UNIVERSITY OF TECHNOLOGY. Department of Mathematics PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP. Memorandum COSOR 76-10

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1 EI~~HOVEN UNIVERSITY OF TECHNOLOGY Departmet f Mathematics PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP Memradum COSOR O a class f embedded Markv prcesses ad recurrece by F.H. Sims Eidhve, July 1976 The Netherlads

2 O a class f embedded Markv prcesses ad recurrece by F.R. Sims Abstract. By meas f a geeral type f embedded prcess we shall give a shrt deducti f sme recurrece prperties f the Markv shift. 1. Prelimiaries Let (X,L,m) be a a-fiite measure space. Let M be the space f (equivalece classes f almst everywhere equal) egative exteded real valued measurable fuctis X. A Markv peratr is a mappig P f M it itself such that PI :s; I, ad P( I ex f ) = =1 I a Pf =1 f E: H, ex ~. The dmai f P ca be exteded t L such that P is a psitive liear ctracti i L Such a peratr is always the adjit f a psitive liear ctracti i L 1, which we shall als dete by P, but w writte t the right f the fucti symbl. The acti f this psitive liear ctracti P L 1 ca, by meas f mte apprximati, be exteded t the space M It fllws that <fp, g> = <f,pg> M fr all f, g E: Here <f,g> stads fr!fgdm. With respect t a give Markv peratr P (X,!:,m) we ca decmpse the space X it a cservative par.t C ad a dissipative part D. This decmpsiti is e t E. Hpf, ad ca e.g. be fud i [1], chapter II, [5], chapter 4, 2. Fr later use we cllect the results which we shall eed i tw lemma's ad sme crllaries. Lemma 1. The fllwig statemets are equivalet. i. The cservative part f X with respect t P is C. ii. The set C is the (md m) largest set such that fr all subsets A we have A. iii. If 0 ~ f < ad Pf ~ f C, the Pf = f C.

3 - 2 - Lemma 2. The fllwig statemets are equivalet. 1. The dissipative part f X with respect t P is D. ii. The set D is the (md m) largest set such that there exists a fucti g ~ 0, with {g > O} = D ad I:=pg is buded. (This equivalece ca be btaied e.g. frm (2,5) i [1] ad the maximum priciple, chapter 2, therem 1.12 i [5J). Crllaries. I 2. PI = PI = I C, ad therefre PI D = C. C Fr every f E M we have L pf = 0 r C. =1 c If C = { I pf = O} C, ad C I = { I pf = } C, the PIC. = "'l =l 1 C I., i=o,i If A c C ad m(a) > 0, the < Ic,PI A > > O. It fllws that {lcp > O} = c. 4. Fr every s ~ I the cservative part f X with respect t p S equals C. 2. The embedded prcesses s-i Let P, Had H' be Markv peratrs, ad assume H H' P fr sme s ~ I. We defie the peratr Q by Q = L (PH' )ph =O Obviusly Q is a a-additive mappig f M by substitutig f = ) that Q is a Markv peratr. t itself. The ext lemma implies L~maa 3. If fr f E M we have Pi $ i, the we als have Qf $ f. Prf. Usig psf $ f, we easily verify by writig ut -I I (PH,)rpHf (PH,)f $ f. r=o Hece it fllws that Qf $ f.

4 - 3 - If H is the multiplicati by the characteristic fucti f a set A~ the multiplicati by la" the Q is the embedded prcess the set A. The situati that H is multiplicati by a fucti f, 0 ~ H' f s 1, ad H' multiplicati by the fucti 1 - f. is studied i [2J, [4J. I bth cases we have H H' = P, hece s = I. The situati with s > I ccurs whe we are ivestigatig recurrece prperties f the Markv shift, as We shall see i the ext secti. Therem 1. Fr every f E M we have I Qf = I pslhf. =1 =O Prf. p(-l)si Hf = P(H' H)P P(H' H)PHf = L (PH') IpH{PH ' ) 2pH l kk= (PH') kphf ad therefre = L L (PH') IpH =l O kk= (PH') kphf.. ~ L (PH I) IpH (PH') "'phf ~=O Therem 2. The cservative part f X with respect t Q is {leh > O} C, where C is the cservative part f X wi th respect t P.

5 - 4 - Prf. L Suppse A c {I H > } C. c By therem 1 we have It fllws by crllary 4 ad crllary 2 that this sum is 0 r ~ C. Put C = { I Q 1A = } C, =l C 1 = { I Q 1A = } C, =l the psi C = 0 C 1 ' ad sice PHI A we btai I ~ ps ph1a =O 0 c ~ I ps ph1a = C =O ~ I pspsl c = 0 C I, =O 0 hece i particular PHI A = 0 C. This meas <lcph1a > = 0, ad therefre by crllary 3 S~ce A O c A, we have m(a) = 0, ad A. Hece by lemma 1 the set {lch > O} C is a subset f the cservative part f X with respect t Q.

6 A that O} C, the <ICHI A > = 0, hece RIA = 0 C. It fllws C, hece A is a subset f the dissipative part f X with respect t Q. iii. Let g be a fucti with {g > O} == D ad r is buded. The =OP g 0:> L Qg = L psphg ~ L psg ::; L p g <, =1 =O =1 =O ad D is a subset f the dissipative part f X with respect t Q. Crllary. If s = 1 ad H is multiplicati by a fucti h with 0 ~ h ~ I, the {lch > O} C = {h > O} C ad we btai a result f Li [4]. 3. Recurrece prperties fr the Markv shift I this secti we shall give, with the aid f the peratr Q f the previus secti, a fast deducti f sme recurrece prperties f the Markv shift. These results g back t a paper f Harris ad Rbbis [3J. Let S be a measurable trasfrmati a measure space (,F,M). A set A is said t be uaderig uder S if pits f A retur t A uder the acti f S, ad recurret if M-almst all pits f A retur t A uder the acti f S. A set is said t be dissipative if it is (md M) waderig sets, ad cservative if every subset is recurret. Obviusly, if A the ui f cutablymay is the subset f A f the pits which retur exactly times uder S t A the A is waderig. Hece, if almst all pits f A retur fiite,. may times t A, the A is dissipative. Nw let (~,F,M) be the realizati space f Markv prces P (x,i> with iitial prbability m, i.e. (,F) :=(X,L), ad... PIA I>

7 - 6 - fr all AO,,A E E, where X. detes prjecti 1 -I Let F dete the a-algebra geerated byx L:,, shift trasfrmati i (,F). the i-th crdiate. X-IE ' ad let S be the Suppse that the iitial measure is such that F ca als be c'sidered as a Markv peratr M (X,E,m). (This is the case if ad ly if mea) = 0 ~ - P(,A) = 0 m-a.e.) Let C be the cservative part f X with respect t P, ad defie C = {X ~ C fr all }. Therem 3. i) The set \C is dissipative. ii) Fr every ad every A E F, the set A COO is recurret. Prf. Because f Pi = 0 C, we have \C = {X E D} =u {X E O i=1 O D } i, where D I,D 2,... is a partiti f D such that tece f such a partiti easily fllws frm I pl is buded. The exis~ D. = O 1. lemma 2. We the have L M(X E D. ) I 1 := <IP I.> < D, =O =O 1. hece by the Brel-Catelli lemma we btai M{X E D. i..} = O. Almst all pits f {X O E 1. D i } retur t this set uder S ly fiitely may times, s {X E D i } is dissipative, ad therefre {X O E D} = u {X O E D i dipative. } is als disi=1 ii) Withut lss f geerality we may assume that X = C, ad therefre C. Ch A F d d f f f c M ~ E I' a e le r every ~ s- where we csider the FO~easurable fuctis i the right-had side as fuctis X. The peratrs Had H' are Markv peratrs (X,E,m) ad satisfy (H H')f = E(f(X _ ) s I I F ) = ps-if. O

8 - 7 - Let A O be the set f pits f A which retur t A uder SS The at least ce. u =l s } i ~, S W E: A <Xl M(A O ) = l <IH(PH,)-l ph1 > = <IHQ1> =1 sice by therem 2 Q ~s cservative {IH > a}, we have QI = I {IH > OJ, ad therefre M(A O ) = <IH1> = M(A) < <Xl, hece A O = A (md M). Hece the set. s A ~s recurret uder S, ad therefre uder S. Remark I. Let P be cservative. If A E F s ' the A E: F t fr every t z s, ad we have actually shw that A is recurret uder st fr every t. Hece almst all pits f A retur t A ifiitely may times uders. Remark 2. The crucial pit i the paper f Harris ad.rbbis is that if there exists a algebra f recurret sets geeratig F ad a fiite r a-fiite equivalet ivariat measure, the must be cservative. Therefre, if there exists a fucti u with 0 < u < <Xl C, u = 0 D ad up = u~ the the meadm' sure M' defied by --- = u(x ) is (-)fiite ad ivariat uder S, ad the dm O set C is cservative. Refereces [IJ Fguel, S.R.: The ergdic thery f Markv prcesses. New Yrk: Va Nstra~ Reihld Cmpay, [2J Fguel, S.R., Li, M.: Sme rati limit therems fr Markv peratrs. Z. Wahrsch. verw. Geb. ~, (1972). [3J Harris, F.E., Rbbis, H.: Ergdic thery f Markv chais admittig a ifiite ivariat measure. Prc. Nat. Acad. Sci. U.S.A. ~, (1953). [4J Li, M.: O quasi-cmpact Markv peratrs. The Aals f Prbability l, (1974). [5J Revuz, D.: Markv Chais, Amsterdam: Nrth-Hllad Publishig Cmpay, 1975.