is b independent identicallytkemp/280a/16.1.kolmogorov-ext-i-after.pdfin a ge kyj, theorem:...
TRANSCRIPT
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 ]