valued not difficult weak independencetsunoda/lecture/2020/ref/...2020/12/02 · possible extension...
TRANSCRIPT
Weakon Independence (not treated ) 1. Path
leve.la/tention2.Higherlevelattention
= {W : [0 . i ] → R : w isconti.no ) = 0 )
Fact a is a Bana de sp.by the norm N. - Walk = suplw.lt ) - wilt ) 1
, W.inEG.AE0 , T]
( Banadi Exercise )
Let A G.fi BCG) .pt
.tw?CER.Thml.9cWienermeas.
0 = t tk
P (mero : Wai) www..IE Ai )
P (waiwai ) -Want EA i ) ftp.ti.tn
Def 1 .
1 0
a Brown i an Motion if its
distrib.at/nisPo.FixanR-valuedi.i..dr.v!sK.X2.
Imai
Note that 2mcan.be regarded as a Coraluedr.ir .
Under reasonable conditions , the following holds .
• LLN : 2 0 1 E ) = 0 .
weakly
Dansker '
s
.
LDP : satisfiesa.sn LDP with speed n and some rate function .
Higher Level attention • PIR) : the set of allprob.meas on Run ,
• R - valuedi.i.dr.is •
Letir Sxias wo' Sci Dirac meas.at xER.K.x.in random
. DM : random
theories) Rmeas.MX 2 X 3 ×4
Several known facts
These knit theoremsimply Ones fort.I.d.sum.es Further higherlevellimittheorems.HN/LDP) Level ← Level 2 ← Level 3
ii. d sum empirical empirical measure Process (m)
.
• Let × called Configuration sp
. . . Here is a Particle at
7101=714=111:43 0 0 h no y
• R 12 1 ) (torus)
' Milk ) : the set of cell measure on T
with Total mass bounded by 1 .
e For each REI , define a : → R
.
Exercise : {en ) ortho nominal System in
e Define the distanceonlhbya.aem d ) 1 ek) - ek) 1
Where e ) eck.timeas.eifunc.at?Fact:eMtiscpt.andPdish.eTnDRinM+
cont.fm ma
T.EC f )
define the Product measure on Xw by
.
=
- Ticoterre oo
Seem ,
random ( XN
Under we regard Ear as Mevaluedr.ve
→ prob.CA 1 . P )
Thm 2.1
Under VJ, Mt valuedr.v.nu convergestopcnduinprob.i.e.VE?O d ( pas du ) )
= 0
• random but pas du : non - random
.
d jliy.tk) - du.tk > 1
) dul
,indepundervjiE.ir[ ( ( E
p 1 - p = 0 )
E Evj [ ) 1 E 2
as →
2 . 2 CLT for empirical measure (
We Want to consider the fluctuat Ion of T
• Due to the Singularity of fluctuati.ms , We Can
'
→ Need to
Note -4G .
Hn { fe t.ee?rRusJ.hth-orderSobdevsp.
For each MEN , H /
Facts :
.
(G) ( G .
e Can be regarded as an Hi valuedr.ir Ei 1
)1hshhd f 2
2 22'2 @RER
Thm 2.2
conveges weakly to an Hi -valued
Gauss r.v.ywhosecovariance.is given by E [ (GISH) 1 % 1 1 -
pas ) Gm Hindu G , HE He
Sketch of the Proof . We See the characteristic function of Yw
Eoj [ "" 1
, HEHi
H Hn
IgEEap )du .
This imp.li se that the finite dimensional distribut.cn of the I Init is the des Ired Gaussian .
Note g 1 Claim :
Conti Hi '8 7
E [ exp 11-1131
def [ap
Taylorsince , is a Product meas.eu IgE Eexp { i etmulnee E IgE [ Elg
1 + E [+[Dterror)
= [ + tenh
)
0) +01 " ( 1 - pas ) Hindu
Hi Conti . )
leve.la/tention2.Higherlevelattention
= {W : [0 . i ] → R : w isconti.no ) = 0 )
Fact a is a Bana de sp.by the norm N. - Walk = suplw.lt ) - wilt ) 1
, W.inEG.AE0 , T]
( Banadi Exercise )
Let A G.fi BCG) .pt
.tw?CER.Thml.9cWienermeas.
0 = t tk
P (mero : Wai) www..IE Ai )
P (waiwai ) -Want EA i ) ftp.ti.tn
Def 1 .
1 0
a Brown i an Motion if its
distrib.at/nisPo.FixanR-valuedi.i..dr.v!sK.X2.
Imai
Note that 2mcan.be regarded as a Coraluedr.ir .
Under reasonable conditions , the following holds .
• LLN : 2 0 1 E ) = 0 .
weakly
Dansker '
s
.
LDP : satisfiesa.sn LDP with speed n and some rate function .
Higher Level attention • PIR) : the set of allprob.meas on Run ,
• R - valuedi.i.dr.is •
Letir Sxias wo' Sci Dirac meas.at xER.K.x.in random
. DM : random
theories) Rmeas.MX 2 X 3 ×4
Several known facts
These knit theoremsimply Ones fort.I.d.sum.es Further higherlevellimittheorems.HN/LDP) Level ← Level 2 ← Level 3
ii. d sum empirical empirical measure Process (m)
.
• Let × called Configuration sp
. . . Here is a Particle at
7101=714=111:43 0 0 h no y
• R 12 1 ) (torus)
' Milk ) : the set of cell measure on T
with Total mass bounded by 1 .
e For each REI , define a : → R
.
Exercise : {en ) ortho nominal System in
e Define the distanceonlhbya.aem d ) 1 ek) - ek) 1
Where e ) eck.timeas.eifunc.at?Fact:eMtiscpt.andPdish.eTnDRinM+
cont.fm ma
T.EC f )
define the Product measure on Xw by
.
=
- Ticoterre oo
Seem ,
random ( XN
Under we regard Ear as Mevaluedr.ve
→ prob.CA 1 . P )
Thm 2.1
Under VJ, Mt valuedr.v.nu convergestopcnduinprob.i.e.VE?O d ( pas du ) )
= 0
• random but pas du : non - random
.
d jliy.tk) - du.tk > 1
) dul
,indepundervjiE.ir[ ( ( E
p 1 - p = 0 )
E Evj [ ) 1 E 2
as →
2 . 2 CLT for empirical measure (
We Want to consider the fluctuat Ion of T
• Due to the Singularity of fluctuati.ms , We Can
'
→ Need to
Note -4G .
Hn { fe t.ee?rRusJ.hth-orderSobdevsp.
For each MEN , H /
Facts :
.
(G) ( G .
e Can be regarded as an Hi valuedr.ir Ei 1
)1hshhd f 2
2 22'2 @RER
Thm 2.2
conveges weakly to an Hi -valued
Gauss r.v.ywhosecovariance.is given by E [ (GISH) 1 % 1 1 -
pas ) Gm Hindu G , HE He
Sketch of the Proof . We See the characteristic function of Yw
Eoj [ "" 1
, HEHi
H Hn
IgEEap )du .
This imp.li se that the finite dimensional distribut.cn of the I Init is the des Ired Gaussian .
Note g 1 Claim :
Conti Hi '8 7
E [ exp 11-1131
def [ap
Taylorsince , is a Product meas.eu IgE Eexp { i etmulnee E IgE [ Elg
1 + E [+[Dterror)
= [ + tenh
)
0) +01 " ( 1 - pas ) Hindu
Hi Conti . )