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PII Oliveira Peres S Theorem 2 2012 t act for all ca and 7 positive - PDF document

Aug 28, 2023 •124 likes •176 views

and hitting times for Markov chains Mixing between mixing times and Overview 1 up to constants Equivalence of large sets times hitting different sizes of sets comparison for times 2 flitting Refined mixing and hitting equivalence 3 in

  1. and hitting times for Markov chains Mixing between mixing times and Overview 1 up to constants Equivalence of large sets times hitting different sizes of sets comparison for times 2 flitting Refined mixing and hitting equivalence 3 in a finite state space S X chain Let irreducible Markov be an the P X Let be transition of matrix ti je S Phi jl Picktaj invariant distr STP I Pth y if X aperiodic then x y also t so t step as is prob distr S and Let be 2 u on µ Hp Uktv VIA Anyang 1 µ A ITH w m ax Il Ptc DH minft 70 twixt dit E e t e e 0,1 e turix t twin called reversible X V x.ystcxlpcx.yl utyply.in is minft 0 Xt c A Ex TA where TH 7A x riff lazy version of X R PII Oliveira Peres S Theorem 2 2012 t act for all ca and 7 positive constants reversible s.t ca Markov chains f twixt Catala ca tach lazy twin Katalin

  2. Proof of lower bound twix 3 Catala L a 8 twix Zg t Let E 3 twixt PYX 17 A TCA go Pth A then A ICA t x Take with Iz fo So Geo Zz TA E 16 t t E Za D my reversible lazy Aldous 82 all MC's for tueixkyyqx.CAT fTaI Remark essential Reversibility is Ct laziness biased Exercise 1 RW Eu Consider on a R PII 3 c P 2 Zn ITI show twixt m2 and H x TH a kn the then If theorem is false Remark 12 a Exercise 2 2 cliques on u vertices joined an edge by kn kn Show that and tula twixt m2 kn if a of Theorem 1 Proof Clipperbound Definition at geometric time Mixing a Let Ze be taking values in of parameter 1 a geometric v.v and X of indep myx 11 Px X7t Ilku Define DG Ctl doit e I and ta t 0 geometric mixing ruin

  3. instead of geometric we take Remark If Ut to be uniform on 7 t the Cesaro mixing time this to then rise gives Show that dolt t Exercise 3 is in decreasing Aldous lazy For all reversible chains taktaix Ideas Theorem 2 was and For all Winkler chains ta Katield Theorem 3 t a c 12 Immediate from Thur's Pf of Thou 1 2 and 3 D 2 tula Pf of Thur 3 to easy Ip to constants to ha tuk we I a prove 8 a CBI 3 Iz t c ta find want to a set B with We s t Let 3 Ot for max Ex 2B positive constant O some t c ta Pz Htt C A C ICA A F s t t Ig ICA Pg Htt c A B 3 ICA f y 3 f Claim ICB t ftp.cotlyl PylXZtyA P ICA A yButyPg ft CA Z EI EI c TCB TIBI t ICA CA E stCBI f f D e Dt for Eaters that a suitable constant 0 we will prove assuming leads contradiction to a Markov's ineq Rz By 20Mt 2B 7 E 1 2M ME IN Pz Xzt c At Zt Pz XZt e Al 2B L 20Mt Pz 7 3213,213 28 Mt 2B memoryless property of Zt ymeifg Py XZt EA and strong Markov at EB

  4. Pz 201ft CBC 20Mt It HCA Ig Patt 20Mt Pz TBC 20Mt 20Mt L gu ICA I Lg 1 20Mt L Fm fun g I 20M D qq.rs PzCXztC A 74TH f 2 I Choosing shows Pz Xzt EA ICA M large I enough Taking which contradiction is I a Idea of geometric mixing due to Oded Schramm Ex Ax Ax is randomised stopping time tstop m ax ruin a s.tl CXn 7 stl I Baxter and Chacon 76 filling rule Lou'asz Winkler Aldous reversible Thou 2 tstop k twixt stop E 8 twin easy The hard to 1 direction stop 2 twix show is Prove that for reversible chains tstop E 8twix Exercise 4 flint to define Use separation distance appropriate stopping time an u

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