Recall Thm OliveiraPeres S

t a ez 7 Ca and ca O s t t finite reversible lazy Mc's

ca trial twixt ca thxRecall tHCxl Max

A CatsEx Ta

Removing the reversibility assumption

q twixt NZ tula ku ta72

In 3ngagEx Ty kn

For a E 6,1 define x A Att o At x ft 70

and define also 2A infft 0 Xt CAt for A At t o

Theorem Winkler S

fix act Then there exist two positive constants ca and ca st

for all irreducible finite Markov chains iftmovCxl sup ExITA then

Ac AG

CatmovCa k twixt catmoulx

Hitting times Griffiths Kang Oliveira Patel

Theorem I Let 0cal PEI Then for every irreducible finite Markov chain

tak t tulp t 1a 1 THA p t ghttpRecall that ye a Extra

Remark These 2 ineq are sharp i e t Ocacpetz there exists an irred finiteMarkov chain for which all three terms are equal

B 1a is a boundary case in the sense that xp 12 there exists a class

of ironed finite Mc's sit talalltalp can be made arbitrarily large

lemma 1 For an irred finite MC for any A B Egstatespace

define dtCA B yeaf Ex 2B and dCA B y.ie ExlTBThen ICA E dtCA B

DHAB dCBANon rigorous explanation TCA dtCAB d CBAD E dtCABLet xeA best Except DIA B

ICA Exterdt dCBAD E dtCABergodic theorem by time t the chain visits theset A A t times

Define TB A minft 2B Xt cA

By time 2psA the chain spends time ICA Exftp.A in A

Also the time spent 4 ExITB after MB no more visits to A

ICA ExtMBA L ExCEB DCA BisExters d B A

Proof of Theorem I Fix and a set A with ICA 2

Want to show Extra E TH t 1a 1 tulip

Want to define a set B with aCBI p so that we first wait tohit B and then starting from there the time to hit is controlled

by the second term

Define B y Eg Za E Z 1 THApIf we show TCB B then we are done because

Extra t Exceed yegg Eyka stuff t Z 1 THE pClaim stCB pSuppose not i.e ICB p and let C Bc Then STCC 1 pCA E d A C using lemma

DHA C d GA

d CA C e tha p and d Cc A 1a a tall fSo ICA 2,1 a which is a contradiction becauseICA x B

Proof of Lemuria 1

Lemuria 2 Let be an irreducible finite Markov chain withvalues in S

Let p be a pub distribution and 2 a stopping time s t

Prue x pGl Tx

Then ftp FI 1CXic At ICAI Erft t AE S

win t 1 XtXo

Proof Exercise1

flint vCx7 Ex Lali x vs v up ns.t

Define Text ftpFI ICXi x Show F JP J has to be a multipleof itPf of C I Define

73pA minft 2B Xt c Aand an auxiliary Markov chain with transition matrix Q

QKyl R ftp.A y x y cA

This is a finite ironed MC it possesses an invar distr µLet uly Ppr Xz y ye B Call 2 2,3ACount the number of visits to A up until 2,3A starting fromfr7 2

Epl EICK c At E Epl2Bi o

Because µ is invar for Q Tf te x pk Hx

So the conditions of Lemma 2 are satisfied

ftp oE1CXieAD stCA7ErlcI stCAItErl2BTtEuE2AD

So ICA Ep TB EuITA ErtBICA Fulla E f HADErfTB

This now completes the proof because

ICA E dtCAB HAI d B.A tht HADITHBdtfAB dCBA

and Eoka dCBA and Ertl c DHAB

v is supported on B and µ is supported on A D