Chapkrt win ? can I : we distribution ps.iq So with iid Last time - - PowerPoint PPT Presentation

SLIDE 1

Chapkrt

I :

can

we

win ?

SLIDE 2

Last time :

let

{ G) new

be

iid

with

distribution ps.iq So

.

let

Xu

at ?g

,

2 Ci

• 1

.

Define

"

hitting tines

"

Te

• int { n 30

: Xn

• c }

and

To

= inf { n > O : Xu

• O }

.

±um:i:÷ini÷¥÷::::

PIT.

• Ta) =/

peek

p

• Yz

c

SLIDE 3

Ex

p

• 0.49

a

• i

:f÷:f""÷÷n

99900

100000

1,8%

900001000002×10-17+17

keawwg

:

if

you

increase your

but

size

to

$1000 in

The

last

case

,

you

significantly

increase your

• dds of winning

If yer bet

$10000

at

a

time

, you're in the

a-

9 ,

c

• lo

scenario

→

88%

chance !

SLIDE 4

tMf

In

the

coin flip game

,

can we charge

• ur

gambling

strategy

to

" do

better

" ?

SLIDE 5

Dein ( Gambling Policy)

keeping

the

nothin

from

last

time

, let

Xu

• at .EE?Y::.Imae
• f ith
• f

" win "

bet

• r

" losing "

where Wi 30

and

Wi

• fi .! 9,4,
• -
• Ici - i)

.

SLIDE 6

Exe

( Bold

Play)

Suppose

we

want to reach

c

dollars

.

let

Wi

=

min ( Xi - i

, c

• Xin )

.

idea : bet the

most you

can

without going

• ver c

.

Deep theorem : this

strategy

maximizes

IPL To < To)

.

SLIDE 7

EI

( Double til you

win)

let

W , =L ,

and

Wi . {

2

" '

it

c.

=

• - -
• Ci. ,
• o

O

else

if

4--1

,

then

X

, = att = Xz=Xs : Xy

• Xs

=

• else

cut

, then

X ,

• _ a - I

and

Xz

• a-1+2

= att

• X,
• Xy
• Ys

=

• -

else

4=1 , then

X

,

• a - l

,

Xa=

a -I -2

• a-3

,

X,

=

a

• 31-4
• att

± Xy = Xs = Xie :

• you

can

check

if

N'

• influxN

: Cn -4 ) , Then

Xn

att

• Xut X.at?

SLIDE 8

Note

,

in

this

scenario

we

get

IP( "Y X n

• att )
• I

Question

c .

does

this

mean

that

" double til

you

win

"

gives

a

winning strategy

in

this game? IE ( "nm Xn )

• IE (att )
• at I

Answer

:

no

, because

your

pockets aren't deep enough

.

SLIDE 9

let

KEN

be

the

largest

integer

with

azitzt-e.tk lie ,

we

can

play

" bdovble lil you

win "

K

times)

we

have

Xµ= att

if

inffnza.cn

• MEK

else

Xk

.

• a
• I
• 2-
• --
• 2K

if iuxffn > : Cal )

> k

we

see

⇐ (Xk)

• qkfa
• I -2 -
• - -
• 2K) th
• qk) ( att )

= att

• 2(2q)k

This

is

less

than

a

if

pet

.

SLIDE 10

We've

seen

double

til you

win

doesn't actually

" beat

the

casino

" .

will

any

system ?

Recall

Xu

= at

÷2

,

Wilhoit)

= at II ,

wi @Ci - t )

t Wn @Cn

• i)

=

Xu , t Wn ( kn

• 1)

But

we

knew Wn

• tn ( Ci ,

. -, Cn

• i )

where

C

. . - - , Cn - i , Cn

are

independent

.

SLIDE 11

So

Wn

and

2cm - I

are

independent,

so

⇐ ( Xn)

=

⇐ ( Xu - it win @Cn -t))

=

IE ( Xn

. . )

t

⇐ ( Wn @ en

• t))

=

IE ( Xu - i) t

⇐(Wn) ⇐ ( kn -t)

" IE (Xn . .) t

IELWN )lp

• q)
• 30

So : if

p

• I

,

then

a - IECX, )

• IEC#

=

• if

petz , then

a > IECX,)3IECXz) 's

• if

p >

'

z , thin

as

(X.Is # ( Xz) E

SLIDE 12

Nate this gives

bum IE( Xn) sa

if

pet

.

In

contrast

, with

double til you win

, we

saw

⇐ (

"nm Xn )

• att

.

when

is

by # Xn)

• tellin Xa) ?

We

know

• f

EXIT

I;mXn

, then

MCT

says

linm # ( Xn)

• ⇐ ( "

nm Xa)

.

SLIDE 13

Thou

( Bounded convergence

Theorem)

Ippon

that

lynx.

exists

almost surely

. If

KEIR

so

for

all

n we

have

• IX. Is k

, then

"

nm Elk)

=

E- (

"nm Xu)

.

PI

Dude

in Xu

• X

.

The triage inequality

say, for;4

I ⇐(x)

• IE (Xn) I

=

I ⇐ ( X

• Xn ) /

E IE( IX

• Xml)

.

We

want

: for

all

e > 0

we

have

NEIN

so for all

n z N

we get

IIECX)

• E-( Xn ) l K E

.

we

do this

by

shewing

El IX

• Xal) -

s O

as

a -50

.

SLIDE 14

let E >0

.

Define

An

• { wer

:

I Xlw)

• Xnlwll > e }

.

We

have for

all

we.r

we

have

I Xlw)

• Xnlw)Is et 2K Han (w)

Take

expectations

⇐ ( lxlw)

• Xnlw) l )

E

IEC et 2K Iancu ))

=

et 2K IPC An)

Naw

take

kmsup

• n

both sides

"map

⇐ ( IX

• Xml)

E

Et 2K

"TYP MAN) set 2K Pl

''m:P An)

SLIDE 15

But limsup

An

• "must { wer : lxlw)
• X.Cull > e)

is the

set

• f

w

with

line Xnlw) t Xlw)

,

which

is

just

{ wer :

hnmxnlw)

DNE)

.

But

by hypothesis

this

set is

measure

zero .

Hence

"mhm

⇐ ( IX

• Xml)

E

Et 2K

"TYP MANI

• ' et 2K Pl

''m:P An)

E

E

.

S .

"msn.PE/lX-Xal) < E

fer

all

s , s.

"I'm IEHHXND

= Off