for Xt C At A At t 2A 0 o Winkler Theorem S ca and Then there - PDF document

Oliveira Peres S Recall Thm t finite reversible lazy Mc's a ez Ca and s t t 7 O ca twixt thx ca trial ca Recall Ex Ta tHCxl Max A Cats Removing the reversibility assumption tula twixt NZ ku ta q 72 ngag Ex Ty 3 kn In A Att o a E 6,1 ft 70 define At For x x and define also infft for Xt C At A At t 2A 0 o Winkler Theorem S ca and Then there st positive constants fix exist two ca act all finite irreducible Markov chains for if Ex ITA then tmovCxl sup A c AG catmoulx CatmovCa k twixt Griffiths Kang Oliveira Patel Hitting times 0cal PEI every irreducible finite Markov chain Then for Let Theorem I t g http t tulp 1 THA p tak t 1a Recall that ye a Extra an irred finite t Ocacpetz there exists Remark These 2 ineq are sharp i e Markov chain for all three terms which are equal sense that case in the a class there a boundary exists B 1a is xp 12 ironed finite be made arbitrarily large Mc's talalltalp of sit can

state space lemma 1 for A B Eg For irred finite MC an any d CA B y.ie ExlTB define dtCA B yeaf Ex 2B and dt CA B Then ICA E d CB A DHA B d CB AD E dt CA B Non rigorous explanation dtCAB TCA DIA B Except Let xeA best Exterdt d CB AD E dt CA B ICA visits the set A the chain t by time ergodic theorem A t times TB A minft 2B Xt c A Define spends time ICA Ex ftp.A chain in A time the By 2ps A Ex ITB the more visits to A Also time 4 after MB spent no ICA Ext MB A L Ex CEB D CA B is Exters d B A and Proof A with Fix ICA of Theorem I a set 2 1 tulip Want to Extra show E TH t 1a we first a CBI p with so that a set Want define wait to to B hit hit there the time and to is controlled B then starting from the second by term E Z 1 THA p Eg Za Define B y are done because If show B then TCB we we yegg Eyka stuff t 1 THE p Extra t Exceed Z Claim stCB p p and let C Bc Then 1 p not ICB STCC i.e Suppose d A C lemma E CA using d GA DHA C

a tall f d CA C tha p and d Cc A e 1a So which is because ICA ICA a contradiction a x B 2,1 Proof of Lemuria 1 irreducible finite Markov chain with values in S Lemuria 2 Let be an Let a pub distribution and s t be a stopping time 2 p Prue p Gl Tx x ICAI Erft ftp FI 1CXic At t AE S Then win t 1 Xt Xo Proof Exercise 1 vCx7 Ex flint Lali up vs v x ns.t Text ftp FI ICXi F JP J has to be Show Define a multiple x of it Pf of C I Define 73pA minft 2B Xt c A with transition matrix Q and chain auxiliary Markov an R ftp.A Q Kyl x y c A y invar distr ironed MC finite This it is an µ a possesses Ppr Xz Call 2 2,3 A Let ye B uly y number of up until 2,3 A starting from fr A the to Count visits 7 2 E Epl 2B Epl EICK c At i o x pk invar for Tf te Because Hx Q is µ conditions of So the satisfied Lemma 2 are stCAItErl2BTtEuE2AD ftp oE1CXieAD stCA7ErlcI Eu ITA ErtB So Ep TB ICA E f HAD ErfTB Fulla ICA

This the proof because completes now dtCA B HAI d B.A tht HADITHB E ICA d CBA dtfA B and c DHA B Ertl Eoka and d CB A and on B A is is supported D supported on v µ