·Orbita.l A Fproo..c.~ to

Mic rosta.te. Fhe Eh1:rop(to

~b,tkcM i~a.motoo

1. Mtcrost~te Fy-ee Ent~op~__ - . n=-

M: t:A ~h, Ne~mA"'n '"l'a-ebre\.

(o)- W*- ~I3e.b""e0

L : t\. YlOV'WlAI f~i.•.h. t~tAc..i~1

stQ.te on M ..

M~": 1:Ae. se"t of All n )(n­se I f- ~tlJoil'\t MAtri c.e.s

Trt! ~ ( "51.4.111. I) TlrQ.ce. on M •••CCC).

ITv-to( ~ N T)-N

)(, ) .•• , X h: non- COMm. se.lf-Q,tAJoi"'1:"

S • Mr. '-T. "" .

N,me"'-l, ~>o, R>o

rR, Cx, ,·· · , X tl; N, W\ , ~ )

~ { (Als .. ·,Ah) ~(M~Y' Io "At It ~ R , I~;,~n

o ITrN(Ail'" A,,,,)- t..(xtl • ••X:~)I~~

:Tk set 0.J- miC."'OstAte.S

arpl"'OX iMo:tit'\a- X It .•• , X.,.

XR(XI, ..• ~X,,; m,~)

~ lft.ft. slA.....~LNI2.~ I\~h.(r:C><t, ... , Xh; tJ,rn,~~N~to r ~

+T~N ]

:kR(X I, ... , X,,) ~ ~ ~D(XI ... ,X~ lrl,}.~m~GO "J ~/i +0

:x (xl, ..., Xh)~ Sl.\p ~R(XI, ... ,Xt0R>o

: T{e. ( om i(,,\"'0 S't~'te.) !Y"ee.el'ih''Op~

0,+ X~, ... , Xh.

T~ "Le.bes,u.~l'net);s",~ "t-ICln M~-- ~

A == [Al~] EM:N

"AU 1 = Tv: (A2.) =:.~ I A;·,1HS'" 1.)tl k'"

=~Ah + 2. ~ (CReAtit+ (1m Aq )~'."c, ,,<.~ U 0 )

(1-\1•.•5: 'the. H ilb •••.t - Sc.hW\id1: l'\OYm )

Theyefore

Bo.sic ~\'"O pe.t"tie.s of ...~ _

10 (Ind e.peYl cI e.•.•c.e of Utt oH)

(:>lOO(xI,... ,Xrl.)=)~(XIJ")X~:: ~(x'J ... ,Xh)

~rtA}f) 't. R ~ W)o.)( nXi.lltQa ,~~~

1° FOr eve.\'} sin~le X ~'th.. dist. tt,~(x) =jj Jl11x.-~ltlfllC.)dt('it)-tt+{.ko~2.lt..

30 (u~rer "'ou •.•tA )

( ) l\ 12.1te ( 2. 2..~~XIJ ..• ,Xh ~T~\' h "l:X'+"'+X")j ,fO (Sl.\.b~~tli'thT~")

X (x l,••.,X" ,yIp.. ,y",,)

~ :lG(x" ... , Xh)+ X('fa, ••• ,Ym)

S° (U~f'>erSemiUll\'t ir\.u. it1- )

(X~ ...,X~)) (XI~... ,Xn)~dist.

sup n(x~ •.. , x:»"oo <+60.ti

-9 .k(X" ... ,Xn) ~ tt,msCAf. ):(xf~... ,x~lt))r<.-., f>Q •. •

~dd it:i'\1'tt~theoye.tYl .,s.o", ~

SX1) ... "Xh: non-c..otnm .. S.Gl. ~.'\r.

in '" "tV'Q,c,i AI W*- probe Sf.

~) X"." ,x'" O:re ft"eel~ ihele-~.

9' ~(XI J ••• ,Xn)= ~I) + ... + ~(Xh)

(b) Tle c.on,re.vse, ~ (tA) ~.'ds T~Mre..

~l1e.vet"- ~.(Xi.) > - 00 ~or '~i~.,.

UlN) : the I4nito..)-~ 1-~oUf cf ol'der N .(Nf:-n~)

tU{N) : the HCllAY "H>b. rne/U. On U(N),

~ Eo M~A

t\: Ais~,"i b",,1;i on ()f ~Eo M~ 'l.J". r.t tr",.

Q · U'("'I' M~ ·•..•t'," · •..j' ~ .•'" ,. ~u ~ l'\. -&q

Sl-!a~lU) ~ Ut U* £.or UEo U(N) ,

j 0 r!i.(N) --7 ttXt in mOme- •..•ts~s N ')00

;. e.~Tr•.•(tl.lN)"') ') ?:(xi")tA,S N ~.00 foy ~I'mE:""

~)W)E.IW, ~>o

f:\"b(X""" x'" ~ !Ib,a), ... , !••{N); N,rn ,~)

F 8~~~)(r(x" ..., x"'> N ,wn, ~»)= t (U""" Un) € U(N)'" I

3\ltJ~V.,... ,Un)= (tJ."!.{N)Lt.: ••. , tt~hl"')~)f r(XI,~.., Xn; N,M,~ )

:XoYh (X., ... , Xh: \, , •.•, t",)

~ iV.n itm SC4t-> [1. h> _Art ( r (X X ·m~oo......•. ~ .. ~"lU(N) .orb I" • •, h•~~() ~~to.

!IltJ), · • •.)~n (N); "',m .)X'») ]

: 'The ot"bi1;~\ ~'ree. e.htH>t:>'it ot (XI )...,xn)"

R~wn~r k:

'Xo•..b(XI) •••, Xh : ~h ••.) \h) is iodepende •..•"t

o! 'th.e. c..h.oi c..e~of Jt(N) ~ M~C\ J I'l E-IN

So ~o\'"b(xIJ .•• ,Xh~i" ,in) ~ be.

,dehc-ted ·b~ ~O~"(Xl) )Xn)

1°. ~o)-b (x) = 0 ~'" ~ Sh,'t1e ~"':AbJ~ x

2°. (Ne1A.t~uit~) ~o"'b (x." ... " X",) ~ 0

3: ( SlAbAAti itiutt'& )

~O~b (X •)" .. ) X h, Y; , ..... , Ym )

~ ~,..b (X, , ... , X••) + ~o •..b(Y' " ... , YhI)

4: (Upper seWlic,.Ol'lt;;n.LLi~)

(lie> XUU) ( X )..X, s ....• ," ,", > ..X, ) .•. , h

• ..JeIn Q 1St.

=9 'Xor\> (x, )... )Xh)~.lL:m~u.p)!O"'b(X."... "xtt)~~co .

tl

~(x,) ... " Xn)= 1:oril(x,) ....,x,,)+ ~.X(Xi\'=1 :J

Le.Wl~ I~r><.OO , R >0, £>0_m .•

3h\E",' ~:. ~>() S.t.

u V w ,v N Eo"", .) Q,J~.eR~:cli"'~nClo,' mcd:\"I(.(lS

w-i-th "DJI~~.~

I l1rtl(~It.)-+•..,..(O:-)1 <~ ) I~R~m

I ~ II nOli! --, w4 - 2; p)trr-a< £.'"

•

Ske't,h.6f -the.p~oa-5-of MDJn fOYMU,IQ.,

MtA'a- tASSt4me ~

t,,{N) 's A~ediAC)..ot\""\ tJith II \.(N)II ~ lIx all(I 10 eo

B~ Le..rn",o... ,

Vrn~ IN Vb > 0

3,NoE-fN.) 3WlOE-IN with mo ~ ~.

3 ~ . t 't" <:. b0c) > 0 UJI-tn Of) - Ts.t.

V N~ f'lo;) \ 5. Vt ~noSY D ~ (R~

I"'b-•.•(D"') - l:(Xf) \ < ~(), I~k.~\'r) 0

911D- ~i{N)", :5:.?=DWI-I (R= tn~IIXLU:0

'" ~ f'la

(A" "'J An) E rR,(X1 .•... , Xh; N, Mo,rO)

At =Ut Dt Ut' diD..1ono..li2o..t ion.

=> Dt € ~R(Xta" ..W'l(p~o)

:9 I'. Dt - !, (N)"1 ~ ~,,_b--I

\-t~lJB"I '" B~,")- -c(X", · · · X~\'" ) ,

.s. 't\""~(A~I'" A"l'" ) - +v•..•(B1., •. · BioI'") 1

+ 'Tl'" •...(A,,\ · · .A\.\," ) - 't.(x \.1 • • • X i.y) ,

<~+~ <~-1' 00 - o.

=9 (u,.) ...~U-~)

€ r:"b (xI•...•Xtt : \, (N) •... , ~h{~);N.M,i)50 th4.t we f.Ave

(Vi ~...~Un J D, :t ••• , On. )

€ r:rl>(XI •...•Xh: ~("') •..• )~("'); N. Wl.i)/Thn

)(ll,LlR.(Xt; Nt rn,6')•

T~e ceo~diW\~te. c."lU\~e

A Eo M:-~ ~ (Us D) 6 U{I'I)B.)( IR~. '"

I\.tl 3> ~)e /\tJwhere

T""ere.fore)

~(X,~ ... ,Xh)

~ tm [' en( \\- t4-::! fJi 0.0, '\ r;.(X, ....,Xh; N)Wlo)~o~

+~~NJ<: .2: [I (·er\ ""'-.n) ([ (- In!SIAf ;r: l11ur.~)e1\tJ. o~ XIrOO) Xt'I :~460

!,(N)too.,~h(N);N)m'~Y'1rhXft 1J.R.(Xi.; N)m~~~

T~lo ••NJ

= lim s~p [. 1. to 18.,. ([ ( . (N~oa.· N:l. t tfOO ()~b X,~...)X~·~I(N))..., ~h ~)

; lJ,m,~)) ]

+ t t;. r~ 1.,1\••(riiX,,...)Xh~N)rnJ~~,,=, t-a:;J..

1-i:- LIt N J

r\

~ x(x, •...> ~) ~.~'~rtlXh'·" x•..) + ~..lG(Xio)1=1

--!I~

In po..~ic..u.\c::..r:>MiMtl FO"mtA.l~ ~"e.S

-Xo~b(X"y)=~--x.(X~ y)+ X(x) +xt'f°)

-h-r tw-o t •..on - (..0 W\W\.) oS. A. Y. U"'.£ X.> Y

\n (M> t:) w-i-t~"~()() ~ X(.) >--60

i..e. J - ~orb'S ~ kind ot ft""e.e..

p~oa,~b; ';-t'~ C.Ot.th't."'r4'Yt o£- -t"h~

WlIA:tlA.tM in~YW\tA%,on, :t()(iY) ~Ol-'

~O rQQ.' ~dOWl Vl.l~l-.ble.s X ~ Y.

t* (A.; ... ; A~) : the (mic..N)st4te -free)m IA.tu."" I ~re.e i'"£0 'r'm",ti ()~ :foyo S IA.bA.l 'a- :S

At, .... , An.

Nbte: Tftis. is "t)'\le de~ X) YAre..

fyee.'-a- ; t)de,.~'er\dent

i.e. , - l:o\'b (x,Y)= ().= ;.*(W*(x); ~(y))

From "the t/\,bo\Je poin"t of ",t~\S') we t\lre

1:emt=>-t.e.d tb wnr-i-te. ~

t(W'{XI); ... ; W*(x~))~ -~on,(x••...,Xh)Q,nd lAse t-h-e te~m (l W)iCrrO&'t~.e-

, L_ • r _ • 1)lM"'-t&4.~ :rree InTO"'m~t. on. •

1 o. ko~b (X, ) .... J Xh) cl~pe.nds o~1 (f. u.~ol1.

W*(x.), ... , ~(Xn).

10 e.o , ')eon, (x., ..0 , Xn) = koYb (x.', , X0

~"" SoA..s (X ••... ,Xl)) IivntL (xt , , X~)~th, W*(Xt)=w*(xf) ~or i~\'~m

Th.e. ~ihitenQ.ss ~ssr,unr'tion. o~ ~

JG O<;.)'s \s d\'"o pped !~