Chapter Theorem limit Central I : tasting characterizations of - - PowerPoint PPT Presentation

SLIDE 1

Chapter I

:

Central limit

Theorem

SLIDE 2

tasting

Proof

• f equivalent

characterizations of weak convergence

• f

distribution

Observation : weak conveyance of distribution follows theme

• f

how

limits

interact with

expectation

:

MCT : if

{ Xn) TX

Then

"nm IE (Xn) =

IE ( "7 Xn)

weak

III.

when

: it rn

⇒p

then

"

"

nm Emf)

• ⇐impact)

"

for

nice f

SLIDE 3

Recall

• ur goal

to

end

• f

semester :

Thou ( Central

limit

Theorem)

Tt

• X. , Xu,

. . .

be

iid with

finite

mean

m and

finite

variance

v .

let

Sn

• X, t
• t Xn

.

Then

Lfsnjnm ) ⇒

Marion , ← stand:L ,

distribution

SLIDE 4

Good

news :

CLT

wants

us

to

prove

some

weak

convergence

• f

distributions , and

we

knew

a- lot

• f

ways to

measure

that kind

• f

conveyance

.

Bad

news

:

we

need

a

new

tool

to pave at

Good

news :

the

new

tool

is

easy

to

define

and

relates

"

nicely

" to

weak

conveyance

.

Defy ( characteristic faction)

let X

be

a

random

variable

. then

its

characteristic function

is

40×4)

=

E- Leith )

= Efoosltx)) t i E-faultXD

.

SLIDE 5

Nate :

• h

, It)

determined

by the

distribution

• f

X

,

so

if X

and Y

are

identically

d.striated

, Thu

01×14--4, It)

.

What's

so "characteristic "

about the

characteristic fndien ?

Thun

( Continuity Theorem)

Tt

X, Xi

, Xz ,

. . .

be

random variables

. Then

L ( Xn) ⇒ LIX)

iff

"nm Kult)

• oh

, It )

.

SLIDE 6

Pf ( half

• f

continuity

Theorem )

EM

in

'

Assume

L ( Xn ) ⇒ L( x) .

Since

cosine is

a

bounded

Coatney

function , def

'

n of

weak conveyance gives

"

nm IE ( cos ltxn ))

=

"I

Eun ( cos It xD

= IEµ ( cos ItxD

=

IE ( cos ItX ))

Same

idea

follows

with

Kam

IE ( smltxn))

• E ( salt XD

.

SLIDE 7

Huwe

we

have

"

nm Ox. It)

• "Y

IE ( cos ItXa)) t

i IEC Scutt Xa))

"

IE ( eos HX)) t

i IE( Sia LtX))

= oh

, It)

.

µ

with

continuity theorem

in

hand,

• ur

strategy for

at

is

to

show ky4s÷H)=4ma

SLIDE 8

First

:

lets

compute

duraotwsti Anakin

9*14 for Hyun,"

,

By

definition

:

OHH

• IE ( e

it × )

• Erm." ( e

it")

±

§

, it"

e-

"%

Hd x)

= f: ein tr e

• "%

dx

.

We

need

to compute this .

SLIDE 9

let

's

compute It

:

¥ Loh

, Hif

. It!

e ""

'

• E. e
• "

"

dx]

= !! # Lei

"

e

""

dx )

= f? ixeitx # e

• ×%

dx

.

ieitx

du = -teitx

lets integrate

by

parts !

• II. ¥ ×e-×%d*

re:* e-

"

"

SLIDE 10

10×14

'

• f: ix e

"× # e

• "

k

d x

.

= ÷

, cite

• " I?

.

• f?

t t.ci

* e- "

"

dx

=

O

• t oh

, It)

.

So :

• h

,

' H)

It

⇐

• t

. By integrating,

talk

, It))

=

• th

.

SLIDE 11

So

we

get

(after

exponentially )

qft)=e-t%whmXrN#

with this

function

in

hand,

• ur

strategy for

at

is

to show uy9s÷÷lH=

SLIDE 12

How

will

we

do this ?

We'll

need

Them

( Taylor

expansion

for 0×14)

Tet

x

be

a

random

variable

with

E- ( Xk)

is finite

.

Then

01×4)

• Ipo "Y

IEC Xi)

t

junk

where

ttlk (junk) → o

as

t -so

.

SLIDE 13

Pf

( sketch)

serie, expansion for exponential

• h

,

H)

• El e it × ) ! # ( Ecitf÷ )

unit Z

"

Eli)

details

to address

① deal

with complex

numbers

(easy

to

do)

② not

all

terms ar

non -

neg ,

so

we

might

not get

countable

twenty

e't E

.

So : truncate .

D

SLIDE 14

Pf

( CLT

, sketch form )

Theall :

X, Xy

. . .

are

iid

with

mean

m

and variance v

.

Want :

fr

Su

• X , t
• - t k

we

have

L ( sniff) ⇒

Mmm,

.

Equivalent

to :{ %y÷H=et%

Deline

Ii

• Xi

.

Then

Etxi)

and NAH

.

Then

the

random

variable

In

• It

. - -TXT

should

have

hum &

It)

• e
• t%

. If

so,

same ravttforgn

,

SLIDE 15

So :

WLOG

assume m:O

and

r

• I

.

Want :

• lsyr

. It)

→

e

• th

.

Observe

Olsyr

.lt/--fE(eit" ) .

⇐ ( eitxtrneitx.la

. . - eitxnlrn)

i#t=

⇐ ( eitxtra) # feit Mra)

. . . E feit 'T

")

i:# µei÷

") )

"

• ④×.lt/rn ))

"

SLIDE 16

So

:

"in 0% It)

=

"

nm ( Ox.lt/rnl) "

n

Tay

hnm ( ⇐ I

t litY Elk.]tli%EH:) trunk)

=

"

nm (

I + O

• EI

. I

, tjunk)

"

Naw

I

'

Hospital ( after

tu kg

aatml

leg

and writing

a

• IT)

you hid this

equals

e

• t%

.

BEBO