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

SLIDE 1

Mixing

and hitting times

for

Markov chains

Overview 1

Equivalence up to constants between mixing times and

hitting

times

• f

large sets

2

flitting

times

comparison for

differentsizes of sets

3

Refined

mixing and hitting

equivalence

Let

X

be

an

irreducible Markov

chain

in a finite state space S

Let

P

be

the

transition matrix

• f

X

Phi jl

Picktaj

ti je S invariant distr

I

STP

if

X

is

also

aperiodic then

Pth y

step

as

t so

t

x y

Let

µ

and

u

be

2

prob distr

• n

S

Hp Uktv

Anyang 1µA

VIA

DH

m ax IlPtc

ITH w

t

e e

0,1

twixt

e

minft70

dit E e twin

turix t

X

is

called

reversible

Vx.ystcxlpcx.yl utyply.in

TH

x

riff

Ex TA

where

7A

minft 0 Xt c A

lazy version of

X

R

PII

Theorem 2

Oliveira Peres S

2012 t act

7 positive constants ca and ca

s.t

for all

reversible

lazy

Markov chains

Catala

f

twixt ca tach twin Katalin

SLIDE 2

Proof of lower bound

twix 3 Catala

a

L

8

Let t

twix Zg

E 3

twixt

A

PYX17 TCA

go

Take

A

with

ICA

Iz

then

Pth A

fo

t x

So

TA

E

t

Geo Zz

my

Za

E 16 t D

Aldous 82

for

all

reversible lazy

MC's

tueixkyyqx.CAT fTaI

Remark

Reversibility is

essential

Exercise 1

Consider

a

biased RW

• n

Eu

Ct laziness

3

c P

R PII

2

Zn

ITI

show

twixtm2 and H x TH a kn

Remark

If

a

12

then

the

theorem is false

2 cliques on

u verticesjoined

Exercise 2

by

an edge

kn

kn

Show

that

twixt m2

and

tula

kn

if

a

Proof

• f Theorem 1

Clipperbound

Definition

Mixing

at

a

geometric time

Let Ze be

a

geometric

v.v

• f parameter

taking values in

1

and

indep

• f

X

Define

DGCtl

myx 11Px X7t

Ilku

and ta

ruin

t

doit e I

geometric mixing

SLIDE 3

Remark If

instead of geometric

we take

Ut to be uniform on

7

t

then

this

gives

rise

to

the

Cesaro mixing time

Exercise 3

Show that dolt

is

decreasing

in

t

lazy

Aldous

Theorem 2

For all

reversible chains

taktaix

Ideas

was and

Theorem 3

For all chains

ta Katield

t

a c12

Winkler

Pf of Thou 1

Immediate from Thur's

2 and 3

D

Pf of Thur 3

to

2 tula

easy

Ipto constants

we

prove

to ha tuk

a

I

8

Let

t c ta

We want to

find

a set B

with

aCBI 3 Iz

s t max Ex 2B

3 Ot for

some

positive constant O t c ta

F

t

A

s t

Pz Htt C A C ICA

Ig

ICA

B

y

Pg Htt c A

3 ICA

f

Claim

ICB 3 f

P ICA

yButyPg

A

t ftp.cotlylPylXZtyA

EI

ft CA Z

EI

c TCB

CA E stCBI

t ICA

f

TIBI

f

D

we will prove that

assuming

Eaters

e Dt for

a suitable constant 0

leads

to

a

contradiction

By

Markov's ineq

Rz

2B 7

20Mt E

1

2M

ME IN

Pz XZt e Al

Pz Xzt cAtZt

2B 2B L 20Mt Pz 73213,213 28Mt

ymeifgPy XZtEA

memoryless property of Zt

and strong Markov at EB

SLIDE 4

HCA

Ig

Pz

It

201ft CBC 20Mt

Patt 20Mt Pz TBC20Mt

ICA

Lg I 20Mt

L gu

20Mt 1

fun g

I

20M

L Fm

Choosing

D qq.rs PzCXztC A 74TH f

I 2

Taking

M

large

enough

shows Pz Xzt EA

ICA

I

which

is

a

contradiction

I

Idea of geometric mixing due to

Oded Schramm

tstop

max ruin

Ex Ax

Ax is

a

randomised stopping time

s.tl

CXn 7

stl I

filling rule

Baxter and

Chacon

76

Aldous

Lou'asz Winkler

Thou 2

reversible tstop k

twixt

stop E 8twin

easy The hard

direction is

to

show

1

stop 2 twix

Exercise 4

Prove that for

reversible

chains tstop E8twix

flint

Use separation distance

to define

an

appropriate stopping time

u