Chapter distributions Conveyance of I : goal for weeks : - - PowerPoint PPT Presentation

Chapter

I

:

Conveyance

• f

distributions

Our

goal for

last

two

weeks : understand

the

Thon

( Central limit

theorem)

Ippon

Xi, K2 ,

• . .

are

iid

with

mean

m kn )

and

variance

v ka)

Lf Hth?-mI)

weakly converge

to

Marian

NwstvfffrtodayWDef.me

weak

conveyance

for

distribution .

We're

already

discussed

convergence

• f

random

variables :

strong

says

IP ( im Xu

• X)
• l

l"

weak

says

"I Pl Hn

• Xl > e)
• O

(Teresa!)

But

what

is

anyone

in

distribution ?

Det

(weak

conveyance

• f

distribution)

let pi , µ ,

be

distributions

• n (IR , 93)

{un) weakly

convey

to

µ , denoted

pun ⇒ µ

if for

all

banded, continuous f: IR-

SIR

we

have

"I Eun tf)

• T

T

i

same

as

£ fit) Mulde)

§

, HttpCdt)

EI

let

D= Laid

and

R

• X .

let A

, Az

• (

As :[ oily)

Ay

• Ey

As :( I. 4 ) Aa :(4,11

Aa

• lo ,'s]

Ag

• l's, IT,

• Define

Xu

• HA.

we've

seen

Xn away weakly to

Claim : if

we

let

µ.

then pin ⇒ µ

where

µ

is

the distribution of

the tea function .

lie.

So )

let

f

be

a

bounded, continuous fraction

Then

Eplf )

• IE,( flat)

f- to)

(here

X

zero

function

,

and

XY )

On

the

• ther

hand

ftp.lfl-IEnff/XnD--fIo)1fAnc)tfll)HAn)

As

a -3N,

we

knew

Alan )

• so

and

Hoti ) -11

s .

by Ep . (f)

= Emf).

Bd

Hau

can

we

determine weak

conveyance

• f distributions ?

we'll

stale

a theorem

wth lots

• f equivalent

formulations . Recall : if

AER ,

then

The

bounty

at A

is

JCA)

• { XEIR

(x

• E, #E)AA-70

and (

x

• fete)AA¥oB

,

Tqm ( Equivalent

characterizations for

weak distribution comymu ) The

following

are equivalent

("

Mn ⇒ M

'"

"I

pin IN

• petal

for

all

A

with platt)

• O

l "

by

pen

(tax)

• pull
• aid)

for all

x with

MIKI)

• O

(4) ( Skorohod's theorem) there

random variables

Y , Ya, Ya,

• ( ou

(oil]

under

lebesgue

measureI

with LIY)

• M

and Llyn)

• fun

and

Yn → Y

strongly

(5) "

for

all

banded, Boel - measurable functions f

with

µ ( LxHRsdif)=0

Df

Preotstntegy

:

( t)

( 2)

w

(5)

( 3)

\, as

For

now, we

prove

Cor ( weak

convergence

implies

weak convergence)

Ippon

X, Xi, Xz ,

are

random

variables

• s. that

His

convey weakly

to

• X. Then

L ( Xn) ⇒LIX)

PI will

prove

this

by

showing

: for

any

z with NWo

we

have

"nm Mn (t -ah)

• pelts

where

un

• L( Xn)

and

µ

• LH)

let

z

be

given

with

a ( IH)

• O

let

E

Then

{ Xu Ez)

E l IX.

• X Ise}

u { Xszte}

Taking

probability gives

IP (

Xu Ez ) E

IP ( l Xu

• Xl > e)

Take

4ms up

• f

both

sides

"m:P p ( xn ez) stuns

• r p ( kn
• Xl >e) t

IP (XE ate)

Since

Xu

convey

to

X weakly ,

we

have "

nm p( Ha

• Xl >e) =P

"m:P lplxnez) shuttle) t lpfxezte) Now

let

E

go to

"m:P

Ip ( Xu ez)

H

H

"m:P yall

• ah)

Mt

• a. ED

Da

the

• ther

hand

{ Xs z

• e)

E { Hn

• Xl > E}

u { Xnsz )

So :

IP ( X Ez

• e) E

IP ( l Xu

• XI 're) t

RC Xasz)

Take

kouinf :

IP ( xez

• e)

s

• Hse) t

"mint Blintz)

since

Xn convey H

X weakly

:

IP ( xez

• e) ⇐

"mint RANEE)

As

e

goes

to

,

we

get

A ( Xcz)

E

"mint

p( Xn et)

since

we've

assumed

MH)

• O ,

we get

MHz)

• lplxez)

So

we

have "

must plxnez)

E IPCXEE)

• IP ( X Cz) ' "mint lpfxnee)

Heme

km

:P pans -2)

• "mint RANEE)

implies

that

"

nm Mall

• at) - "I RLXNEZ)