i:*ha:i : in probability conveyance almost Tests for sure - - PowerPoint PPT Presentation

Chapters

II :

weak

versus

Strong

lastlimci

Two

notions of conveyance

①

sure

means lP({ w

his XnH=XlwB) =/

①

means

He > 0

"

IP( Hn

• Hss)
• ①

We

saw

an example of

random variables which converge

to

in probability

not

almost surely

Motivation

more

nuanced

ways to

talk about

random variables

certain

behavior

Then I WLLN)

If

X , ,Xn,

variables and

there

exist

m , ve IR so

That IECXi)

• m

and

W ( Xi) EV

fer

all

i

Ruden variables

Sn

• I (x , t
• - - t Xa) converge to

m

in probability

V-E > 0

hun

IP ( Isn

• ml >E) =D .

i:*ha÷:÷÷÷÷÷i

:

conveyance

in probability

② Tests for

almost

sure

convergence

Thm_(weakversusStney6nuegm

Suppose

Xi , XL ,

• -

convey to

X

almost surely

Then Xi, XL;

• -

converge to

X

in

probability

¥ let

es o

be given

want "I lP(Hu

• Xl > e)
• O

we

know

Pl

' 'am Xn

let

Bn

w Er

I Xnlw)

• Xlw ) I > E)

"nm

IPL Bn)

• O

like te

un

continuity of probability

Problem

sets {Bn}

aren't

nested .

Solution : define An

• { wer

I Xmlw)

• Hulke}

We then

get

{ An} I ? An

Further :

Bns Ain

Hence if IPC ? An)

• O,

we get

a-

'

am MAN)='

So

we

WTS

that

IP ( fan)

let

we 1 An

This

means

"um X. ( w) * Nw)

( Since

the

point

{ Xn ( w)) regularly

jump away

them

Kw) )

But All

w Er

"I Xnlw) t Kw))) =D

since

the

Xn

Convey to

X

almost surely

IPC

'im Xa

• X)
• I

Heme

? An

= { wer

Xnlw) 't Kw)) and

so

PC ? An)

TBH

Notational

consequence

:

almost

sure

convergence is

strong

and

convergence

in probability is

Since

almost

sure

convergence

is

so desirable

we'd

like

so

new

ways

to test for it .

keltesttorslneyconvergmletxxi.kz

,

• -

be

random

variables

all

a > 0 we

have IP ( Hn

• Xl > E
• i. o)

Then

Xi , Xt ,

• converge to

X almost surely .

If we

want

BC

':X.

• X ) - l

Now

{ wer

"I

Xnlw) = Xlw) }

lxnlw)

• Hulke}

the > o IX. (w)

• Xlv) KE
• a. a . }

• IX. Lw)
• Xlw)l3E
• i. o . }

'

But earn

arch

'M

{ wer

lxnlw)

• Kw)l3E
• i. o . }

"N%= ¥

{ wer

lxnlw)

• Hwy > g.
• i. o.}

Jtag

By

assumption

q

we

get

IP (

I Xn

• Xl > q

i

• o . ) =D

Hence

by

countable subadditivity

and

monotonicity :

IP ( { wer

I Xnlw)

• Xlw>I > E
• i. o

'

• IP ( Y

{ wer

lxnlw)

• Xml > q

• i. o .})

E §

Ipf Ewer

• Hull sq

• Eo

Cor ( Bord- Cartelli

test for stroy

convergence)

Ft

X , Xi, Xz,

• i

be

random variables

so

That

for

all

E > 0

we

have §

lptxn

• Xl >e) <is ,

Then

the

{Xa)

conveys

to X

almost surely

PI By

Borel

• Cartelli , for

all

E > 0 we

have

Rl

l Xu

• Xl > E
• i. o . )
• O

The

last result ,

we get

almost

sure

conveyance

④