Luật mạnh số lớn trong đại số von Neumann.

8/2/2019 Lut mnh s ln trong i s von Neumann.

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Mc lcLi m u 1

1 Kin thc chun b 41.1 i s von Neumann v vt . . . . . . . . . . . . . . . . 4

1.1.1 i s Banach . . . . . . . . . . . . . . . . . . . . 41.1.2 Php tnh lin hp . . . . . . . . . . . . . . . . . 51.1.3 i s von Neumann . . . . . . . . . . . . . . . . 61.1.4 Phim hm tuyn tnh dng v biu din . . . . 6

1.2 Ton t o c theo mt vt . . . . . . . . . . . . . . . 71.2.1 Cc tnh cht c bn ca ph . . . . . . . . . . . 71.2.2 Khi nim v ton t khng b chn . . . . . . . 91.2.3 M u v php chiu . . . . . . . . . . . . . . . 111.2.4 L thuyt v ton t o c . . . . . . . . . 12

1.3 Khng gian Lp theo mt vt . . . . . . . . . . . . . . . . 211.3.1 Hi t hu u trong i s von Neumann . . . . 241.3.2 Cc kiu hi t hu chc chn trong i s von

Neumann . . . . . . . . . . . . . . . . . . . . . . 261.3.3 Dng khng giao hon ca nh l Egoroff . . . . 281.3.4 Khi nim v lut s ln . . . . . . . . . . . . . . 29

2 Lut mnh s ln trong i s von Neumann 312.1 Tnh c lp . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Hi t hu y trong i s von Neumann . . . . . . 322.3 nh l gii hn mnh cho dy trc giao . . . . . . . . . 342.4 M rng khng giao hon ca nh l Glivenko-Cantelli . 412.5 Bt ng thc Kolmogorov i vi vt v mt s h qu 442.6 Lut mnh s ln i vi vt . . . . . . . . . . . . . . . . 47

i


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1

2.7 Tc hi t trong lut mnh s ln . . . . . . . . . . . 622.8 Ch v ch thch . . . . . . . . . . . . . . . . . . . . . 68

Kt lun 70

Ti liu tham kho 72


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Li ni uCc ti liu khoa hc hin ti a ra nhiu bng chng rng cc

phng php i s vn cch mng ha ton hc thun thy th nayli ang gy nh hng tng t i vi khoa hc vt l. Tip cn is i vi c hc thng k v l thuyt trng lng t l mt v dcho nh hng mi ny.

Gn y nhiu tc gi m rng cc nh l hi t im c bntrong l thuyt xc sut v l thuyt ergodic sang (ng cnh) i svon Neumann.H cung cp mt s cng c mi cho vt l ton vng thi to ra nhiu k thut hp dn cho l thuyt i s ton t.

Mc ch chnh ca ti l trnh by bn cht ca mt s tngv kt qu t lnh vc ni trn,chuyn cc kt qu c in bit trongl thuyt xc sut n cc phin bn khng giao hon ca chng, avo cc ng dng trong l thuyt trng lng t.

i s von Neumann l mt s tng qut ha khng giao hon rt

t nhin ca i s L v nhng cu trc tt ca n em li kh nngthu c cc phin bn hu chc chn ca cc nh l gii hn.Trong i s von Neumann, ta c th a ra khi nim hi t huu tng ng vi khi nim hi t hu chc chn trong i sL.Kiu hi t ny s l nn tng cho ton b ti.

Ni dung ca ti gm hai chng :Chng 1 Trnh by mt s kin thc chun b cho vic nghin

cu chng 2 bao gm cc kin thc nn tng v gii tch v xc

sut.Mt s tnh cht ca hi t

hu u

trong i s von Neumann.Chng 2 Ni dung chnh ca ti: Lut mnh s ln trongi s von Neumann

Trnh by cc kt qu ca Batty ,cng mt s kt qu khc. Nu nhtrong xc sut c in cc dng hi t theo xc sut ,hi t hu chcchn v hi t trung bnh ca dy bin nhin ng vai tr then cht

2


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Li ni u 3

trong Lut s ln th y chng ta c tip cn vi 1 kiu hi thon ton mi hi t hu u .Cc nh l c chng minh i vitrng thi , i vi vt. V mt trng thi th khng cng tnh di trndn cc php chiu nn cc qui tc v cc k thut i vi trng thikhng vt khc nhiu v kh khn hn nhiu so vi cc qui tc i vivt. ng ch y l cc lp lun cn dng i vi vt thng rt

ging trng hp c in. Nhng trong mt s thng hp th chng tacn hng tim cn mi .Hu ht cc kt qu c trnh by trong ti l kt qu mi.

Mt s ni dung ca ti l mt trong s cc bi ging ti iHc Tenessee Knoxville v ti Trung Tm Qu Trnh Ngu Nhin(Centerfor Stochastic Processes) ti i hc North Carolina ChapelHill (bi R.Jajte)

Ni dung ca ti vn ang l hng quan tm ca nhiu nhton hc v vt l hc trong cc lnh vc i s ton t v cc ngdng ca chng nh Lance 1976-1978 ; Goldstein 1981; Watanabe 1979;Yeadon 1975-1980; Kiinrinerre 1978;.......

Trong qu trnh tm hiu , nghin cu ni dung cc cng trnh cangi khc chng ti h thng , gii thiu nhm phc ha trin vngng dng cc kt qu nghin cu ca mnh trong tng lai.

Tc gi xin c gi li cm n su sc n ngi thy, ngi hngdn khoa hc ca mnh l PGS. TS. Phan Vit Th, ngi a ra

ti v hng dn tn tnh trong sut qu trnh nghin cu bn lunny. ng thi tc gi cng chn thnh cm n cc thy c trong khoaTon - C - Tin hc trng i hc Khoa hc T nhin, i hc Qucgia H Ni, v cc thy c gio Vin Ton hc -Vin KHCNVN tn tnh ging dy ,to iu kin thun li trong sut thi gian Tc gihc tp v nghin cu.

H Ni, nm 2009Hc vin

V Th Hng


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CH NG 1. KI N TH C CHU N B 5

A c php nhn:A A A

(x, y) xy

tha mn cc tnh cht sau :

1. x(yz) = (xy)z;

2. (x + y)z = xz + yz; x(y + z) = xy + xz;

3. (xy) = (x)y = x(y), x,y,z A, CKhi , A c gi l mti s phc. Hn na , nuA l mtkhng gian Banach (vi chun ||.||) tha mn cc tnh cht sau :

4. ||xy|| ||x||.||y||; (x A, y B)

5. A cha phn t n v e sao cho xe = ex = x (x A);

6. ||e|| = 1;thA c gi l mti s Banach.

Nu php nhn giao hon th gi l i s (i s Banach) giaohon .

1.1.2 Php tnh lin hp

nh ngha 1.1.2. Gi sA l mt i s phc , nh x x A x

gi lphp lin hp nu tha mn cc iu kin sau :

(i) (x + y) = x + y

(ii) (x) = x

(iii) (xy) = yx

(iv) (x) = x

i s phc A ng i vi php lin hp gi l i s . Nu Al i s Banach ng i vi php lin hp th gi l mt i sBanach .NuA l mt i s Banach tha mn iu kin ||x|| = ||x|| thAgi li s Banach lin hp. Nu i s Banach tha mn iukin ||xx|| = ||x||2, x A th gi l mtC i s


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Phn t x A (A l i s ) gi l chun tc nu xx = xx.Gi l Hermit nu x = x, unitar nu xx = xx = e, (e l n v caA). Nu A l C i s th ||x|| = ||x||, x A. Vy mi C i su l i s Banach lin hp.

1.1.3 i s von Neumannnh ngha 1.1.3. Gi sH l khng gian Hilbert, B(H) l i s ccton t b chn. A B(H) l mt i s con.i sA B(H) gi l i s von Neumann nu:

(i) A l kn i vi php ly lin hp;

(ii) I A

(iii) A ng i vi topo hi t yu, tc lAnw

A nu :< Anx, y >< Ax, y > vi mix, y H

Nh vy i s von Neumann l mt C i s.

1.1.4 Phim hm tuyn tnh dng v biu din

nh ngha 1.1.4. (*) Gi sA l i s von Neumann. Phim hmtuyn tnh :

: A C

gi ldng nu:(xx) 0, x A

(*) gi lphim hm dng chnh xc (ng) nu:

(xx) 0

v t(xx) = 0 suy rax = 0

c bit : |||| = (I)(*) Nu(I) = 1 th gi ltrng thi

nh ngha 1.1.5. K hiu : A+ = {x A : x 0} . nh x : A+ [0, ] tha mn tnh cht :

(i) (x + y) = (x) + (y), x, y A+


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(ii) (x) = (x), 0; x A+ (quy c 0. = 0)

(iii) NuU l unitar th : (U xU1) = (x), x A+

Khi gi l vt ca i sA .Nu :

(x) < , x A+

th gi lvt hu hn.Nu :

(x) = sup{(y)|y x; (y) < }

th gi l vt na hu hn .Nu : (x) = 0, x 0 m suy rax = 0 th gi lvt chnh xc (hayvt ng)Vt gi lchun tc (Normal) nu:

(T) = sup (T)

trong T l dy cc ton t tng tiT.

i s von Neumann gi l hu hn (hay na hu hn) nu vi mix A, x = 0 tn ti vt chun tc hu hn sao cho (x) = 0. Trn is von Neumann na hu hn lun tn ti mt vt chun tc , chnhxc na hu hn.

1.2 Ton t o c theo mt vt

Phn ny ta s nh ngha khi nim v tnh o c theo mt vt trn mt i s von Neumann A v ch ra rng tp A cc ton t oc l mt i s topo y .Gi s A l mt i s von Neumann na hu hn hot ng trnkhng gian Hilber H v l trng thi na hu hn chun ng trn A.

1.2.1 Cc tnh cht c bn ca ph

nh ngha 1.2.1. Gi sA l i s Banach . G = G(A) l tp hptt c cc phn t kh nghch ca i s A . Khi G lp thnh mt


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nhm . Ph (x) ca x A l tp hp tt c cc s phc sao choe x khng c kh nghch . C/(x) c gi l tp hp chnh quy caphn tx.

C/(x) = { : (e x)1}

S(x) = sup{|| : (x)}

c gi l bn knh ph ca phn t x.

Ta lun chng minh c rng (x) = , x A.

nh ngha 1.2.2. Gi s a l ton t tuyn tnh vi min xc nhD(a) . K hiu

D(a) = {g : ! g, < g|af >=< g|f > f D(a)}

vi gi thit D(a) = H. Khi D(a

) l khng gian con v ton tag = g, g D(a), g l phn t duy nht

< g|af >=< g|f >

l mt ton t tuyn tnh. a gi l lin hp ca ton ta.

Nu a a th a gi l ton t i xng . Nu a = a th a gi lt lin hp.Khi a l ton t ng v D(a) = H th khi D(a) = H v a = a.

Ch . i s con A ca i s B(H) gi l chun tc nu n giaohon v nu T A th T A.

nh ngha 1.2.3. Ton t P c gi l ton t chiu nu D(P) =H; P = P = P2

Nh vy , ton t chiu P l b chn v c s tng ng mt - mtgia cc ton t chiu v cc khng gian con ng trong khng gianHilbert.

nh ngha 1.2.4. Xt H l khng gian Hilbert . (, ) l mt khnggian o, l trng. P l tp hp cc ton t chiu trong khnggian HilbertH. nh xE : P c gi l mtkhai trin n vtrn (, ) nu cc iu kin sau c tha mn :

1. E() = 0, E() = I


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2. E(AB) = E(A)E(B)

3. E(A B) = E(A) + E(B) nuAB =

4. x, y H hm tp hp Ex,y xc nh bi cng thc

Ex,y(A) =< E(A)x, y >

l mt o phc trn

nh l 1.2.5. (V biu din Ph) Nu T B(H) v T l ton tchun tc th tn ti ng mt khai trin n v trn cc tp con Borelca ph(T) ca ton tT sao cho

T =

(T)

dE()

NuT l t lin hp th(T) R. Khi

T =

ba

dE

NuT l ton t unitarT T = TT = I . Khi (T) nm trong vngtrn n v . Khi

T =

20

eidE()

1.2.2 Khi nim v ton t khng b chn

Vi cc ton t (tuyn tnh) a, b trn H ta c th nh ngha tng a + bv tch ab l cc ton t trn H vi min xc nh :

D(a + b) = D(a) D(b)

D(ab) = { D(b) b D(a)}

Cc php ton ny c tnh cht kt hp, v th a+b+c v abc l cc tont c nh ngha tt. Hn na ,vi mi a,b,c ta c : (a + b)c = ac + bcv c(a + b) ca + cb


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nh ngha 1.2.6. Mt ton ta trn H l ng nu thG(a) can ng trongH H ;a l trc ng nu bao ngG(a) ca th ca n l th ca mtton t ng no ( gi l bao ng ca a , k hiu l [a]) ;a l xc nh tr mt nu D(a) tr mt trongH.

Nu a, b v ab xc nh tr mt th :

(ab) ba

ng thc xy ra nu a b chn v xc nh khp ni.Ton t ng , xc nh tr mt a c biu din cc:

a = u|a|

y |a| l ton t t lin hp dng , v u l mt ng c ring vi

supp(a) l php chiu u ca n v r(a) , php chiu ln bao ng camin gi tr ca a , l php chiu cui ca n.

nh ngha 1.2.7. Nu tnga + b ca hai ton t xc nh tr mtavb l trc ng v xc nh tr mt , th bao ng [a + b] c gi ltng mnh caa vb. Tng t , tch mnh l bao ng [ab] nu ab ltrc ng v xc nh tr mt.

Ta vit||a|| = sup{||a||

|||| 1}

vi mi ton t xc nh khp ni a trn H, b chn hoc khng. Khi c lng sau y ng :

||a + b|| ||a|| + ||b||; ||ab|| ||a||.||b||

K hiu A l hon tp ca A (hon tp A ca i s von Neumann Al tp tt c cc b trong B(H) giao hon vi a trong A). nh l hontp 2 ln von Neumann khng nh A = A.

nh ngha 1.2.8. Ton t tuyn tnh a trn H c gi l kt hpviA (v ta vitaA) nu:

y A : ya ay

Ta k hiu A l tp tt c cc ton t ng , xc nh tr mtkt hp vi A


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1.2.3 M u v php chiu

K hiu Aproj l dn cc php chiu ( trc giao ) trong A . i vih (pi)iI cc php chiu trc giao trong A, k hiu :

iI

pi; (iI

pi)

l php chiu ln iI

piH; (iI

piH)

Ta c :(iI

pi) =

iI

pi ; (iI

pi) =

iI

pi

y p = 1 p l php chiu trc giao vi p. Hai php chiu p v q

l tng ng nu p = uu v q = uu vi u A no . Ta k hius tng ng l . Cc php chiu tng ng c cng vt.

Mnh 1.2.9. Gi sa l ton t ng, xc nh tr mt kt hp viA. Khi :

supp(a) r(a)

yr(a) k hiuphp chiu ln bao ng min gi tr caa.Vi cc php chiup, q A ta c:

(p q) p q (p q)

ko theo :(p q) (p) + (q)

Tng qut hn :(iI

pi) iI

(pi)

i vi h ty (pi)iI cc php chiu trongA ( nuI hu hn th iuny ko theo bng qui np; i vi trng hp tng qut , s dng tnhchun tc ca ).

Nhn xt 1.2.10.

p,q Aproj : p q = 0 = p 1 q


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( y c ngha tng ng vi mt php chiu con ca ).Tht vy :

p = 1 q = (p q) p

= (p q) p q (p q) q = 1 q

1.2.4 L thuyt v ton t o cnh ngha 1.2.11. Gi s, R+ . Khi ta k hiuD(, ) l tptt c cc ton taA sao cho tn ti php chiup A tha mn :

(i) pH D(a) v ||ap||

(ii) (1 p)

KhipH D(a) th ton tap xc nh khp ni , i hi ||ap|| ko

theo ap b chn .

Mnh 1.2.12. Cho 1, 2, 1, 2 R+ . Khi :

(i) D(1, 1) + D(2, 2) D(1 + 2, 1 + 2)

(ii) D(1, 1).D(2, 2) D(12, 1, 2)

Mnh 1.2.13. Gi s, R+:

(i) Nua l ton t trc ng , th:a D(, ) [a] D(, ).

(ii) Nua l ton t ng , xc nh tr mt vi biu din cc a = u|a|th:

a D(, ) u A, |a| D(, )

B 1.2.14. Cho a A v, R+ . Khi :

a D(, ) (],[(|a|))

( y],[(|a|)

k hiuphp chiu ph ca |a| tng ng vi khong ], [ ).


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Chng minh.

t :p = [0,](|a|)

. Khi : pH D(|a|) v

|| |a|p||

Vi p Aproj no ta c :

|| |a|p||

v (1 p) Gi s:

|a| =

0

de

l phn tch ph ca |a| . By gi: pH ta c:

|| |a|||2 2||||2

v : (1 e)H{0}

ta c:|| |a|||2 > 2||||2

V:

|| |a|||2 =

0

2d(e|) =

],[

2d(e|)

nn:(1 e)HpH

phi l {0} ,tc l (1 e) p = 0Vy 1 e 1 p, do

(1 e)


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Mnh 1.2.15. Cho a A v, R+. Khi :

a D(, ) a D(, )

Chng minh. Gi s a = u|a| l biu din cc ca a. Khi u l mtng c ca

]0,[(|a|) = supp(a)ln

]0,[(|a|) = supp(a) = r(a)

Do tnh duy nht ca phn tch ph suy ra vi mi R+, u l mtng c ca

],[(|a|)

ln ],[(|a|). p dng b trn c iu phi chng minh.

nh ngha 1.2.16. Mt khng gian con E ca H c gi l trmt nu R+ , tn ti php chiu p A sao cho pE E v(1 p) .

Mnh 1.2.17. Gi sE l khng gian con tr mt caH . Khi tn ti mt dy tng (pn)nN cc php chiu trongA vi

pn 1, (1 pn) 0,

n=1pnH E

.

Chng minh. Ly cc php chiu qk A, k N sao cho :

qkH E

v (1 qk) 2k. Vi mi n N , t :

pn =

k=n+1

qk

Khi

pnH =

k=n+1

qkH E


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v

(1 pn) =

k=n+1

(1 qk)

k=n+1

(1 qk)

k=n+1

2k = 2n

. Ko theo: pn 1.Tht vy, k hiu p l supremum ca dy tng pn , ta c

n N : (1 p) (1 pn) 2n

do (1 p) = 0 v p = 1. Hn na

n=1pnH E

Vy nu E l khng gian con tr mt ca H th E tr mt trongH.

B 1.2.18. (i) Cho p0 Aproj .Gi s rng :

R+, p Aproj : p0 p = 0

v(1 p) . Khi : p0 = 0.

(ii) Cho p1, p2 Aproj. Gi s rng

R+, p Aproj : p1 p = p2 p

v(1 p) . Khi : p1 = p2.

Mnh 1.2.19. Cho a, b A vE l khng gian con tr mt ca

H cha trongD(a) D(b). Gi sa|E = b|E . Khi a = bChng minh. Xt trong khng gian Hilbert H2 = H H i s von

Neumann A2 =

A A

A A

c trang b vt na hu hn chun ng 2


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xc nh bi 2

x11 x12

x21 x22

= (x11) + (x22). K hiu pa v pb l cc

php chiu ln th G(a) v G(b) ca a v b .V a v b kt hp vi Ann G(a) v G(b) bt bin di tt c cc phn t ca

A2 = { y 00 y |y A}

v do pa, pb A2 . Gi s R+ khi tn ti mt php chiu p A

vi pH E v (1 p) /2. t p2 =

p 0

0 p

Khi : 2(1 p2) . Hn na

pa p2 = pb p2

V a v b thng nht trn pH E nn

G(a) (pHpH) = {(,a)| pH,a pH}

= {(,b)| pH,b pH} = G(b) (pHpH)

Theo b trn , ta suy ra pa = pb, do a = b.

nh ngha 1.2.20. Mt ton t ng , xc nh tr mt kt hp viA

c gi l o c nu vi mi R+ , tn ti mt php chiup A sao cho pH D(a) v(1 p) .K hiuA ltp tt c cc ton t ng , o c ,xc nhtr mt

Nhn xt 1.2.21. 1. Nu a, b A v a b th a = b.

2. Nu a A, v a l i xng th a t lin hp .

3. Nu a ng v p Aproj tha mn pH D(a) th ton t , xc nh

khp ni ap cng ng v b chn.

nh ngha 1.2.22. Ton t aA c gi tin o c nu vimi R+ tn ti php chiup A sao cho

pH D(a), ||a|| <


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v(1 p) .Hay tng ngGi saA . Khi a l tin o c khi v ch khi

R+, R+ : a D(, )

Mnh 1.2.23. (i) Ta c A

A.(ii) Via A tha A.

(iii) Cho a, b A khi a + b vab xc nh tr mt v tin ng , v[a + b] A, [ab] A.

(iv) A l mt i s i vi tng mnh v tch mnh.

T y ta s b qua k hiu [ ] trong k hiu tng mnh v tch

mnh.nh ngha 1.2.24. Vi mi, R+ , ta t

N(, ) = A D(, )

tc l , N(, ) l tp cc a A, o c sao cho tn ti php chiup A tha mn ||ap|| v(1 p) .

nh l 1.2.25. (i) N(, ), vi , R+ to thnh c s cho cc

ln cn ca 0 i vi topo trn A bin A thnh khng gian vc ttopo.

(ii) A l i s topo Hausdorff y vA l mt tp con tr mtcaA

Chng minh. iu kin (i) l hin nhin . Ta chng minh (ii) :(1) . A l Hausdorff , ta s chng minh rng

,R+

N(, ) = {0}

Lya

,R+

N(, )


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Khi : R+, R+ : (],[(|a|))

V l ng (faithful) , iu ny ko theo

],[(|a|) = 0

, do a = 0.(2). Tip theo ta s chng minh A l i s topo.Theo kt qu trn php ton lin hp l lin tc. Ly a0, b0 A v, R+ . Chn , R+ sao cho

a0 N(, ), b0 N(, )

Khi vi mi a, b A tha mn a a0 N(, ) v b b0 N(, ) ,ta c

ab a0b0 = (a a0)(b b0) + a0(b b0) + (a a0)b0

N(, )N(, ) + N(, )N(, ) + N(, )N(, )

N(2, 2) + N(, 2) + N(, 2)

N(( + + ), 6)

Do (a, b) ab l lin tc.(3).A l tr mt trong A.Tht vy , gi s a A , ly cc php chiu pn A sao cho

pn 1, (1 pn) 0,nN

pnH D(a)

Khi apn A v apn a trong A v

||(apn a)pm|| = 0

vi mi m n v (1 pm) 0 khi n (4). Cui cng ta s chng minh rng khng gian vc t topo A l y.V A c mt c s m c cho cc ln cn ca 0 ( chng hn s dng

N(1/n, 1/m), n, m N


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) ta ch cn chng minh rng mi dy Cauchy (an)nN trong A hi t .V vy gi s (an)nN l mt dy Cauchy trong A v

n N : an+1 an N(2(n+1), 2n)

Ly php chiu pn A sao cho

||(an+1 an)pn|| 2(n+1)

v (1 pn) 2n. Vi mi n N ,t:

qp =

k=n+1

pk

V

(1 qn) = (

k=n+1

(1 pk))

k=n+1

(1 pk)

k=n+1

2k = 2n

vm n + 1, l N : ||(am+l am)qn|| 2

m

V qn pkvi mi k m n + 1 v do

||(am+l am)qn||

m+l1

k=m

||(ak+1 ak)qn||

m+l1k=m

||(ak+1 ak)pk|| m+l1k=m

2(k+1) 2m

Ly

nN

qnH

.Khi qnH vi n N no ,v do dy(am)mN

l dy Cauchy. ta = limmam


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Nh vy ta nh ngha mt ton t a vi

D(a) =nN

qnH

(ch rng D(a) l khng gian tuyn tnh con v (qn)nN l mt dytng cc php chiu). Theo cc xy dng , a l tin o c . Vimi n N ta c

qnH D(a)

v (1 qn) 2n. Ta khng nh rng a cng tin ng. thy iuny , p dng lp lun trn cho (an)nN. Do tn ti mt ton t tin o c b sao cho

bn = limman, D(b)

Khi D(a), D(b) : (a|) = lim(am|) = lim(|a

m) = (|b)

nn a b. Nh vy a tin ng . Vy [a] A. t: a0 = [a], cui cngta chng minh an a0 trong A. Gi s , N0. Ly n0 N sao cho

2(n0+1)

v 2n0 . Khi vi mi m n0 + 1 ta c

||(a0 am)qn0|| 2(n0+1)

v(1 qn0) 2

n0

v H : (a0 am)qn0 = liml(am+l am)qn0

v||(am+l am)qn0|| 2m 2(n0+1)

do b n0 + 1 : a0 am N(, )


22/74


1.3 Khng gian Lp theo mt vt

Trong phn ny chng ta s nh ngha khng gian Lp , Lp = Lp(A, )vi 1 p v xy dng cc tnh cht c bn ca chng. Segal lm vi p = 1, 2, ( khng gian L chnh l A ) , v Kunze nghincu trong trng hp tng qut. Cch tip cn ca chng ti da trn

khi nim hi t theo o trnh by trong phn trn s n ginhn.Cho trc mt vt na hu hn chun ng trn i s von NeumannA trn khng gian Hilbert. Gi s

L2 = {a A : (aa) < }

Cho a trong L2 v b trong A . Khi

(ba)

ba ||b||a

a

nn ba cng trong L2 . V a v b u trong L2 nn a + b cng trong L2 v

(a + b)(a + b) 2(aa + bb)

Do L2 l mt idean tri . Vy n l t lin hp v do l idean haipha (sau ny ta s ng nht L2 vi L2 L) . t L = L22 . Khi Lcng l mt idean hai pha (sau ny ta s ng nht n vi L1 L).Nu a trong A+ v (a) < th a1/2 trong L2 , nn a trong L. Ngc

li, gi s c 0 vi c trong L. Khi c l tng hu hn c = biai vibi, ai trong L2. V c

1

2

(bib

i + a

i ai) nn ta c (c) < . Do L

cha tt c cc t hp tuyn tnh hu hn cc phn t a trong A+ vi(a) < , v cc phn t ny to thnh L A+. Nh vy m rngduy nht thnh mt phim hm tuyn tnh (vn k hiu l ) trn L.Theo biu din cc th (ba) = (ab), v do (ba) = (ab) vi mia, b trong L. V vy ,nu a 0 v a trong L th

(ab) = (a1/2ba1/2)

nn0 (ba) ||b||(a)

vi mi b 0 trong Lv do , theo tnh chun tc v na hu hn ca ,vi mi b trong A+. Kt qu l |(ba)| ||b||(a) vi mi b trong A.


23/74


Ta cng s s dng k hiu ||b|| cho chun ton t ca phn t b trongA, v ta t ||a||1 = (|a|) vi a trong L. V a = u|a| vi ||u|| = 1 nnta c

|(ba)| ||b||||a||1; b A, a L (1.1)

vi 1 p < v a trong L , t ||a||p = (|a|p)1/p.Ta khng nh rngbt ng thc Holder:

|(ba)| ||b||q||a||p (1.2)

ng , vi a, b trong L v (1

p) + (

1

q) = 1. thy iu ny , gi s u v

v trong A vi ||u|| 1, ||v|| 1 v gi s c 0, d 0, c L, d Ltha mn c v d b chn cch xa 0 trn b sung trc giao ca cckhng gian trng ca chng. Do tnh lin tc nn s gii hn ny v saus b qua, v ta khng nhc n na . Khi s (udsvc1s) lin tc

v b chn trn 0 Res 1 v chnh hnh phn trong. Theo v dv nguyn l Phragmn-Lindelof c bit n nh l nh l 3 ngthng ,ta c :

|(udkvc1p)| supRes=1|(udsvc1s)|ksupRes=0|(ud

svc1s)|1k

vi 0 k 1, v theo (1.1) v phi ||d||k1||c||1k1 . p dng iu ny

vic = |a|1/k, d = |b|1/(1k)

y a = u|a|, b = v|b|, k = 1/q, v 1 k = 1/q , iu ny ko theo(1.2).Vi a = u|a| trong L, t

b =|a|p1u

||a||p/qp

y 1/p + 1/q = 1 . Khi d thy ||b||q = 1 v (ba) = ||a||p. Tc l

||a||p = sup||b||q=1|(ba)| (1.3)

v supremum l t c. T y ta d dng thu c bt ng thcMinkowski

||a + b||p ||a||p + ||b||p (1.4)


24/74


vi a, b trong L. V v tri l (c(a + b)) vi c tha mn ||c||q = 1 ,nhngn bng (ca) + (cb), do nh hn v phi theo bt ng thc Holder.Cui cng ,ch rng ||a||p = 0 ko theo a = 0 do tnh ng ca . Dovy L l khng gian tuyn tnh nh chun vi chun ||.||p . Gi Lp lkhng gian Banach tng ng vi L sau b sung cho y .Nu a 0 trong L vi biu din ph

a =

0

de

th

||a||pp = (ap) =

0

pd(e)

nn ||a||pp p(e) (1.5)

vi mi 0. Vy , nu an trong L l dy Cauchy trong Lp th n ldy Cauchy theo o. Do tn ti mt nh x tuyn tnh t nhinlin tc ca Lp vo A. Ta chng minh c rng nh x t nhin ny ln nh.

nh ngha 1.3.1. i vi ton t dng t lin hp a kt hp vi Abt k , ta t

(a) = supnN(

n0

de)

y

a =

0

de

l biu din ph caa . Khi vi mip ]1, [ ta c th nh ngha

Lp(A, ) = {a A(|a|p) < }

v||a||p = (|a|

p)1/p < , a Lp(A, )


25/74


.(Lp(A, ), ||.||p) l cc khng gian Banach trong

I = {x A(|x|) < }

l tr mt , v tt c chng c cha trong ( hoc thm ch l nhnglin tc trong ) A

Li bnh Khi nim v ton t o c c a ra bi I.E.Segal[17] v to thnh c s cho vic nghin cu l thuyt tch phn khnggiao hon , tc l l thuyt tch phn . L(X, ) ( tng ng vikhng gian o (X, ) ) c thay th bi i s von Neumann tng quthn. L thuyt ny ng vai tr chnh trong vic xy dng cc khnggian Lp kt hp vi cc i s von Neumann na hu hn l khng gianc th ca cc ton t ng xc nh tr mt .Trong [18] , E.Nelson a ra mt hng tip cn mi i hi t kin thc v k thut i svon Neumann cho l thuyt ny , da trn khi nim v tnh o ctheo mt vt (bt ngun t khi nim hi t theo o c gii thiubi W.F.Stinespring trong [16] ).Ton t o c bt k cng o dctheo ngha Segal ( trong khi iu ngc li ni chung khng ng ). Tuynhin ,tp hp cc ton t o c ln cha cc khng gianLp theo .

1.3.1 Hi t hu u trong i s von NeumannL thuyt xc sut khng giao hon l nn tng ton hc ca c hclng t, n c th coi l m rng t nhin ca l thuyt xc sut cin. Trong c hc c in, vi mi h ht im vt l c mt a tpkh vi U tng ng .Cc trng thi ca h c biu din bi cc imca U, v cc lng vt l (cc quan st c ) s c m t bi cchm (o c) trn a tp U. Trong c hc lng t, vi mi h vtl c mt khng gian Hilbert H tng ng. Vi h c s bc t do hu

hn , cc trng thi hn hp c cho bi cc ton t lp vt dng (ton t tr mt ) .Cc quan st c s c biu din bi cc ton tt lin hp hot ng trn H. i vi h ht c s bc t do v hn ,ngi ta ng nht trng thi ca h vi trng thi ( ton hc ) trnmt i s ton t A thch hp . Trong hu ht trng hp ta c th lyA l i s ton t von Neumann hot ng trn mt khng gian phc


26/74


(kh ly) . i s tt c cc ton t tuyn tnh b chn trn H l mt is von Neumann. Trng hp c in dn ta n i s von Neumanngiao hon L(U, Bu, ) cc hm o c b chn trn mt khng giano (U, Bu, ) . Khi cc hm o c khng b chn s c gn vo L(U, Bu, ) mt cch t nhin. o sau khi thc trin duynht thnh tch phn

U

f(d)

l mt trng thi chun tc ng trn L. Do trng hp c inc coi l sn ca i s von Neumann (giao hon).

i vi mt khng gian xc sut (, F , ) , gi L(, F , ) l i s(cc lp tng ng ) tt c cc hm nhn gi tr phc, b chn ctyu v F o c trn . N c th coi l mt i s von Neumanngiao hon hot ng trong L2(, F , ) nu ta ng nht hm g Lvi ton t nhn ag : f f g, vi f L2. i s A = L(, F , ) ctrng thi vt chun ng ( cho bi

(f) =

f d

). Theo nh l Ergoroff, s hi t hu chc chn ca dy (fn) tA l tng ng vi s hi t hu u ca n. Tc l ta c th pht

biu li hi t hu chc chn bng chun trong L , trng thi vcc hm c trng.

nh ngha 1.3.2. Gi s A l i s von Neumann vi trng thichun tc ng. Ta ni rng dy(xn) cc phn t caA hi t huu ti phn tx A nu vi mi > 0 ,tn ti mt php chiup Avi(1 p) < tha mn ||(xn x)p|| 0 khin .

Ch . nh ngha trn khng ph thuc vo cch chn do hit hu u tng ng vi hai iu kin sau:(*) Trong mi ln cn mnh ca n v trong A , tn ti php chiu psao cho ||(xn x)p|| 0 , khi n (**) Vi mi trng thi chun ng trn A v , tn ti php chiup A vi (1 p) < tha mn

||(xn x)p|| 0


27/74


Nhn xt. Nu l mt trng thi chun tc ng th topo mnh tronghnh cu n v S trong A c th metric ha bi khong cch

dist(x, y) = [(x y)(x y)]1/2

nh l 1.3.3. Gi s A l mt i s von Neumann vi trng thi

chun ng. Vi dy b chn cc ton t (xn) tA s hi t hu uko theo s hi t mnh ( mnh) ca(xn).

1.3.2 Cc kiu hi t hu chc chn trong i

s von Neumann

Khi nim hi t hu u l s tng qut ca khi nim hi t hu chcchn cho i s von Neumann . Ta c th xt cc phin bn khng giaohon khc ca khi nim ny.

nh l 1.3.4. Gi s A l mt i s von Neumann vi trng thichun ng .Vi mi dy b chn (xn) trong A, cc iu kin sau ltng ng

(i) Vi mi > 0 , tn ti php chiu p trong A vi (1 p) < vs nguyn dngN sao cho

||(xn x)p|| <

vin N.

(ii) Vi mi > 0 , tn ti php chiup A vi(1 p) < sao cho

||(xn x)p|| 0

khin .

(iii) Vi , tn ti dy cc php chiu (pn) trong A tng ti 1.( trongtopo mnh ) sao cho

||(xn x)pn|| <

vin = 1, 2,...


28/74


(iv) Vi mi php chiu khc khng p trong A tn ti php chiu khckhngq A sao cho q p v

||(xn x)q|| 0

khin

R rng trong trng hp i s von Neumann giao hon L(, F , )c 4 iu kin va thnh lp u tng ng vi hi t hu chcchn.

nh l 1.3.5. Nu l mt trng thi vt chun ng ( tc A l is von Neumann hu hn ) th c 4 iu kin trn l tng ng.

Gi s l mt vt ( hu hn hoc na hu hn ) . Xt i s

A cc ton t o c i vi (A, ) theo ngha Segal Nelson. Hi thu u (hay hi t gn u khp ni )cng c th c xt i vi dytrong A (c th l i vi dy (xn) trong L1(A, )).

nh ngha 1.3.6. Mt dy(xn) trongA c gi lhi t hu utix nu vi mi > 0, tn ti php chiup A sao cho

(p) < , (xn x)p A

vin > n0 v ||(xn x)p|| 0

khin .

nh l 1.3.7. Dy(xn) trongA ( trongA nu l mt vt ) c gil hi t hu u hai pha ti x A hay(A) nu vi mi > 0 ,tn tiphp chiup A sao cho (1 p) < v

||(xn x)p|| 0

khin


29/74


1.3.3 Dng khng giao hon ca nh l Egoroff

Mnh 1.3.8. Gi sA l mt i s von Neumann hot ng trongkhng gian Hilbert H. Nu dy (xi) trong A hi t mnh ti x0 th vimi > 0 , tn ti dy(pi) ProjA sao cho pi 1 mnh v

||(xi x0)pi|| <

vii = 1, 2,...

nh l 1.3.9. (nh l Egoroff khng giao hon) Gi sA l mt is von Neumann vi trng thi chun ng ; xn l dy trongA hi tnx theo topo ton t mnh . Khi vi mi php chiup A v mi > 0, tn ti php chiu q p trong A v dy con (xnk) ca (xn) saocho (p q) < v ||(xnk x)q|| 0 khik .

B 1.3.10. Gi s {xn} l mt dy ton t dng t A v {n} lmt dy s dng . Nu

n=1

1n (xn) < 1/2

th tn ti php chiup A sao cho

(p) 1 2

n=1 1

n (xn)

v ||pxnp|| 2n vi min = 1, 2,....

nh l 1.3.11. nh l Rademacher-Menchoft. Gi s 1, 2, ....l dy cc bin ngu nhin trc giao vc1, c1, .... l dy s thc tha mn

k=1c2k(lgk)

2 <

Khi chui

k=1

ckk

hi t vi xc sut1.


30/74


Cc k hiu s dng trong ti : A k hiu 1 i s vonNeumann hot ng trong mt khng gian Hilbert phc H ; A l hontp ca A; l mt trng thi trn A ; A+ l nn cc ton t dngtrong A ; ProjA l tp hp tt c cc php chiu trc giao trong A .Vip ProjA lun lun p = 1 p. Ton t n v trong A l 1, i vitp con Borel Z ca ng thng thc v ton t t lin hp x , k hiu

eZ(x) l php chiu ph ca x tng ng vi Z. Vi x A th |x|2 = xx.A l tp cc ton t ng , o c , xc nh tr mt. A l tp ccton t ng , xc nh tr mt v kt hp vi A

1.3.4 Khi nim v lut s ln

Mt bin c ngu nhin c th xy ra m cng c th khng xy i vimi php th. i lng ngu nhin c th ly mt trong cc gi tr c

th ca n. Nhng khi xt mt s ln nhng bin c ngu nhin hay ilng ngu nhin , ta c th thu c kt lun no m trn thc tc th xem l chc chn. Trong l thuyt xc sut ngi ta gi nhngnh l khng nh dy no nhng i lng ngu nhin hit theo xc sut v hng s l nhng nh l lut s ln . Nhngnh l lut s ln c in ( lut s ln i vi hi t theo xc sut )nh nh l Bernoulli v nh l Chebyshev , Khinchin...v lut s lni vi hi t hu chc chn nh nh l Kolmogorov.

nh ngha 1.3.12. Lut yu s ln cn c gi l nh lKhinchin Xt n bin ngu nhin X1, X2,....,Xn c lp , cng phnphi vi phng sai hu hn v k vng E(X). Khi vi mi s thc dng , xc sut khong cch gia trung bnh tch ly

Yn =X1 + X2 + ... + Xn

n

v k vngE(X) ln hn l tin v0 khin tin v v cc.

limnPX1 + X1 + .... + Xn

n E(X) = 0

nh ngha 1.3.13. Lut mnh s ln Kolmogorov Xtn bin ngunhin c lp X1, X2, ........., Xn cng phn phi xc sut vi phng saiE(|X|) < . Khi trung bnh tch ly

Yn =X1 + X2 + .... + Xn

n


31/74


hi t hu nh chc chn vE(X) .Tc l

P

limnYn() = E(X)

= 1


32/74

Chng 2

Lut mnh s ln trong

i s von Neumann

Trong chng ny chng ta s cp n mt s kt qu c coi lm rng ca nh l c in cho dy bin ngu nhin c lp ( haykhng tng quan ) . D nhin ta s cn khi nim tng qut tng ngv tnh c lp trong i s von Neumann. Vic thit lp li nh nghac in khng kh. Nhng iu cn nhn mnh y l tnh c lplin quan n trng thi l mt iu kin rt hn ch ,c bit khi

khng phi l vt.Khi nim c lp trong xc sut khng giao khngng vai tr qu quan trng nh trong xc sut c in. l l do vsao cc nh l v dy ton t c lp dng nh t quan trng so vicc nh l martingale hay ergodic . Rt may l i vi trng thi vt(tracial state) th cc k thut vn tng t nh trng hp c in.Vvy ta thu c rt nhiu kt qu ng cho c trng hp giao hon vkhng giao hon theo cch lm khng khc nhiu cch lm c in .

Thay cho vic nghin cu tnh c lp, ta s nghin cu tnh trcgiao ( lin quan n trng thi ) vi iu kin km cht hn nhiu. Ccnh l lin quan n dy trc giao dng nh c ng dng nhiuhn, v s c nghin cu trong chng ny. Cc nh l nh vy clin h vi l thuyt tng quan trong cc qu trnh ngu nhin lngt ( xem [14] ) v cho ta mt s thng tin v bin thin tim cn cady quan st c khng tng quan.

31


33/74

CH NG 2. LU T MNH S L N TRONG I S VON NEUMANN 32

Nu trng thi l vt th ta s thit lp c mt s nh l chody ton t o c. Chnh xc hn, ta s xt dy (xn) A ,vi A l*- i s t p cc ton t o c theo ngha Segal-Nelson . Cc thutng v mt s kt qu lin quan n ton t o c c th xem thmti liu trong phn ph lc.

2.1 Tnh c lp

Cho A l mt i s von Neumann vi trng thi chun tc ng (faithful normal state ) . K hiu A1, A2 l cc i s von Neumanncon ca A . Theo Batty [11] ta c 2 phin bn ca khi nim clp i vi dy ton t .

nh ngha 2.1.1. Cc i s conA1, A2 c gi l c lp ( lin quann ) nu(xy) =(x)(y) vi mix A1, y A2 .

R rng quan h c lp c tnh cht i xng.

nh ngha 2.1.2. Cc phn tx, y A ( hay tA nu l trng thivt ) c gi lc lp nu cc i s von NeumannW(x) vW(y)ln lt sinh ra bi x vy l c lp.

Dy {xn} cc phn t t A ( hay A nu l mt vt ) c gil c lp lin tip nu vi mi n , i s W(xn) c lp viW(x1, x2,....,xn1) .

nh ngha 2.1.3. H {B, } ca cc i s von Neumann concaA ( hay trongA nu l vt ) gi lc lp yu nuB c lpviW{B; {}}

2.2 Hi t hu y trong i s vonNeumann

T y ta s s dng mt s kiu hi t trong ATa cng nhn nh ngha sau:


34/74


nh ngha 2.2.1. Dy {xn} trong A c gi l hi t hu y tix nu vi mi > 0 ,tn ti dy (qn) cc php chiu trongA sao cho

n

(1 qn) <

v ||(xn x)qn|| < ; n = 1, 2, ....

Trc ht ta lu rng nu l vt th hi t hu y ko theohi t hu u.

Chng minh. Tht vy , gi s xn 0 hu y . Khi tn ti dycc php chiu qn sao cho ||xnqn|| < , n = 1, 2,... v

n(qn ) <

t :

pn =

s=n

qn

, ta c :

(1 pn)

s=n

(1 qn) 0

Tc l xn 0 hu khp ni . Theo nh l 1.3.5 th xn 0 hu u.

Khi l mt trng thi , ta c kt qu sau:

nh l 2.2.2. Gi sA l i s von Neumann vi trng thi chuntc ng , v (xn) l dy b chn trong A . Nu xn x hu y thxn x hu u.

Chng minh. Gi s ||xn|| 1 v x = 0 . Cho trc > 0. Ta s tmdy (qn) cc php chiu trong A sao cho :n

(qn ) <


35/74


v ||xnqn|| < vi n = 1, 2,...C nh dy s dng (n) tha mn:n 0 v

n=1

(1 qn)1n < /2

Theo b 1.3.10 , tn ti php chiu p A vi (p

) < v tha mn||pqnp|| = ||qnp

||2 < 2n; n = 1, 2, ....

Khi ta c:

||xnp|| ||xnqnp|| + ||xnqnp|| ||xnqn|| + ||q

np|| < + (2n)

1/2 < 2

vi n > M0()V vy iu kin sau c tha mn:

(*) Vi mi > 0 , tn ti php chiu p vi (p) 1 v tha mn||xnp|| < vi n > n0();Theo nh l 1.3.4 th xn 0 hu u ;

2.3 nh l gii hn mnh cho dy trc

giao

Trong mc ny ta s chng minh nh l gii hn mnh sau y v dytrc giao lin quan n mt trng thi .

nh l 2.3.1. [15] Gi s A l i s von Neumann vi trng thichun tc ng , v (xn) l dy trc giao tng i trong A (tc l(xnxm) = 0 vim = n)Nu :

n=1(lg n

n

)2(|xn|2) < (2.1)

Th cc trung bnh :

Sn =1

n

nk=1

xk (2.2)

hi t hu u ti0


36/74


chng minh nh l ny ta s bt u vi kt qu nh sau .

Mnh 2.3.2. Gi s (yn) l dy trc giao tng i trong A . t:

tn =n

k=1

yn (2.3)

Khi tn ti dy ton t dngBm trongA sao cho :

|tn|2 (m + 1)Bm 1 n 2

m (2.4)

v

(Bm) (m + 1)2mk=1

(|yk|2) (2.5)

Chng minh. Vic chng minh da trn tng ca Plancherel [ 7] vc trnh by trong l thuyt v chui trc giao ( xem [8] ). Ta s btu vi biu din nh phn ca ch s n ; ta chia khong I = (0, 2m]thnh cc khong (0, 2m1] v (2m1, 2m] ,mi khong ny li tip tcc chia i .... v ta s thu c dy phn hoch ca I . Cc phnt ca phn hoch u tin c di 2m1 ,ccphn t ca phn hochth r c di 2mr . i vi s nguyn dng n 2m , ta c biu dinnh phn ca n. Khi khong (0, n] c th vit thnh tng ca nhiunht m khong ri nhau I(n)j mi khong thuc mt phn hoch khc

nhau , tc l :

(0, n] =m

j=0

I(n)

j (2.6)

Vi I(n)j l tp rng hoc l khong c di |I(n)

j | ; (j=1,2,...m) Tac th vit :

tn =m

j=0

kI

(n)j

yk (2.7)

(ng nhin , ta s t kI

(n)j

yk = 0


38/74


y Bk l ton t dng , khng ph thuc vo N (2k, 2k+1] v thamn :

(Bk) (k + 2)2k+1

s=2k+1

(|xs|2) (2.12)

Do , vi 2k < N 2k+1, ta c:

|SN S2k|2 Dk (2.13)

Vi Dk A+, v

(Dk) 212k[

2ks=1

(|xs|2) + (k + 2)

2k+1s=2k+1

(|xs|2)] (2.14)

Theo gi thit ca nh l , ta c :

k=1

(Dk)

k=1

212k22k

k2

2ks=1

(|xs|2)(

lg s

s)2

+

k=1

212k22k

k2(k + 2)2

2k+1s=2k+1

(|xs|2)(

lg s

s)2

s=1 (|xs|2

)(

lg s

s )

2

(

k=12

k2 + const) (2.15)

Hn na:

k=1

(|S2k|2) =

k=1

1

22k

2ks=1

(|xs|2)

k=1

22k22k

k2

s=1

(|xs|2)(

lg s

s)2 < (2.16)

Cho trc > 0.T (2.15) ,(2.16) suy ra ta c th tm c dy s dng(k) sao cho k 0 v :

k=1

1k (|S2k|2 + Dk) < /2 (2.17)


39/74


Theo b 1.3.10 tn ti php chiu p A vi (p) 1 tha mn:

||p|S2k|2p|| < 2k; ||pDkp|| < 2k (2.18)

Nh vy ,vi 2k < N 2k+1 , ta c c lng sau :

||SNp||2 + ||(SN S2k)p + S2kp||

2

2[||(SN S2k)p||2 + ||S2kp||

2]

= 2[p|SN S2k|2p|| + ||p|S2k|

2p||]

2[||pDkp|| + ||p|S2k|2p||]

< 8k 0 khiN . iu ny c ngha l :

SN =1

N

N

s=1 xshi t hu u v 0nh l c chng minh xong.

Ta s thit lp phin bn r chiu ca nh l 2.3.1

nh l 2.3.3. Gi s (x(i)n ); i = 1, 2,....,r l mt h hu hn cc dytrc giao tng i trong A ( tc l((x(i)n )x

(i)m ) = 0 vi m = n v

i = 1, 2,...,r). Gi thit :

n=1

(lg n

n)2(|x(i)n |

2) < ; i = 1, 2,...,r

Khi , vi mi > 0 , tn ti php chiu p A vi (1 p) < vtha mn:

Max1ir||1

N

Nn=1

x(i)n p|| 0; N

tc l, cc trung bnh1

N

Nn=1

x(i)n

hi t ti0 hu u v u theo 1 i r;


40/74


D nhin nu l mt vt th nh l ny l h qu tm thng canh l 2.3.1. Nu l trng thi chun tc ng tng qut , chng minhc th thu c bng cch xem xt k chng minh ca nh l 2.3.1.

Chng minh. t :

S(i)N =

1

N

N

n=1 x(i)n

V p dng mnh 2.3.2 , ta c c lng :

|S(i)N S

(i)2k

| D(i)k ; (i = 1, 2,...,r)

Vi D(i)k A+ no c tnh cht ging Dk . n y ta c th t :

Dk =r

i=1D

(i)k ; SN = (

r

i=1|S

(i)N |

2)1/2

V tip tc chng minh nh trong nh l 2.3.1

Ta s so snh nh l 2.3.1 vi cc kt qu c in .nh l Rademacher - Menchoft v s hi t hu chc chn ca chuitrc giao [8] cng vi b Kronecker cho ta lut mnh s ln sau y:

nh l 2.3.4. i vi dy(Xn) cc bin ngu nhin khng tng quan, nu :

n

(lg nn

)2var(Xn) <

Th:1

n

nk=1

(Xk EXk) 0

vi xc sut1

R rng nh l 2.3.1 c th c coi l s m rng ca lut mnhs ln ni trn cho trng hp khng giao hon .Vi mt vi iu kin mnh hn trong nh l 2.3.1 ta s thu c shi t tt hn ca dy trung bnh . Do d dng chng minh 2 nhl sau:


41/74


nh l 2.3.5. Gi s(xn) l dy trongA trc giao lin quan ti trngthi . Nu

s=1

as(|xs|2) <

khi0 < as 0 v

s=1

1s2as

<

th1

n

ns=1

xs 0

hu y .

Chng minh. t:

Sn =1

N

Ns=1

xs

, khi :

(|SN|2) =

1

N2

Ns=1

(|xs|2)

1

N2aN

Ns=1

as(|xs|2)

Do vy : (|SN|2) < Vi > 0 , ta t :

qN = e[0,2](|SN|2)

Khi :||SNqN|| < vi N = 1, 2,... Hn na :

N=1

(qN) 2

k=1

(|Sk|) <

Kt thc chng minh.nh l tip theo l kt qu mnh hn ca Batty [11] (nh l 2.3.1)


42/74


nh l 2.3.6. Gi s(xn) l dy c lp yu b chn u trong A vi(xk) = 0 . Khi

1

n

ns=1

xs 0

hu y .

Chng minh. Cho trc > 0. t :

SN =1

N

Ns=1

xs

D dng ch ra rng :

(|SN|4) N4(3N2 N)

Do : N

(|SN|4) <

t :qn = e[0,4](|SN|

4)

suy ra: N

(qN) <

v ||SNqN||4 || |SN|4qN|| < 4

Vi N = 1, 2,... suy ra iu phi chng minh.

2.4 M rng khng giao hon ca nh l

Glivenko-Cantelli

Ta cn thm mt nh ngha na.

nh ngha 2.4.1. Gi s v l 2 ton t t lin hp c st nhpviA . Gie(.) ve(.) l cc o ph ca v .Ta ni rng v l cng phn phi nu (e(Z)) = (e(Z)) vi mi tp con Borel Zca ng thng thc.


43/74


Ta s chng minh kt qu tng qut ca nh l Glivenko-Cantelliv phn phi thc nghim.

nh l 2.4.2. [9] Gi s n l dy ton t t lin hp , c lp tngi v cng phn phi , st nhp viA. Khi vi mi > 0 , tn timt php chiup trong A sao cho :

sup


44/74


Nhng t nh ngha ca i,k, ta c :

(i+1,k 0) (i,k + 0) 1

k

V kt qu l :

N(i,k + 0) (i,k + 0) 1

k

N( 0) ( 0) N(i+1,k 0) (i+1,k 0) +1

k(2.19)

Nu 1,k th:

1

k N( 0) ( 0) N(1,k 0) (1,k 0) +

1

k(2.20)

V nu > 1,k th:

N( 0) ( 0) = 0 (2.21)

thun tin ta t :

N(0,k + 0) (0,k + 0) = N(1,k 0) (1,k 0) = 0

Theo (2.19) ,(2.20) ,(2.21) th vi mi s thc v k = 1, 2,... tn timt i gia 0 v k 1 sao cho :

N(i,k + 0) (i,k + 0) 1k

N( 0) ( 0)

N(i+1,k 0) (i+1,k 0) +1

k;

do vy vi php chiu p ty t A ta c:

p[N(i,k + 0) (i,k + 0)]p 1

kp

p[N( 0) ( 0)]p

p[N(i+1,k 0) (i+1,k 0)]p +1

kp

V ta thu c kt qu:

||p[N( 0) ( 0)]p||


45/74


max1ik,=0 ||p[N( + ) (i,k + )]p|| +1

k

Ko theo :

sup


46/74


v:||(sn (sn))(1 q)|| n 1

Chng minh. Ta c th gi s (xn) = 0. Ta s nh ngha cc dy phpchiu pn, qn trong A bng quy np. t q0 = 0 . Cho trc qn1 ,t:

pn = e(2,){(1 qn1)snsn(1 qn1)}

v :qn = pn + qn+1

R rng cc pn trc giao ,

qn =N

r=1

pr

, pn v qn thuc W{xn : r n}

.T tnh c lp v cc tnh cht ca vt th vi r n:

(pr|sn|2pr) = (pr((sn sr) + sr)

((sn sr) + sr)pr)

((sn sr)srpr) + (prs

r(sn sr)) + (prs

rsrpr)

= ((sn sr)srpr) + (prs

r(sn sr)) + (prs

rsrpr)

= (pr(1 qr1)srsr(1 qr1)pr) 2(pr);v:

(qn) 2

nr=1

(snsnpr) = 2(snsnqn)

2(snsn) = 2

nj=1

||xj||22

t :

q =

r=1

pr

, khi :

(q) 2n

j=1

||xj||22


47/74


, v:||(1 q)snsn(1 q)||

||(1 pn)(1 qn1)snsn(1 qn1)(1 pn)||

2

Nh l h qu ca nh l trn, ta nhn c dng suy rng sau yca nh l Kolmogorov.

nh l 2.5.2. Gi sxn l dy c lp lin tip trongL2(A, ) sao chochui

n=1

||xn (xn)||22

hi t , v gi

sn =

n

k=1 xk. Khi dysn (sn) hi t hu u.

Chng minh. Ta c th gi s rng xn l t lin hp v (xn) = 0 . Chns nguyn nk sao cho

n=nk

(x2n) < 8k

Theo nh l 2.5.1 tn ti cc php chiu qk vi (qk) < 2k

tha mn||(sn sm)(1 qk)|| < 2

(k1)

vi m, n nkNn {sn} l dy Cauchy theo o . Gi s l gii hn theo o can . Khi ta c:

||(sm s)(1 qk)|| < 2(k+1)

vi n > nk.V vy sn s trn cc tp ln v do hi t hu u .

S dng b Kronecker ta thu c :


48/74


nh l 2.5.3. Gi s{xn} l dy c lp lin tip trongL2(A, ) saocho

n=1

n2||xn (xn)||22 <

. Khi

n1

n

k=1

(xk (xk))

hi t hu u .

Ch rng t kt qu va nu ta c th suy ra m rng sau y calut mnh s ln Kolmogorov.

nh l 2.5.4. Gi s {xn} l dy cc ton t cng phn phi , clp lin tip v t lin hp trongL1(A, ) . Khi :

n1n

k=1

xk (x1)

hu u.

Ta b qua chng minh ca nh l ny v trong phn sau ta s nura kt qu tng qut hn.

2.6 Lut mnh s ln i vi vt

Ta s bt u vi nh l 2.5.2 t suy ra m rng ca phn

ca nh l ba chui Kolmogorov.

B 2.6.1. Gi s{k} l dy ton t o c c lp lin tip t A.Vid > 0 , t

k = ke[0,d]{|k|}

. Khi cc iu kin :

(i)

k (e(d,){|k|}) <

(ii)

k (kk) <

(iii)

k (k) hi t .


49/74


Ko theo s hi t hu u ca chui

k k;

Chng minh. Theo nh l 2.5.2 v (ii) chuik

(k (k))

hi t hu u , v do theo (iii) chuik

k

hi t hu u . iu ny cng vi (i) cho ta s hi t hu u cak k.

Tht vy, t :

sn =

n

k=1 k;

sn =n

k=1

k

R rng ch cn chng minh {sn} l dy Cauchy hu u , tc l :Vi mi > 0 , tn ti p ProjA (tp hp cc php chiu ca A) vi(p) 1 v s nguyn dng N sao cho

||(sn sm)p|| <

vi n, m N.V chui k

k

hi t hu u nn vi mi > 0 , ta tm c q ProjA v N sao cho

||( sn sm)q|| <

vi n > m N v (q) < /2. t :

p = q k=m+1

e[0,d]{|k|}


50/74


theo (i) ta c:

(p) (p) +

k=m+1

(e(d,){|k|}) <

Vi m ln (m N) . Khi :

||(sn sm)p|| ||( sn sm)q|| <

vi n > m NDo {sn} l dy Cauchy hu u . Kt thc chng minh.

Bin i mt cht nh l trn ta c kt qu sau:

B 2.6.2. Gi s {n} v {n} l cc dy ton t o c t A v{cn} l dy s dng . t:

n = ne[0,cn]{|n|}.

Nu n

(e(cn,){|n|}) <

th chui n

(n + n)

hi t hu u khi v ch khi chuin

(n + n)

hi t hu u.Ni ring nu

n

(e(cn,){|n|}) <

, th chuik k hi t hu u khi v ch khi chuik

k

hi t hu u .


51/74


Chng minh. t :

sn =n

k=1

(k + k)

;

sn =n

k=1(k + k)

V x l cc hiu (sn sm), ( sn sm) nh cch ch ra trong chngminh nh l trn.

nh l 2.6.3. Gi sn : R+ R+ l dy hm khng gim sao chon()/ v2/n() khng gim vi min .Gi s

0 < xn

Nu {k} l dy ton t c lp lin tip lin kt vi A v tha mn(j) = 0 v

k=1

(n|k|)

(xk)< (2.22)

Th chui

k=1

kxk

(2.23)

hi t hu u.

Chng minh. t :

n = ne[0,xn]{|n|} = ne[0,1]{|nxn

|}; n = 1, 2, ....

Khi ta c :(e[xn,]){|n|})

(n|n|)

n(xn)

T (2.22) ta thu c :

n=1

(e(1,){|nxn

|}) < (2.24)

Hn na , v :2

x2n

n()

n(xn)


52/74


vi 0 xn nn ta c:

(|n|2)

x2nn(xn)

(

xn0

n()ed{|n|})

x2nn(xn)(n|n|)

V do :

n=1

n

xn

2= n=1

n

xne[0,1]{|

nxn

|}2< (2.25)

V : (n) = 0 nn ta c :

|(n)| (|n|e[xn,){|n|}) (

|

n|)

(xn)xn

(V

n()

n(xn)xn

vi xn). Do vy :

n=1 n

xn

n=1(n|n|)

(xn)< (2.26)

Theo b 2.6.1 v (2.24) ,(2.25) ,(2.26) ta thu c (2.23). Kt thcchng minh.

By gi ta s chng minh lut mnh s ln m khng cn gi thitv tnh hu hn ca cc m men thng ca n.

nh l 2.6.4. Gi sn : R+ R+ l dy hm khng gim tha mn

2

/n() khng gim , v gi s0 < xn

, {n} l dy ton t o c c lp lin tip . t:

n(d) = (ed{|n|})


53/74


Nu :

n=1

0

n()

n(xn) + n()n(d) < (2.27)

Th tn ti dy hng s (Ck) sao cho :

1xn

nk=1

(k Ck) 0 (2.28)

hu u.V trong trng hp ny ta c th t :

Ck = (ke[0,xk]{|k|}) (2.29)

Chng minh. i vi hm khng gim : R+ R+ v ton t vi

|| =

0

e(d)

ta t(.) = (e(.))

th s c c lng sau ( vi x > 0 )

0

()

(x) + ()(d)

1

2(x)

x0

()(d) +1

2

x

(d)

=1

2(x)((||e[0,x]{||})) + (e(x,){||})

=1

2(x)((||e[0,x]{||})) + (e(x,){||})

V vy ta c : n

(n|n|)

(xn)< (2.30)

V: n

(e(xn,){|n|}) < (2.31)


54/74


Vi:n = ne[0,xn]{|n|} (2.32)

Hn na :

(|n|2)

x2nn(xn)

(n|n|)

V do : n

(|n|2)

x2n<

iu ny ko theo s hi t ca chui :

n

n (n)

xn

V do vy theo b 2.6.2 ta suy s hi t hu u ca chui

n

(n)

xn

By gi ta ch vic p dng b Kronecker.

Hai nh l sau y chnh sa mt s kt qu ca W.Feller ( i vibin ngu nhin thc)

nh l 2.6.5. Gi s{k} l dy ton t t lin hp , cng phn phi

i xng v c lp lin tip , lin kt vi (A, ) .Gi s rng l hmkhng gim sao cho 2/() khng gim.Gi thit: 0 = x0 < x1 < x2, .... v

k=n

1

(xk)= 0(

n

(xn)) (2.33)

Khi iu kin :

n (e(xn,){|1|}) < (2.34)Ko theo:

1

xn

nk=1

k 0 (2.35)

hu u .


55/74


Chng minh. t:n = ne[0,xn){|n|}

. Khi :

n=1

(|n|)

(xn)=

nk=1

(|1|e[xk1,xk){|1|})

n=k

1

(xn)

C

k=1

k[xk1,xk)

()

(xk)(ed{|1|})

C

k=0

(e(xk,){|1|}) <

Do (2.30) ng. Hn na (2.31) tha mn theo gi thit v (n) = 0.( v n phn phi i xng ). Lp li phn chng minh ca nh l (2.6.4)

sau cng thc (2.31) ta thu c (2.35) . iu phi chng minh.Trong trng hp n khng phn phi i xng ta cn hn ch thm

iu kin i vi {xk}

nh l 2.6.6. Gi s{n} l dy cng phn phi , c lp lin tip, v{xn} c m t nh trong nh l (2.6.4)Nu thm vo

(1) = 0;xkxn

C0k

n; k n (2.36)

th (2.34) ko theo (2.35).

Chng minh. Tng t nh l (2.6.5) ta ch ra rng :

1

xn

nk=1

(k (k)) 0

hu u. V do :1

xn

n

k=1

(k) 0

Ta c c lng sau:

1

xn

nk=1

(n) 1

xn

nk=1

xk

(ed{|1|})


56/74


Do s dng phn tch cc :

k k = uk|k k|

ta c:|(k k)| ||uk||(|k k|) = (|k k|)

v|k k| = |k|e(xk,){|k|}

V vy :1

xn

nk=1

(k) 1

xn

nm=1

mk=1

+

m=n+1

nk=1

xm+1xm

(ed{|1|}) S1 + S2

Vi :

S1 =1

xn

nm=1

mxm+1(e[xm,xm+1){|1|})

S2 =1

xn

nm=n+1

nxm+1(e[xm,xm+1){|1|})

S1 0 v :n

m=0

(m + 1)(e[xm,xm+1){|1|})

=

m

(e(xm,){|1|}) <

Theo gi thit v do ta c th p dng b Kronecker.Lp lun tng t cho S2 . Chng minh c hon thnh.

By gi ta s chng minh kt qu cho trng hp khng giao hon

ca lut s ln Marcinkiewicz. Ta bt u vi 2 mnh sau:Mnh 2.6.7. Gi s{n} l dy c lp lin tip trongA v

n=1

10

(e[,){|n|})d <


57/74


Khi chui

n=1

( (n))

hi t hu u , y :

n = ne[0,1){|n|}

Chng minh. Tch phn tng phn , ta c:

10

(e[,){|n|})d =1

2(e[1,){|n|})

+1

2

1

02(ed{|n|})d =

1

2(e[1,){|n|}) +

1

2(|n|

2)

V do , theo gi thit ta suy ra:n

(|n|2) <

v n

(e[1,){|n|}) <

Bt ng thc th nht cng vi nh l 2.6.2 cho ta s hi t hu uca chui

n

(n (n))

v cng vi bt ng thc th hai v b 2.6.2 suy ra s hi t huu ca

n=1

(n (n))

.


58/74


Mnh 2.6.8. Gi s{n} xc nh nh trn v

n=1

10

(e[bn,){|n|})d <

vi{bn} l dy hng s dng , bn . Khi :

(i)

n=1

b1n (n (n))

hi t hu u , v

(ii)

b1n

n

k=1(k (k)) 0

hu u

yn = ne[0,bn){ n}

Chng minh. Ta c :

1

0(e[bn,){|n|})d =

1

0(e[,){

n

bn })d

V theo mnh 2.6.7 chui

n=1

nbn

n

bn

hi t hu u ,vi :

nbn

=

n

bn e[0,1){|

n

bn |}

=nbn

e[0,bn){|n|}

V vy ta c (i)p dng b Kronecker ta thu c (ii).


59/74


By gi ta s chng minh kt qu tng t ca Marcin Kiewicz theoLuczak [9]

nh l 2.6.9. Gi s{n} l dy ton t c lp lin tip , cng phnphi , tLr(A, ), 0 < r < 2. Khi

n

1/rn

k=1(k k) 0hu u. yk = 0 vir < 1 vr = (1) vi1 r < 2

Chng minh. Vi phn phi xc sut trn na ng thng thc:

F() = (e[0,){||}), 0, A,

ta c c lng sau:

r

n=1

(e[n1/r ,){||}) (||r) 1 + r

n=1

(en1/r,){||}) (2.37)

Chng hn xem [10] . S dng (2.37) ta thu c:

n=1

10

(en1/r,){|1|})d

(|1|r)

1

0

1rd <

Nu r < 2 bt ng trn cng vi mnh 2.6.8 cho ta :

n1/rn

k=1

(k (k)) 0

hu u vi :k = ke[0,n1/r]{|k|} 0

Lp lun tiu chun ch ra rng :

n1/rn

k=1

((k) k) 0, n

(Chng hn , xem [10], chng minh hon tt.)


60/74


Ta s kt thc phn ny vi mt s lut yu s ln trong i svon Neumann.

nh l 2.6.10. Cho {n} l dy cc ton t t lin hp o c , cngphn phi , c lp lin tip t A . Nu :

limn

n(e(n,){|1|}) = 0

th :

n1

k=1

(k n) 0

theo o, vi :n = (1e[0,n){|1|})

Chng minh. Gi s A tc ng trong khng gian Hilberte H, t :

Sn =n

k=1

k; (n)k = ke[0,n){|k|}

Sn =n

k=1

(n)k ; mn = (Sn) = n(

(n)1 )

Vi s > 0 bt k , ta c:

p = e[2,)(|Sn mn|) e[0,)(|Sn mn|)

n

k=1

e[0,n){|k|} = 0

Tht vy, nu vi x no c chun bng 1 , px = x th

x e[0,n){|k|}(H)

v do kx = (n)k x

vi k = 1, 2,...n.Ko theo Snx = Snx .V th ta thu c :

2 || |Sn mn|e(2,){|Sn mn|x}||


61/74


= ||(Sn mn)x|| ||(Sn Sn)x|| + ||(Sn mn)x||

= || |Sn mn|e[0,){|Sn mn|x}||

iu ny khng th xy ra, vy p = 0 v do (xem ph lc)

e[2,){|Sn mn|} e[,){|Sn mn|} n

k=1 e[n,){|k|} (2.38)Theo bt ng thc Tchebyshev v t kt qu :

(| ()|2) (||2)

Ta thu c :

(e[2,)(|Sn mn|)) 2(|Sn mn|

2) + (n

k=1e[n,)(|k|))

2n

k=1

(|(n)k |

2) +n

k=1

(e[n,))

= 2n(|(n)1 |

2) + n(e[n,)(|1|)) (2.39)

By gi ly > 0 ty , t (2.39) vi =n

2, ta c:

e[,)(|Snn n|)= e[,)(|Sn mnn |)=

e[n,)(|Sn mn|)

4(|

(n)1 |

2)

2n+ n

e[n,)(|1|)

= 42n1

[0,n)

2

ed(|1|)

+n

e[n,)(|1|)

Tch phn cui cng c th c lng nh sau:

[0,n)

2

ed{|1|}) n1k=0

(k + 1)2(e[k,k+1){|1|})


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=n1k=0

(2k + 1)(e[k,n){|1|}) (e[0,n){|1|})

+3n1k=1

k(e[k,n){|k|}) 1 + 3n1k=1

k(e[k,){|1|})

Do vy cui cng:(e[,){|

Snn

n|})

42n1[1 + 3n1k=1

k(e[k,){|1|})] + n(e[n,){|1|}) 0

khi n Chng minh c hon tt.

S dng k thut tng t ta c th chng minh kt qu sau calut yu s ln.

nh l 2.6.11. Cho {n} l dy phn t t lin hp c lp lin tiptL1(A, ). Nu:

(i)n

k=1

(e[n,){|k k|}) 0

khin

(ii)n1

nk=1

(|k k|) 0

khin ,vi || = ||e[0,n){||} v :

(iii)

n2n

k=1

([|k k|2]) 0

khin vik = (k) th :

n1n

k=1

(k k) 0

theo o.


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2.7 Tc hi t trong lut mnh s ln

Cc kt qu trong phn ny c trong t tng ca cc nh l Spitzer-Hsu-Robbins-Katz i vi bin ngu nhin thc. Chng ta s i theophng php da trn c lng tt ca cc m men ca binngu nhin b cht ct nu ra bi Dugue [12] (xem thm [13])

Trong phn ny {n} s k hiu cho dy ton t c lp lin tip t A .Mt vi kt qu s lin quan ti trng hp n t lin hp v cng phnphi .Trong cc nh l khc ta s gi thit n khng nht thit t lin hpnhng cc ton t (|k|, k = 1, 2, ....) cng phn phi vi r > 0 v t > 0ta t :

(n)k =

(n)k,r,t = ke[0,nr/t]{|k|}

, 1 k n V :

(n)k =

(n)k,r,t =

(n)k,r,t (

(n)k,r,t)

Ta s thit lp hai mnh ph tr sau:

Mnh 2.7.1. Cho {n} c lp v tha mn |k| cng phn phi

(A) Nu0 < t < 2 vr > 0 th iu kin(t1) < ko theo

n=1

nr2(e(nr/t,){|n

k=1

nk |}) < (2.40)

(B) Nu0 < t < 1, r 1 th iu kin(|1|t) < ko theo

(|(n)1 |) = 0(n

r/t1) (2.41)

Chng minh. Gi s

o < t < 2, r > 0, , (|1|t) <

Khi theo bt ng thc Tchebysev ta c :

n=1

nr2(e(nr/t,){|n

k=1

(n)k |})

const

n=1

nr2n2r/tn(|(n)1 |

2)


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i vi o xc sut trn [0, ) ,

0

xt (dx) <

v r

t > 0

ko theo:

n=1

1

n+1

nr/t0

xt+ (dx) (2.42)

const

k=1k(r/t)

kr/t

(k1)r/txt (dx)

( so snh [12] ). Trong trng hp ca chng ta th :

(|(n)1 |

2) = nn(r/t)(2t)+1

vi 1

n <

V do ta c (2.40)By gi ta s chng minh (2.41) vi gi thit (B) . Tht vy, t:

||1 =

0

ue (du)

(biu din ph) v (du) = (e(du)). Ta c :

n1r/t(|(n)1 |)

nr/t0

xt( xnr/t

)1t (dx) 0

v r 1 v (||t) <


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Mnh 2.7.2. Gi s {n} c lp vi {|k|} cng phn phi v(||2) < . Khi ta c:

m=n

(e(n,){|n

k=1

(n)k |}) < (2.43)

Chng minh. Hin nhin ta c:

(|n

k=1

(n)k |

4) constk=1

(|(n)k |

2|(n)1 |

2)

+n

k=1

(|(n)k |

4)

Hn na:(|(n)k |4) = nn3 + (|(n)1 |4)

Vi:

n=1

n <

(v chng hn: (|

(n)k |

2(n)k )

(1

2||

(n)k ||) const.n

). S dng cng thc (2.42) ta thu c :

(|(n)1 |

4) n3n

vi

n=1

n <

V vy:

n

k=1

(n)k

4 nn4vi

n=1

n <


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v do :

nn4

nk=1

(n)k

4 (n)k 4e(n,){n1

(n)k

}

const.n4e(n,){n

1 (n)k }

Cui cng ta thu c:

n=1

e(n,){ n

k=1

(n)k

}<

nh l 2.7.3. Cho {n} c lp v |k| cng phn phi , 0 < t < 1.Khi iu kin (|1|t) < ko theo

n=1

n1

e(,){ 1

n1/t

nk=1

k

}< (2.44)vi mi > 0

nh l 2.7.4. Cho{n} c lp , t lin hp v cng phn phi. Githit : (|1|t) < , 1 t 2 v(1) = 0. Khi :

n=1

nt2

e(,){1

n

nk=1

k

}< (2.45)ng vi mi > 0.

Chng minh. By gi ta s thit lp 2 iu kin :

(a) {n} c lp v |k| cng phn phi

0 < t < 1, (|1|t) < , r 1

(b) {n} c lp ,cng phn phi ,t lin hp

1 t 2, (|1|t) < , (1) = 0, r = t


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R rng chng minh cc nh l trn ,ta ch cn chng minh rngmi mt trong cc iu kin (a),(b) ko theo:

n=1

nr2

e(nr/t,){ n

k=1

k

}< (2.46)

lm iu ny ,ta s t vi r > 0, t > 0, n = 1, 2, ....

pn = pn,r,t = e(nr/t,){|n

k=1

k|}

qn = qn,r,t =n

k=1

e(nr/t,){|k|}

n = n,r,t = e((n1)r/t,){|

n

k=1

(n)

k,r,t|}

Ch rng :pn q

n

n = 0

vi mi n. Tht vy, nu c :

0 = x = pn qn

n x

th ta s c :

||n

k=1

kx|| = || n

k=1

k

pnx|| nr/t||x||V cng lc :

||

nk=1

kx|| ||

nk=1

(n)k

x|| < (n 1)r/t||x||

L iu khng th xy ra. V vy, ta c:

pn qn n

v do (pn) (qn) + (n)


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n y ta cn ch ra c:

n=1

nr2(qn) < (2.47)

V:

n=1

nr2

(n) < (2.48)

Nu t > 0, r > 0 v (|1|t) < th r rng

n=1

nr2(qn,r,t)

n=1

nr1(e(nr/t,){|1|}) <

V vy ch cn chng minh (2.48) .Gi s (a) ng, ta s chng minh :

Qn = n e[0,(nr/t)/4]{|n

k=1

(n)k |} = 0 (2.49)

Thc ra nu tn ti 0 = x = Qnx th ta s c :

||n

1(n)k x|| (n

r/t)/4||x||

v:

||n1

(n)k x|| (n 1)

r/t||x||

t:(n)k = (

(n)k ), n = max1kn|

(n)k |

p dng phn tch cc ca (n)k v theo mnh (2.7.1.b) ta c :

n (|(n)1 |) < 14

nr/t1

V vy vi n , ta thu c :

||n1

(n)k x|| ||

n1

(n)k x|| + n|n|.||x||

nr/t

2||x||


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iu ny l khng th ,do (2.48) ng. iu ny ko theo:

n e((nr/t)/4,){|n1

(n)k |}

Xem xt ti (2.40) ta thu c (2.48) .

By gi ta gi s (b) ng. Nu 1 t < 2 th (2.40) ng (vi r = t vp dng mnh (2.8.a)) .Trong trng hp t = 2 ta c (2.43) theo mnh (2.7.2).Ch rng

n = |((n)1 )| 0

, ta c th chng minh (2.49) ( vi t = r) v kt thc chng minh nhtrng hp (a).

2.8 Ch v ch thch

Batty a ra rt nhiu khi nim v dy ton t c lp. Chng ta chx l dy c lp yu v c lp lin tip.

nh ngha 2.8.1. H {B, } cc i s von Neumann con caAc gi l c lp mnh nu W{B, 1} c lp vi W{B, 2} vi mi tp con ri nhau 1, 2 ca . Dy ( hay h ) ton t

trong A (hay A nu l vt) l c lp mnh nu cc i s vonNeumannW() c lp mnh.

Trong [11] ta c th tm thy trnh by c th lin quan ti mi quanh gia cc kiu c lp. y ta cp n mt lut s ln mnh c nu ra bi S.M.Goldsteinlin quan n dy ton t tha mn iu kin Rosenblatt.

nh ngha 2.8.2. Cho A l i s von Neumann vi trng thi chun

tc ng . Dy {n} t A c gi l tha mn iu kin Rosenblattnu tn ti dy s dng{an} gim v0 sao cho:

|(xy) (x)(y)| an||x||.||y||

vi mi x W(1, 2, ....k) v y W(k+n, k+n+1, ....); (k, n =1, 2,...)


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Dy{n} c gi l tha mn iu kin ||.||2 (||.||) hay tim cn giaohon nu tn ti dy s dng{bn} gim v0 sao cho :

||xy yx||2 bn||x||||y||

(hay tng ng

||xy yx|| ||x||||y||) vi mi

x W(1,....,k), y W(k+n, k+n+1, .....)

(k, n = 1, 2, ....).

nh l 2.8.3. Cho {n} l dy t A tha mn iu kin Rosenblattcng vi dy{an} v t :

bn = sup|ij|n||ij ji|| (n = 1, 2, ....)

Gi s rng() an c1n

1, bn c2n2

vic1, c2, 1, 2 l cc hng s dng, n = 1, 2,... v :

() sup||n|| <

Khi :

n1

n

k=1

(k (k)) 0

hu u .

Chng minh. Ta cng nhn nh l trn.


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Kt lunLut s ln l mnh khng nh trung bnh s hc ca cc bin

ngu nhin hi t theo xc sut. Lut mnh s ln l mnh khngnh trung bnh s hc ca cc bin ngu nhin hi t hu chc chn.

Lut s ln u tin c cng b vo nm 1713 bi Jamer Bernoulli.Sau kt qu c Poisson ,Chebyshev ,Markov ,Liapunov m rng.

Lut mnh s ln c pht hin bi E.Borel nm 1909 v c Kol-mogorov hon thin nm 1926.Lun vn cp n mt vn hon ton mi trong xc sut

hin i , l s quan h cht ch gia i s ton t , vt l lngt v xc sut. Cc kt qu , phng php , m hnh trong xc sut cin c thay th bi cc k thut chng minh, cc khi nim ,nhngha mi. Nghin cu lut mnh s ln da trn cc ton t trc giao,vt , trng thi ,i s von Neumann,...cc dng hi t hu u ,

hu chc chn ...Ni dung ca ti vn ang l hng nghin cu cacc nh khoa hc trn th gii, nhiu cng trnh khoa hc khng ngngc cng b nh:L thuyt tch phn khng giao hon c sng tobi I.Segal (1953), c p dng vo l thuyt biu din cc nhm com-pact a phng ( Ray Kunze....) . Ngy nay n c nhiu ng dng quantrng trong l thuyt trng lng t ,iu ny c d bo trongcc cng trnh ca Segal ...Hay l thuyt v khng gian Lp tru tng

ca J.Dixmier ; Khi nim hi t theo o (tnh o c theo mt vt)c gii thiu bi W.F.Stinespring (1959), E.Nelsen(1972)...V gn y

nht l cc l thuyt v chng minh khi ngun ca Haagerup(1974) vkhng gian tru tng Lp kt hp vi i s von Neumann. Sau s xuthin cc khng gian Lp ca Haagerup, A.Connes a ra nh nghav khng gian Lp da trn khi nim o hm khng gian. Nhng khnggian ny c nghin cu bi M.Hilsum.....

Vn c cp trong lun vn tng i mi v phc tp.V

70


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Kt lun 71

vy lun vn khng trnh khi nhng hn ch .Tc gi mong mun nhnc kin ng ca cc thy c gio ,cc ng nghip b sung ,hon thin ti.


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Ti liu tham kho[1] V Vit Yn-Nguyn Duy Tin (2001) ,L thuyt xc sut, Nh xut

bn gio dc.

[2] Nguyn Vit Ph-Nguyn Duy Tin (2004), C s l thuyt xc sut, Nh xut bn i hc Quc gia.

[3] Nguyn Duy Tin (2000), Gii tch ngu nhin ,tp 3, Nh xut bni hc Quc gia H Ni.

[4] Ryszard Jajte (1984), Strong Limit Theorems in Non-CommutativeProbability, Springer -Verlag, Berlin New York Tokyo.

[5] Marianna Terp(1981), Lp Spaces Associated with von Neumann Al-gebras , Universitetsparken.

[6] Edward Nelson(1972) ,Notes on Non-commutative Integration,Princeton University, New Jersey.

[7] M.Plancherel (1913), Sur la convergence des series de fonctions or-thogonalles, Acad . Sci. Paris.

[8] G.Alexits (1961), Convergence problems of orthogonal series, NewYork- Oxford-Paris.

[9] A.Luczak, Some limit theorems in von Neumann algebras, Studia

Math.

[10] M.Loeve(1960), Probability theory , New Jersey.

[11] K.Batty(1979), The strong law of large numbers for states and tracesof aW algebra, Z.Wahrscheinlichkeitstheorie verw .

72


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T I LI U THAM KH O 73

[12] D.Dugue(1958), Traite de statistique theorique et appliquee, Paris.

[13] S.Goldstein(1981), Theorems in almost everywhere convergence invon Neumann algebras, J.Oper.Theory 6.

[14] L.Accardi(1980), Quantum stochastic processes, Dublin Institute forAdvanced Studies, Ser.A29.

[15] R.Jajte , Strong limit theorems for orthogonal sequences in von Neu-mann algebras, Proc.Amer .Math.Soc.

[16] W.Stinespring(1959), Integration theorems for gages and duality forunimodular groups, Trans.Amer.Math.Soc.

[17] I.E.Segan(1953), A non-commutative extension of abstract integra-tion, Ann.of Math.57.

[18] A.Zygmund (1959), Trigonomtric Series, Vol.II, Cambridge Uni-vesity Press, London, New York.

Luật mạnh số lớn trong đại số von Neumann.

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Transcript of Luật mạnh số lớn trong đại số von Neumann.