RANDOM SYSTEMS WITH COMPLETE CONNECTIONS AND THE GAUSS PROBLEM FOR THE REGULAR CONTINUED FRACTIONS

Transcription

1 RNDOM SYSTEMS WTH COMPETE CONNECTONS ND THE GUSS PROBEM FOR THE REGUR CONTNUED FRCTONS BSTRCT Da ascu o Coltescu Naval cademy Mcea cel Bata Costata lascuda@gmalcom coltescu@yahoocom Ths pape peset the mpotat ole that adom system th complete coectos played solvg the Gauss poblem assocated to the egula cotued factos Hece usg the egodc behavo of homogeeous adom system th complete coectos e ll solve a Gauss Kuzm type theoem Keyods: cotued facto Gauss poblem adom systems th complete coectos MSC : 5 K55 6 DEFNTONS et deote the collecto of atoal umbes the ut teval cotued facto (RCF) hee the N - valued fuctos o a a a a : ak Wte as a egula () ae uque ad called complete quotets of We ll usually dop the depedece o the otato of complete quotets Defe o the tasfomato as follos ; a : N : a a a a3 : () t follos fom () ad () that fo e have a (3) Cosequetly e ca te the tasfomato of as hee get th : (4) deotes the floo fucto Usually ths tasfomato s called the Gauss map Fo e a ad a a ad N (5) Fo ay N tg fo abtay detemates e have lm a a (6)

2 The pobablty stuctue of the sequece equatos hee ad Thus ude the sequece a N a a : a a P s ude the ebesgue measue s descbed by the N (7) N (8) P a N s a a (9) -algeba of Boel subsets of thee s a pobablty measue o called Gauss measue hch makes vaat e N () s ethe depedet o Makova The f deotes the d () a N RNDOM SYSTEM WTH COMPETE CONNECTONS Defto quaduple to a stctly statoay sequece Moeove s - () W X u P s amed a homogeeous adom system th complete coectos (RSCC) f () W ad X ae abtay measuable spaces; () u : W X W - measuable fucto; () P s a tasto pobablty fucto fom W to X Net deote the elemet X th Defto The fuctos : u W X W N ae defed as follos By coveto e ll te s a u stead of u f u u f N u Defto 3 The tasto pobablty fuctos P N ae defed by P fo ay W N ad Defto 4 ssume that fo ay W N ad P X X X () f P P () P f hee s the chaactestc fucto of the set X The e defe P P X (3)

3 Theoem 5 (Estece theoem) et W X u P W The thee est a pobablty space P ad to chas of adom vaables ad N defed o th values X ad W espectvely such that () (a) P P m P m m P P m m m P (b) (c) fo ay m N ad hee be a homogeeous RSCC ad let P P -ae -ae N deote the adom vectos ad espectvely () s a homogeeous Makov cha th tal dstbuto cocetated N ad th the tasto opeato U defed by Uf P f (4) X fo ay f eal W - measuable ad bouded fucto Remak 6 ettg m ()(b) e obta P P P -ae (5) that s the codtoed dstbuto of by the past depeds actually by ths though u Ths fact justfes the ame of cha of fte ode o cha th complete coectos used fo N Remak 7 O accout of (4) e have U f P f N (6) X fo ay f eal W - measuable ad bouded fucto Remak 8 The tasto pobablty fucto of the Makov cha N + s Q P P (7) X hee X : W Makov cha s N hee X : Defto 9 et t follos that the tasto pobablty afte paths of the Q P (8) Q be the tasto pobablty fucto defed by k fo ay W ad Defto et U be the Makov opeato assocated th k Q Q (9) Q The () f thee ests a lea bouded opeato U fom W to fo ay f W () f fo ay f W W such that lm U f U f () th f e say that U s odeed lm U f U f () th f e say that U s apeodc

4 () f U s odeed ad U W s oe-dmesoal space the U s amed egodc th espect to W (v) f U s egodc ad apeodc the U s amed egula th espect to W ad the coespodg Makov cha has the same ame Defto f W X u P s a RSCC hch satsfes the popetes () W d s a metc sepaable space; () hee () hee (v) thee ests k N such that k k d k k sup Pk k N ; () X d k d P P R sup sup k hee d y y fo ay y Theoem 3 et W d be a compact space ad ; (3) the e say that ths RSCC s a RSCC th cotacto W X u P a RSCC th cotacto The Makov cha assocated to the RSCC s egula f ad oly f thee ests a pot W such that lm d (4) fo ay W hee supp Q hee supp deotes the suppot of the measue emma 4 Fo ay m N W e have m (5) m hee the le desgates the topologcal adeece Defto 5 et ay N thee ests a pobablty W X u P be a RSCC The RSCC s called ufomly egodc f fo P o such that lm hee sup P P W N Theoem 6 et has egula assocated Makov cha the t s ufom egodc W d be a compact space f the RSCC 3 THE GUSS KUZMN TYPE THEOREM Poposto 3 The fucto P P to N N Poof We have to vefy that P the N W X u P th cotacto fom () defes a tasto pobablty fucto fom fo all Sce P N P Defto 3 Poposto 3 ad elatos (8) ad (9) allos us to cosde the adom system th complete coectos W X u P th

5 ad u Remak 33 Fo ad P W X th ths RSCC cocde th the sequeces N N P (see Theoem 5) the sequeces ad s N a N N N ad assocated N s defed (5) ad (9) Poposto 34 The RSCC fom Defto 3 s a RSCC th cotacto ad ts assocated Makov opeato U s egula th espect to (the collecto of all pschtz fuctos) Poof We have to vefy the codtos fom Defto We have ad Hece fo ay ad N e have ad d P d d u d d sup P d d sup u d Thus R ad To poof the egulaty of U th espect to N th agumet lead to the cocluso that let us defe ecusvely Clealy ad theefoe emma 4 ad a ducto Fally the egulaty of U th espect to N But lm fo ay Hece d follos fom Theoem 3 No by the vtue of Theoem 6 the RSCC fom Defto 3 s ufom egodc Moeove Q coveges ufomly to a pobablty measue Q ad that thee ests to postve costats q ad k such that hee ad s the om ove U f U f kq f N f (3) U f f yq dy U f f y Q dy f sup f (3) f f sup Poposto 35 The pobablty Q cocde th the Gauss measue defed () Poof By the vtue of uqueess of Q e have to pove that Q d (33)

6 Sce the tevals u geeate t s suffces to check the above equato just fo u u We have Q u P The thus : u Q u P u u d d d Q u d u u u log u u u u u u Poposto 36 et be a abtay o atomc pobablty measue o f th F F F N (34) the fo ay N F satsfes the follog Gauss Kuzm type equato F F F N Poof Sce t follos that a (35) ssumg that fo some m N the devatve F m ests eveyhee ad s bouded t s easy to see by ducto that F m ests ad s bouded fo all N Ths allos us to dffeetate (35) tem by tem obtag F F (36) Futhe te N f F N The (36) becomes f Uf m th U beg the lea opeato defed as Uf P f (37) No let be a pobablty measue o such that The t ca be sho that hee F F F a a N N N + U f d N (38) f F (39) d th F d Theoem 37 (Gauss Kuzm Theoem) et be a pobablty measue o such that f the desty F of s a Rema tegable fucto the

7 lm log (3) f the desty F of s a pschtz fucto the thee est to postve costats q ad k such that fo all ad N + hee th k Poof et F be a pschtz fucto The q log (3) f ad by the vtue of (3) U f f Q d f d F d (3) ccodg to (3) thee est to costats q ad k such that th T f kq Futhe cosde th the supemum om Sce fo f U f U f T f N + (33) C the metc space of eal cotuous fuctos defed o s a dese subset of C e have lm T f (34) C Theefoe (34) s vald fo measuable f hch s -almost suely cotuous that s fo Rema tegable f Thus Equato (3) s equvalet th hch esults fom 4 REFERENCES lm lm U f u du U f u du u u log log u F q F N + N + U f q U f [] OSFESCU M vey smple poof of a geealzato of the Gauss-Kuzm-èvy theoem o cotued factos ad elated questos Rev Roumae Math Pues ppl37 pp [] OSFESCU M GRGORESCU S Depedece th Complete Coectos ad ts pplcatos Cambdge Uvesty Pess Cambdge 99 [3] OSFESCU M KRKMP C Metcal theoy of cotued factos Klue cademc Publshes [4] KPZDOU S O a poblem of Gauss-Kuzm type fo cotued facto th odd patal quotets Pacfc J Math Vol3 No pp [5] ROCKETT M SZÜSZ P Cotued factos Wold Scetfc Sgapoe 99 [6] SCHWEGER F Egodc theoy of fbed systems ad metc umbe theoy Claedo Pess Ofod 995 [7] SEBE G O a Gauss-Kuzm-type poblem fo a e cotued facto epaso th eplct vaat measue Poc of the 3-d t Coll "Math Egg ad Numecal Physcs" (MENP-3) 7-9 Octobe 4 Buchaest Romaa BSG Poceedgs Geomety Balka Pess pp