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

SLIDE 1

Recall

Thm

OliveiraPeres S

t

a ez

7 Ca and

ca

O

s t

t finite reversible lazy Mc's

ca trial

twixt

ca

thx

Recall tHCxl

Max

A Cats

Ex Ta

Removing the reversibility assumption

q

twixt NZ

tula

ku

ta

72 In

3

ngag Ex Ty

kn

For

a E 6,1

define

x

A

Att o

At

x

ft 70

and

define also

2A

infft

Xt CAt

for

A

At t

• Theorem

Winkler

S

fix

act

Then there

exist two

positive constants ca and ca

st

for

all

irreducible

finite

Markov chains

if

tmovCxl 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 ghttp

Recall that ye

a Extra

Remark These 2 ineq

are sharp i e

t Ocacpetz there exists

an irred finite

Markov chain for

which

all three terms

are equal

B 1a

is

a boundary

case in the

sense that

xp 12

there

exists

a class

• f

ironed finite

Mc's

sit

talalltalp

can

be made arbitrarily large

SLIDE 2

lemma 1 For

an

irred finite

MC

for

any

A B Eg

statespace

define dtCA B yeaf Ex 2B and

dCA B

y.ie ExlTB

Then ICA

E

dtCA B

DHAB

dCBA

Non rigorous explanation

TCA

dtCAB

d CBAD E dtCA B

Let xeA best

Except

DIA B

ICA

Exterdt d CBAD E dt

CA B

ergodic theorem

by time

t

the chain

visits theset A A t

times

Define

TB A minft 2B

Xt c A

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 B

is

Exters

d

B A

Proof

• f 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 to

hit

B

and

then

starting from

there the time

to

hit

is controlled

by

the

second

term

Define

B

y

Eg Za

E Z

1 THA p

If

we

show

TCB

B then

we are done because

Extra

t Exceed yegg Eyka stuff t

Z

1 THE p

Claim

stCB

p

Suppose not

i.e

ICB

p and let

C Bc Then

STCC

1 p

CA

E

d A C

using

lemma DHA C

d GA

SLIDE 3

d CA C

e

tha p

and

d Cc A 1a

a tall f

So 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 Xt Xo

Proof

Exercise

1

flint

vCx7 Ex

Lali

x

vs

v

up

ns.t

Define

Text ftpFI ICXi

x

Show

F JP

J has to be

a multiple

• f it

Pf of C I Define

73pA minft 2B Xt c A

and

an

auxiliary Markov

chain

with

transition matrix

Q

QKyl

R ftp.A

y

x y c A

This

is

a

finite

ironed MC

it

possesses

an

invar distr

µ

Let

uly

Ppr Xz

y

ye B

Call

2 2,3A

Count

the

number of

visits to

A

up until 2,3A starting fromfr

7 2

Epl EICK

c At

E Epl 2B

i 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

ErtB

ICA

Fulla

E f HADErfTB

SLIDE 4

This

now

completes

the proof because

ICA E dtCA B

HAI d B.A tht HADITHB

dtfAB

dCBA

and

Eoka

dCB A

and

Ertl

c DHAB

v

is

supported

• n B

and

µ

is supported

• n

A

D