http://math.berkeley.edu/~ilc/talks/2019/ymcstara.pdf
Free Stein IrregularityYMC * A 2019 - University of Copenhagen
Ian Charlesworth ( UC Berkeley )Joint with Brent Nelson ( Michigan State University )
What is free probability ?
÷::÷:::*⇒⑤von Neumann algebras
with states
( Fln ,µ , E) ← a)
IR - valued bounded random variable self-adjoint xeM ,
Independence ← tensor product free product
Hit ii. d. IGXI -0µm,s ) [email protected]
← #Information theory
Shannon entropy Free entropyFisher 's Information measure Free Fisher Information
LT1R ' , D8 ) I ilpi ,d8m) ? ? ?
II. → In random variables-7 Law : {" nice " test functions} → ①
* s.in
onyR"
→ HI , ,→In)dµ
X , , . . . > XnE Msa
→ this . . . ,xn : ① LT , . . . ,TnY → ①"non-commutative law
"
p→ 4( ptxs.mn))
When n - I , µ×.lp ) = 91pA)) = ↳ It )dR¥trd measure )
Regularity properties
( " niceness " of µ , ,→×n) ⇒ (interesting properties of WH.int )
④ xi , .no , Xn) > I W*fx , , . . . ,xd has no( " x , ,.→xu are easy to
approximate ⇒ Cartan,is non -M , Lvoicuksa
by matrices"
) is prime kitFree skew field (
"
norational functions
"
)
x. , - → xn )=n ⇒ 64Th .→ 1¥
.
d)
( " 4. → xn become nice afterwith T , ↳ × ,
" iated " " " ""
small perturbations by free [ MaiSpeicher Yin]
Brownian motion ")
( classical ) Fisher InformationIf I know the distribution of X
- ELXI, how much do I
learn about ELXI from sampling × once ?
X - EG3 - no , D - not much X - ELX] - it IS , - lots
## -1.1The smoother the distribution of X , the less you
learn .
ILX ) = 118*711 ? where D: HRidpY→MdtdpDp → ( djpi )
is viewed as an unbounded operator
What is the non-commutative Jacobian ?
Jj : ① CT , , . . . . 7→ CUT 's . - ⇒ ④ CNT . . . - →TN3P
linear,
Leibniz rule,
with 7. Ti = of 1×01 .
( Note , if nil , and viewing ECT ,I ① Ls ,t3 ,
J.pt , ) = pls ) - pit)
It. )
Set I : ECT . . . .in#MnLehTs.-TnY0EhT....iTnY)pi-l2;Pi)
For A- EMILY MOD) , if JHELUM )"
so that
{ H , PIX " . . ,kD= LA , If ix. . . . . ,XnDV. Peek . .
thensay
Aedomtfx with J×*A=H .
The free Fisher information of X
IE4X ) : = {11051115if I :-( . . edomI¥otherwise
.
What if we only ask for It to be close to doing ?
The free Stein irregularityof X
8*1×1 := dist 11 , domtfx ) .
Also,free Stein dimension
→
JLX) : - n - ETLXT = dimmµo×n⇒dom8×
Example : X - * c- Mule )set p ;fIk-H]/lx - I ;)
set gdtt-2.IE . Then lkxa,
to
iffa-j-blgapl-j.EI.ITa) " " Also , pjlx )=o .Then if he doin ,
= ? } " "=fI dply.mil Hence has :D -0It follows that domJ*= face
,dp ) where adti.li ) ;
is supported on the off - diagonal
That is ,* 2
Ill - act ? = In , so EIX) ← th o idimmndom # I 1
- I
Hence • LX) =L .
Theorems : 2*1×5+5145 I [ * IX. Y ) with equality if X andY are free
TIX ) ± 89×1 with equality if a- I
TIXI is an algebra invariant
%xyE
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