Chapters IID and II a : SUN new and sun proofs of WUN Last - - PowerPoint PPT Presentation

Chapters

II

:

IID

and

a

new

SUN

Last time

:

proofs of

WUN

and sun

Downside

:

hypotheses

are

awkward

and

hard to

verify

.

To fix this

:

create

a

" new

"

SUN

with

modified

hypotheses

"

I

• dds:*. entails

③ State

a

modified

SUN

Defy

( Identically

Distributed )

A

collection

• f

random

variables { Xilie±

is

identically

distributed it

for

all

seek

and

any

i.je I

we

have

IP ( Xi ex)

• IPCX; ex)

Note :

slightly

different

definite'm

than

• ne

in

the

text.

EI

suppose

• A. B

have

MA )

= RCB)

. Then

we

get

Alka Ex)

= { play

it

"

a.* ,

1

if

x > I

= { far,

it

"

a.* ,

1

if

x > I

=

IPC IBEX)

I

let f- EAD

under

lebesgue

measure

, and

let

An

• {

w Elo , D

: nth

decimal

• f

w

is

7 }

So A

,

=

[ Ho , %)

Az

• [ Hoo, shoo ) v C'Huo,

'shoo) v

• - - u [9%0,9%0)

Az

• [ Hmo , %.o) u (

'7ham,

'%ooJu

• - - u [ 94%..

. 99%..)

Each

An

has

BC An)

=

'to

By the

last

example ,

the

{ An) new

are

identically

distributed.

x

{ Ian} NEIN

theorem

{Xilie #

is

identically distributed

iff

fer ay

measurable

f : Rt IR

we

have

CECHXI) )

• Iefffxj))

for

all

i.j

c-I

FI (⇐) let

seek

be given

We want

IPC Xis x)

=

IP ( Xj Ex )

for

all

i

C- I

Note

for

f

• Hea

,

, we

have Efffxi))

= ( Xi Ex)

(⇒ )

we have

lP( Xi ex)

• IPCXJ Ex)

fer all

KEIR

and

all

i.j

C- I

Using The

uniquest

result

them

the

extension Theorem,

This

is enough

to

prove

that PIXIE B) =lP( Xj EB)

for ay

Borel

B EIR .

We

proceed

to

show

IEC flxi) )

Eff (Xj )) for

ay

Borel measurable

function f

by

working through

cases :

① indicate- fxns ④simpletons

③ non

• negative fxn,

④

general

fins

Suppose

f- =

HB

where

B

is

Borel

Tun

f-(Xi )

= {

1

if

Xlw) EB

O

if

Xlw) # B

"

# Xi

• ' CB)

( Hxi))

• lP( Xi

' IB))

• IPCXIEB)
• RCXJEB)
• IEHIX;))

Now if f

is

simple lie,

limff) Ica )

then

f- = E

Ci IB ;

where

Bi

= f-

'(Ci)

use

linearity

• f

expectation

to

show

(Efflxi ))

• IECHXJ))

i

i

If f

is

a non

• negative function ,

we

can tried

a

sequence of

simple,

non

• negation fn

: IR → R

u.li

{f n) Tf

( recall

:

• f. (x) {

co

• " Lion Hx)) if ftxkn

n

if flx) > n

Use MCT

to

argue

IECHXI) ) = "T

IE ( fnl Xi)) =

"I Elfnlxj ))

• Eff(Xj))

Finally , handle

general f

by decomposing

f- :

ft

• f
• .

Dan

Cer ( identical distribution implies equal

moments)

If

{Xilie *

are

identically

distributed, then all

moments and

central

moments

agree

:

IEC

( Xi

• Mk )

= Efcxj

• m )

")

II

choose

f-(x)

• ( x
• y )

"

and

apply

previous

result .

④

Define Iiit)

A collection

{Xi lien

are

called

" did

" if

they

are

independent

and

identically

distributed .

⇐

let

An

= { we lo ,D :

ath decimal

is

7 }

.

Then

{ Han)

are

iid

therm ( SLLN ,

version 2)

Suppose

Xi, Xz,

• -

are

iid , and suppose that MEIR

is

the

common

mean

for The

random

variables

Sn

= I ( X, t

• - t Xu)

convey to

m

almost surely

:

pl

hnmsn

• m )
• 1

PI

Technical

and

dense .

p④

,