i id Random Variables

A sequence { Xn} E.,

of random variables

Xn : .BE, IP) → IS,B) = ( Rd, Bard ))

is called iid -

- independent and identically distributedif all the Xn are independent , and Mxn -- fix , the N

.

xp Yp Yip.

But how do we know such things exist ?In general

,we would like to construct sequences

{Xnln?,of independent random variables 1 vectors

with any proscribed laws : fun3nF , on CSB )

Mxn-

- MnFor finite sequences , this is easy , and instructive .

Lemma : Let µ .. . .pe , be probability measures

on IS, ,Bd , - . - , Csn ,BD . Definer = S

,x -

- - t Sn

F = M,

- - - Q BN

IP = µ ,- - - ④ MN

.

Then the random variables Xn : r→ Sn

Xu = In CB)= kn

Ia,xD

are indendent , and Mxn-

-

un .

Pf.

P ( X,t Bi,→

X neBD =

µ ,⑦ - -gun ( x. esp

- -- xsn : nie Bi

,.

-sane BD

B ne Bin-

- Mio--

qin ( B , x-

- x Bn)

g= Mil BD -

- -

Mn IBD

apply A Bj - Sj # n

IPI Xnt Bn ) = M¥94 -- -

Mn IBD.-M¥50 -

-ten CBD

Eg . To construct d iid 16,1 ) random variables,

set 8 boy = both e-""L

,and dm = j dy

en ( IR,BURD

.

Then equip (Rd,Blind)) with p -- mold

.

It

Burjd

i. I -- hi

,Xd) with Xnlxi

,. .

.xd)-

- kn ane it'd . 16,D .

Since Mx; has a density 8 wrt X ,

⇒ p -

-mold has density 8 - - -08 ix. , .> xd) .' 'ke

- "ik... Gift e-

"ik

( AW ) wrt id -

- yd =p dkg's,Ep5Lebesgue en IRD

=#g-df g

- { Hell'

TNd( e,Id )

Kolmogorov 's Extension Theoremwe'd like to construct ii.d. sequences bytaking products .

That means we needto be able to take infinite products ofprobability spaces .

Setup .

Want a probability measure on I say ) pi spends . - - ERN.

IRA = Icardi , i ane IR text do *

"=" fizzled . Rd ↳ RN

04,

-

-Md)↳ CK,

-

→ Xd, 99 .- - )

To take advantage of compactness results, we replace R with cell .

A :-

- Ce,

is"

.

← we give it a topology consistent withthe above inclusions co

,I Id ↳ Q

.

Def : Q is given the topology of pointwise convergence :

as

od -- pinny,x? .

. .

,at e Q converge to a c-Q iff

sokn → In Fn EN .

Theorem :(Tychonaff )Q is lsequentially ) compact .

Ie.

If cam )m7 , is a sequence in Q ,

it has a convergent subsequence Camry! .

Pf.

x? e- Co, it

Patt has a Gnu. snbseq . xY

'D he a,e- Get

xp'

C- ↳ l ]"

it xzmztk) k→ Xz C- 6,1g

:

"

zjnihd → x; c- 6,11 Take gem'D → Kk Fk

sejmiks → xj fi > j . Canek - type argument ) . 1/1

Cor.

C Finite Intersection Property ) m

If km SQ are closed subsets set. f.

,

Kit ¢ tmet,then Kit ¢

.

Pf.

Let xm E E ki. By Ty chernoff, Fant . subseq .

some→ REQ .

sink E.

Ki flak . .

: x - teems week.- Fisk . in a GE kyj,

Theorem : ( Kolmogorov)Let u n be a probability measure on do, it ,Bks NY) ,and suppose these measures satisfy the followingconsistency condition :

Unt , ( Bx co, 11) = Vn CB) FBEBCG.vn)

Then there exists a unique probability measureIP on (Q

,BLQD sit

.

Pl B x Q) = Vn CB) fBEBCG.vn).

Once we prove this , it will generalize almost instantlyfrom co

,is to IR (and then to IRD )

.

111

( g l) E B 6,1 ]