proof of Dobrushin s Theorem for Non-Homogeneous Markov Chains 1

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1 E e c t r o c J o u r a o f P r o b a b t y Vo ), Paper o. 36, pages Joura URL epecp/ A Martgae Proof of Dobrush s Theorem for No-Homogeeous Marov Chas 1 S. Sethurama ad S.R.S. Varadha Departmet of Mathematcs, Iowa State Uversty Ames, IA 50011, USA Ema: sethuram@astate.edu ad Courat Isttute, New Yor Uversty New Yor, NY 10012, USA Ema: varadha@cms.yu.edu Abstract. I 1956, Dobrush proved a mportat cetra mt theorem for ohomogeeous Marov chas. I ths ote, a shorter ad dfferet proof eucdatg more the assumptos s gve through martgae approxmato. Keywords ad phrases: o-homogeeous Marov, cotracto coeffcet, cetra mt theorem, martgae approxmato AMS Subect Cassfcato 2000): Prmary 60J10; secodary 60F05. Submtted to EJP o March 21, Fa verso accepted o September 6, Research supported part by NSF/DMS , NSA-H ad NSF/DMS

2 1 Itroducto ad Resuts Neary ffty years ago, R. Dobrush proved hs thess [2] a mportat cetra mt theorem CLT) for Marov chas dscrete tme that are ot ecessary homogeeous tme. Prevousy, Marov, Berste, Sapagov, ad L, amog others, had cosdered the cetra mt questo uder varous suffcet codtos. Roughy, the progresso of resuts reaxed the state space structure from 2 states to a arbtrary set of states, ad aso the eve of asymptotc degeeracy aowed for the trasto probabtes of the cha. After Dobrush s wor, some refemets ad extesos of hs CLT, some of whch uder more strget assumptos, were proved by Statuavcus [16] ad Sarymsaov [13]. See aso Hae [6] ths regard. A correspodg varace prcpe was aso proved by Gudas [4]. More geera refereces o o-homogeeous Marov processes ca be foud Isaacso ad Madse [7], Iosfescu [8], Iosfescu ad Theodorescu [9], ad Wer [18]. We ow defe what s meat by degeeracy. Athough there are may measures of degeeracy, the measure whch turs out to be most usefu to wor wth s that terms of the cotracto coeffcet. Ths coeffcet has appeared eary resuts cocerg Marov chas, however, hs thess, Dobrush popuarzed ts use, ad deveoped may of ts mportat propertes. [See Seeta [14] for some hstory.] Let X, BX)) be a Bore space, ad et π = πx, dy) be a Marov trasto probabty o X, BX)). Defe the cotracto coeffcet δπ) of π as δπ) = sup πx 1, ) πx 2, ) Var x 1,x 2 X = sup x 1,x 2 X πx 1, A) πx 2, A) A BX) = 1 sup 2 x 1,x 2 X fy)[πx 1, dy) πx 2, dy)]. f L 1 Aso, defe the reated coeffcet απ) = 1 δπ). Ceary, 0 δπ) 1, ad δπ) = 0 f ad oy f πx, dy) does ot deped o x. It maes sese to ca π o-degeerate f 0 δπ) < 1. We use the stadard coveto ad deote by µπ ad πu the trasformatos duced by π o coutaby addtve measures ad bouded measurabe fuctos respectvey, µπ)a) = µdx) πx, A) ad πu)x) = πx, dy) uy). Oe ca see that δπ) has the foowg propertes. δπ) = sup πu)x 1 ) πu)x 2 ) 1.1) x 1,x 2 X u U 1222

3 wth U = {u : sup y1,y 2 uy 1 ) uy 2 ) 1}. It s the operator orm of π wth respect to the Baach sem-) orm Oscu) = sup x1,x 2 ux 1 ) ux 2 ), amey the oscato of u. I partcuar, for ay trasto probabtes π 1, π 2 we have δπ 1 π 2 ) δπ 1 ) δπ 2 ) 1.2) where π 1 π 2 s the two-step trasto probabty π 1 π 2 x, ) = π 1 x, dy)π 2 y, ). By a o-homogeeous Marov cha of egth o state space X, BX)) correspodg to trasto operators {π,+1 = π,+1 x, dy) : 1 1} we mea the Marov process P o the product space X, BX )), P [X +1 A X = x] = π,+1 x, A), where {X : 1 } are the caoca proectos. I partcuar, uder the ta dstrbuto X 1 µ, the dstrbuto at tme 1 s µπ 1,2 π 2,3 π 1,. For < we w defe π, = π,+1 π +1,+2 π 1,. We deote by E[Z] ad V Z) the expectato ad varace of the radom varabe Z wth respect to P. Dobrush s theorem cocers the fuctuatos of a array of o-homogeeous Marov chas. For each 1, et {X ) Marov cha o X wth trasto matrces {π ),+1 = π),+1 ta dstrbuto µ ). Let aso I addto, et {f ) : 1 } be observatos of a o-homogeeous x, dy) : 1 1} ad α = m α π,+1) ). 1 1 : 1 } be rea vaued fuctos o X. Defe, for 1, the sum S = f ) X ) ). Theorem 1.1 Suppose that for some fte costats C, sup sup =1 1 x X f ) x) C. The, f [ m C2 α 3 V f ) X ) ) )] 1 = 0, 1.3) =1 we have the stadard Norma covergece S E[S ] V S ) N0, 1). 1.4) Aso, there s a exampe where the resut s ot true f codto 1.3) s ot met. 1223

4 I [2], Dobrush aso states the drect coroary whch smpfes some of the assumptos. Coroary 1.1 Whe the fuctos are uformy bouded,.e. sup C = C < ad the varaces are bouded beow,.e. V f ) X ) )) c > 0, for a 1 ad 1, the we have the covergece 1.4) provded m 1/3 α =. We remar that [2] e.g. Theorems 3, 8) there are aso resuts where the boudedess codto o f ) s repaced by tegrabty codtos. As these resuts foow from trucato methods ad Theorem 1.1 for bouded varabes, we oy cosder Dobrush s theorem the bouded case. Aso, for the ease of the reader, ad to be compete, we w dscuss the ext secto a exampe, gve [2] ad due to Berste ad Dobrush, of how the covergece 1.4) may fa whe the codto 1.3) s ot satsfed. We ow cosder Dobrush s methods. The techques used [2] to prove the above resuts fa uder the geera headg of the bocg method. The codto 1.3) esures that we-separated bocs of observatos may be approxmated by depedet versos wth sma error. Ideed, may remarabe steps, Dobrush expots the Marov property ad severa cotracto coeffcet propertes, whch he hmsef derves, to deduce error bouds suffcet to appy CLT s for depedet varabes. However, [2], t s dffcut to see, eve at the techca eve, why codto 1.3) s atura. The am of ths ote s to provde a dfferet, shorter proof of Theorem 1.1 whch expas more why codto 1.3) appears the resut. The methods are through martgae approxmatos ad martgae CLT s. These methods go bac at east to Gord [3] the cotext of homogeeous processes, ad have bee used by others mosty statoary stuatos e.g. Kfer [10], Kps ad Varadha [11], Psy [12], ad Wu ad Woodroofe [19]). The approxmato wth respect to the o-homogeeous settg of Theorem 1.1 maes use of three gredets: 1) eggbty estmates for dvdua compoets, 2) a aw of arge umbers LLN) for codtoa varaces, ad 3) ower bouds for the varace V S ). Neggbty bouds ad a LLN are we ow requremets for martgae CLT s cf. Ha- Heyde [5, ch. 3]), ad fact, as w be see, the suffcecy of codto 1.3) s trasparet the proofs of these two compoets Lemma 3.2, ad Lemmas 3.3 ad 3.4). The varace ower bouds whch we w use Proposto 3.2) were as we derved by Dobrush hs proof. However, usg some martgae ad spectra toos, we gve a more drect argumet for a better estmate. We ote aso, wth ths martgae approxmato, that a varace prcpe for the parta sums hods through stadard martgae propostos, Ha-Heyde [5], amog other resuts. I fact, from the martgae varace prcpe, t shoud be possbe to derve Gudyas s theorems [4] athough ths s ot doe here. We ow expa the structure of the artce. I secto 2, we gve the Berste-Dobrush exampe of a Marov cha wth aomaous behavor. I secto 3, we state a martgae 1224

5 CLT ad prove Theorem 1.1 assumg a ower boud o the varace V S ). Last, secto 4, we prove ths varace estmate. 2 Berste-Dobrush Exampe Here, we summarze the exampe Dobrush s thess, attrbuted to Berste, whch shows that codto 1.3) s sharp. Exampe 2.1 Let X = {1, 2}, ad cosder the 2 2 trasto matrces o X, ) 1 p p Qp) = p 1 p for 0 p 1. The cotracto coeffcet δqp)) of Qp) s 1 2p. Note that δqp)) = δq1 p)). The varat measures for a the Qp) are the same µ1) = µ2) = 1. We 2 w be oog at Qp) for p cose to 0 or 1 ad the speca case of p = 1. However, whe 2 p s sma, the homogeeous chas behave very dfferety uder Qp) ad Q1 p). More specfcay, whe p s sma there are very few swtches betwee the two states whereas whe 1 p s sma t swtches most of the tme. I fact, ths behavor ca be made more precse see Dobrush [1], or from drect computato). Let T = =1 1 {1}X ) cout the umber of vsts to state 1 the frst steps. Case A. Cosder the homogeeous cha uder Qp) wth p = 1 ad ta dstrbuto µ1) = µ2) = 1. The, 2 T G ad m 2 V T ) = V A 2.1) where 0 < V A < ad G s a o-degeerate dstrbuto supported o [0, 1]. Case B. Cosder the homogeeous cha ru uder Qp) wth p = 1 1 dstrbuto µ1) = µ2) = 1. The, 2 ad ta T 2 F ad m V T ) = V B 2.2) where 0 < V B < ad F s a o-degeerate dstrbuto. Let a sequece α 0 wth α 1 3 be gve. To costruct the aomaous Marov cha, t w be hepfu to spt the tme horzo [1, 2,..., ] to roughy α bocs of sze α 1. We terpose a Q 1 ) betwee ay two bocs that has the effect of mag the 2 bocs depedet of each other. More precsey et ) = [α 1 ] for 1 m where m = [/[α 1 ]]. Aso, defe ) 0 = 0, ad ) m +1 =. 1225

6 Defe ow, for 1, π ),+1 = Qα ) for = 1, 2,..., ) 1 1 Q 1 ) for = ) 2 1, ) 2,..., m ) Q1 α ) for a other. Cosder the o-homogeeous cha wth respect to {π ),+1 from equbrum µ ) 0) = µ ) 1) = 1 2 Q 1 2 : 1 1} startg. From the defto of the cha, oe observes, as ) does ot dstgush betwee states, that the process tme horzos {) + 1, ) +1 ) : 0 m } are mutuay depedet. For the frst tme segmet 1 to ) 1, the cha s regme A, whe for the other segmets, the cha s case B. Oce aga, et us cocetrate o the umber of vsts to state 1. Deote by T ) = =1 1 {1}X ) ) ad T ), ) = = 1 {1}X ) to respectvey. It foows from the dscusso of depedece above that T ) = m =0 T ) ) ) the couts the frst steps ad steps + 1, ) +1 ) s the sum of depedet sub-couts where, addtoay, the sub-couts for 1 m 1 are detcay dstrbuted, the ast sub-cout perhaps beg shorter. Aso, as the ta dstrbuto s varat, we have V 1 {1} X ) )) = 1/4 for a ad. The, the otato of Coroary 1.1, C = 1 ad c = 1/4. From 2.1), we have that V T ) 1, ) 1 )) α 2 V A as. Aso, from 2.2) ad depedece of m sub-couts, we have that V T ) ) 1 + 1, )) α V B as. From these cacuatos, we see f 1/3 α, the α 2 << α, ad so the maor cotrbuto to T ) s from T ) ) 1 + 1, ). However, sce ths ast cout s vrtuay) the sum of m..d. sub-couts, we have that T ), propery ormazed, coverges to N0, 1), as predcted by Dobrush s Theorem 1.1. O the other had, f α = 1/3, we have α 2 = α, ad cout T ) 1, ) 1 ), depedet of T ) ) 1, ), aso cotrbutes to the sum T ). After ceterg ad scag, the, T ) approaches the covouto of a o-trva o-orma dstrbuto ad a orma dstrbuto, ad therefore s ot Gaussa. 3 Proof of Theorem 1.1 The CLT for martgae dffereces s a stadard too. We quote the foowg form of the resut mped by Coroary 3.1 Ha ad Heyde [5]. 1226

7 Proposto 3.1 For each 1, et {W ) the ested famy G ) Suppose that The, G ) +1, G ) wth W ) 0 = 0. Let ξ ) max 1 ξ ) L 0 ad =1 E[ξ) ) 2 G ) 1 ] 1 L2. W ) N0, 1). ) : 0 } be a martgae reatve to = W ) W ) 1 be ther dffereces. The frst ad secod mt codtos are the so caed eggbty assumpto o the sequece, ad LLN for codtoa varaces metoed the troducto. Cosder ow the o-homogeeous settg of Theorem 1.1. To smpfy otato, we w assume throughout that the fuctos {f ) } are mea-zero, E[f ) X ) )] = 0 for 1 ad 1. Defe Z ) = E[f ) X ) ) X ) ] so that Z ) = { f ) f ) = X ) ) + ) =+1 E[f X ) ) X ) ) for =. X ) ] for ) Remar 3.1 Before gog further, we remar that the sequece {Z ) } ca be thought of as a type of Posso-resovet sequece ofte see martgae approxmatos Namey, whe the array {X ) } s formed from the sequece {X }, f ) = f for a ad, ad the cha s homogeeous, P = P for a, the deed Z ) reduces to Z ) = fx ) + =1 P f)x ) whch approxmates the Posso-resovet souto =0 P f)x ) = [I P ) 1 f]x ) usuay used to prove the CLT ths case cf. p Varadha [17]). Returg to the fu o-homogeeous settg of Theorem 1.1, by rearragg terms 3.1), we obta for 1 1 that whch for 2 1 further equas [Z ) The, we have the decomposto, f ) X ) ) = Z) E[Z ) +1 X) ] 3.2) E[Z ) X) 1 ]]+[E[Z) X) 1 ] E[Z) +1 X) ]]. S = = =1 =2 f ) X ) ) [Z ) E[Z ) X) 1 ]] + Z) 1 3.3) 1227

8 ad so partcuar V S ) = scaed dffereces ξ ) = =2 V Z) 1 [ ) Z V S ) E[Z ) X) 1 ]) + V Z) 1 ). Let us ow defe the E[Z ) X) 1 ]] 3.4) ad the martgae M ) = =2 ξ) wth respect to F ) = σ{x ) : 1 } for 2. The pa to obta Theorem 1.1 w ow be to approxmate S / V S ) by ad use Proposto 3.1. Codto 1.3) w be a suffcet codto for eggbty Lemma 3.2) ad LLN Lemmas 3.3 ad 3.4) wth regard to Proposto 3.1. M ) Lemma 3.1 We have, for 1 <, π, f ) L 2C 1 α ) ad Osc π, f ) ) 2) 2C1 2 α ) ad, for 1 < <, Osc π, f ) π, f ) ) ) 6C 2 1 α ) 1 α ). Proof. As f ) L C ts oscato Oscf ) ) 2C. From defto of δ ) cf. 1.1)) ad 1.2), Because E[π, f ) Oscπ, f ) ) Oscf ) ) δπ, ) 2C 1 α ). )X ) )] = E[f ) X ) )] = 0, the frst boud foows as π, f ) L Oscπ, f ) ) 2C 1 α ). The secod boud s aaogous. For the thrd boud, wrte Osc π, f ) π, f ) ) ) 1 α ) Osc f ) π, f ) ) [ 1 α ) Oscf ) ) π, f ) 6C 2 1 α ) 1 α ). L ] + f ) L Oscπ, f ) ) We ow state a ower boud for the varace proved the ext secto. For comparso, we remar that [2] the boud V S ) α /8) ) =1 V f X ) )) s gve see aso secto [9]). 1228

9 Proposto 3.2 For 1, V S ) α 4 =1 V f ) X ) ) ). 3.5) The ext estmate shows that the asymptotcs of S / V S ) deped oy o the martgae approxmat M ), ad that the dffereces ξ ) are eggbe. Lemma 3.2 Uder codto 1.3), we have that Proof. By Lemma 3.1, Z ) L = The, by Proposto 3.2, E[f ) m sup 1 Z ) L V S ) = 0. X ) ) X ) ] L 2C = 1 α ) 2C α 1. sup 1 Z ) L V S ) 4C α 3 =1 ) 1/2 V f ) X ) )) whch tur by 1.3) s o1). The ext two emmas hep prove the LLN part of Proposto 3.1 for array {M ) }. By the oscato of a radom varabe η we mea Oscη) = sup ω,ω ηω) ηω ). Lemma 3.3 Let {Y ) : 1 } ad {G ) : 1 }, for 1, be respectvey a array of o-egatve varabes ad σ-feds such that σ{y ) 1,..., Y ) } G ). Suppose that m E[ =1 Y ) ] = 1 ad sup 1 where m ɛ = 0. I addto, assume m sup Osc E [ 1 1 =+1 Y ) Y ) L ɛ G ) ] ) = 0. The, m =1 Y ) = 1 L

10 Proof. Wrte E [ =1 Y ) ) 2] = E [ Y ) ) 2] E [ Y ) =1 =1 =+1 Y ) ) ]. The frst sum o the rght-had sde s bouded as foows. From o-egatvty, =1 E [ Y ) ) 2] ɛ For the secod sum, wrte 1 E [ Y ) =1 =1 =+1 E [ Y ) ] = ɛ 1 + o1)) 0 as. Y ) )] 1 = E [ Y ) E [ =1 From the oscato assumpto, we have that Therefore, sup 1 1 sup ω E [ =+1 Y ) G ) ] [ ω) E =+1 =+1 Y ) G ) ]]. ] Y ) = o1). 1 2 E [ Y ) =1 =+1 Y ) )] 1 = 2 E [ Y ) ]E[ =1 = =1 = 1 + o1) E [ Y ) =+1 ]) 2 Y ) ] 1 + o1) E [ Y ) ] =1 =1 E [ Y ) ) 2] + o1) fshg the proof. F ) 1 To appy ater ths resut to v ) = E[ξ ) for 2 we w eed the foowg oscato estmate. ) 2 F ) 1 ] measureabe wth respect to G) = Lemma 3.4 Uder codto 1.3), we have sup Osc E [ 2 1 =+1 v ) ) F ) ] 1 ω) = o1). 1230

11 Proof. From the martgae property, E[ξ r ) ξ s ) F u ) ] = 0 for r > s > u, 3.4) ad 3.2), we have E [ =+1 v ) F ) 1 ] = E [ =+1 = E [ =+1 ξ ) ξ ) = V S ) 1 E [ ) 2 )] F 1 ) 2 X ) ] 1 =+1 = V S ) 1 E [ =+1 f ) f ) X ) ) E [ Z ) +1 X) X ) ) ) 2 )] X 1 ]) 2 )] X 1 V S ) 1 E [ E [ Z ) ] 2 X +1 X) ) 1]. 3.6) By Lemma 3.2, the ast term 3.6) s bouded sup 2 1 V S ) 1 Z ) +1 2 L = o1), ad so ts oscato s aso uformy o1). To estmate oscato of the frst term o rght-sde of 3.6), we wrte Osc V S ) 1 E [ =+1 V S ) 1 f ) X ) ) ) 2 ) X 1] ) 3.7) +1,m Osc E [ f ) X ) )f m ) X m ) ) X 1] ) ). But, for + 1 m, we have from Lemma 3.1 that Osc E [ f ) X ) )f m ) X m ) ) X 1] ) ) 6C 2 1 α ) +1 1 α ) m. The, 3.7) s bouded, uformy, o order V S ) 1 Cα 2 2 whch from Proposto 3.2 ad 1.3) s o1). Proof of Theorem 1.1. From Lemma 3.2, we eed oy show that M ) / V S ) N0, 1). Ths w foow from martgae covergece Proposto 3.1) as soo as we show 1) sup 2 ξ ) L 0 ad 2) =2 E[ξ) )2 F ) 1 ] 1. However, 1) foows from the eggbty estmate Lemma 3.2, ad 2) from LLN Lemmas 3.3 ad 3.4 sce eggbty 1) hods ad =2 E[ξ) )2 ] = 1+o1) from varace decomposto after 3.3) ad Lemma 3.2). 1231

12 4 Proof of Varace Lower Boud Let λ be a probabty measure o X X wth margas α ad β respectvey. Let πx 1, dx 2 ) ad πx 2, dx 1 ) be the correspodg trasto probabtes the two drectos so that απ = β ad β π = α. Lemma 4.1 Let fx 1 ) ad gx 2 ) be square tegrabe wth respect to α ad β respectvey. If fx 1 )αdx 1 ) = gx 2 )βdx 2 ) = 0 the, fx 1 )gx 2 )λdx 1, dx 2 ) δπ) f L2 α) g L2 β). Proof. Let us costruct a measure o X X X by startg wth λ o X X ad usg reversed πx 2, dx 3 ) to go from x 2 to x 3. The trasto probabty from x 1 to x 3 defed by Qx 1, A) = πx 1, dx 2 ) πx 2, A) satsfes δq) δπ)δ π) δπ). Moreover αq = α ad the operator Q s sef-adot ad bouded wth orm 1 o L 2 α). The, f f s a bouded fucto wth fx)αdx) = 0 ad so E α [Q f] = 0), we have for 1, Q f L2 α) Q f L δq)) Oscf). 4.1) Hece, as bouded fuctos are dese, o the subspace of fuctos, M = {f L 2 α) : fx)αdx) = 0}, the top of the spectrum of Q s ess tha δq) ad so Q L2 α,m) δq). Ideed, suppose the spectra radus of Q o M s arger tha δq) + ɛ for ɛ > 0, ad f M s a o-trva bouded fucto whose spectra decomposto s wth respect to spectra vaues arger tha δq) + ɛ. The, Q f L 2 α) f L 2 α)δq) + ɛ) whch cotradcts the boud 4.1) whe. [cf. Thm [15] for a proof dscrete space settgs.] The, πf 2 L 2 β) = π πf, f L2 α) = Qf, f L2 α) Q L2 α,m) f 2 L 2 α) δq) f 2 L 2 α). Fay, fx 1 )gx 2 )λdx 1, dx 2 ) = πf, g L2 β) δπ) f L2 α) g L2 β). 1232

13 Lemma 4.2 Let fx 1 ) ad gx 2 ) be square tegrabe wth respect to α ad β respectvey. The, E [ fx 1 ) gx 2 ) ) 2] 1 δπ)) V fx1 ) ) as we as E [ fx 1 ) gx 2 ) ) 2] 1 δπ)) V gx2 ) ). Proof. To get ower bouds, we ca assume wthout oss of geeraty that f ad g have mea 0 wth respect to α ad β respectvey. The by Lemma 4.1 E [ fx 1 ) gx 2 ) ) 2] = E [ [fx1 )] 2] + E [ [gx 2 )] 2] 2E [ fx 1 )gx 2 ) ] E [ [fx 1 )] 2] + E [ [gx 2 )] 2] 2 δπ) f L2 α) g L2 β) 1 δπ)) f 2 L 2 α). The proof of the secod haf s detca. Proof of Proposto 3.2. Appyg Lemma 4.2 to the Marov pars {X ), X) +1 ) : 1 1} wth fx ) ) = E[Z) +1 X) ] ad gx) +1 ) = Z) +1, we get E [ Z ) +1 E[Z) +1 X) O the other had from 3.2), for 1 1, we have V f ) X ) Summg over, ad otg f ) =1 V f ) X ) )) 4 =1 )) E[ f ) 2E [ Z ) 2E [ Z ) X ) ]) 2] α E [ Z ) +1) 2 ]. )) 2] ) 2 ] [ [ ) + 2E E Z +1 X) ) 2 ] + 2E [ Z ) +1) 2 ]. ]) 2 ] X ) ) = Z ) ad varace decomposto after 3.3), E [ Z ) ) 2 ] 4 [ 1 E [ Z ) +1 α E[Z) =1 +1 X) ] ]) 2] ) + E[Z 1 ) 2 ] = 4 V S ). α Acowedgemet. We woud e to tha the referees for ther commets. 1233

14 Refereces [1] Dobrush, R. 1953) Lmt theorems for Marov chas wth two states. Russa) Izv. Adad. Nau SSSR 17: [2] Dobrush, R. 1956) Cetra mt theorems for o-statoary Marov chas I,II. Theory of Probab. ad ts App , [3] Gord, M.I. 1969) The cetra mt theorem for statoary processes. Sovet Math. Do [4] Gudyas, P. 1977) A varace prcpe for homogeeous Marov chas. Lthuaa Math. J [5] Ha, P. ad Heyde, C.C. 1980) Martgae Lmt Theory ad Its Appcato. Academc Press, New Yor. [6] Hae, A. 1963) Théorèmes mtes pour ue sute de châes de Marov. A. Ist. H. Pocaré [7] Isaacso, D.L., ad Madse, R.W. 1976) Marov Chas. Theory ad Appcatos. Joh Wey ad Sos, New Yor. [8] Iosfescu, M. 1980) Fte Marov Processes ad Ther Appcatos. Joh Wey ad Sos, New Yor. [9] Iosfescu, M., ad Theodorescu, R. 1969) Radom Processes ad Learg. Sprger, Ber. [10] Kfer, Y. 1998) Lmt theorems for radom trasformatos ad processes radom evromets. Tras. Amer. Math. Soc [11] Kps, C., Varadha, S. R. S. 1986) Cetra mt theorem for addtve fuctoas of reversbe marov processes. Commu. Math. Phys [12] Psy, M. 1991) Lectures o Radom Evouto. Word Scetfc, Sgapore. [13] Sarymsaov, T.A. 1961) Ihomogeeous Marov chas. Theor. Probabty App [14] Seeta, E. 1973) O the hstorca deveopmet of the theory of fte homogeeous Marov chas. Proc. Cambrdge Phos. Soc [15] Seeta, E. 1981) No-egatve Matrces ad Marov Chas. Secod Edto, Sprger- Verag, New Yor. 1234

15 [16] Statuavcus, V. 1969) Lmt theorems for sums of radom varabes coected Marov chas.russa) Ltovs. Mat. Sb ; bd. 9, ; bd [17] Varadha, S.R.S. 2001) Probabty Theory. Courat Lecture Notes 7, Amerca Mathematca Socety, Provdece, R.I. [18] Wer, G. 1995) Image Aayss, Radom Feds ad Dyamc Mote Caro Methods. A Mathematca Itroducto. Appcatos of Mathematcs 27, Sprger-Verag, Ber. [19] Wu, We Bao, Woodroofe, M. 2004) Martgae approxmatos for sums of statoary processes. A. Probab