Леоненко Н.Н., Иванов А.В. Статистический анализ случайных полей

pHMa no cOBoKynHocTII nepeaennux (x, 8). n OJIQ)KHM G(x. 't) =

= (£0(x) - g (x, T))2, L

6

('t) =

~

G(x, 't) dx.

6

OueHKOH HaHMeH

bUlHX

KBa

.npa

TO

B napaae rpa 8 Ee Ha3b

IBaeTC

H

/\

A A

/\

C. B. 8

6

= (8

16

, . ••,

Oq6 ) C Ta

KHM

CBO

HC

TBO

M, L

6

(8

6

)

= inf L

6

(r

).

t Ee '

ECJIH g (x, 0) - H

eJI

HHeHHaR

epY

HKUHll.

JIH60

g (x , 8) JIHHei'iHa,

HO

e

=1=

R

q

II He onncu saercs npO

CT

blMH

OrpaHH'leHl

i51MIl

.

TO

"

ouenxy 06' sooome roeops, HeJIb3H sannca

rs

B

HBHOM

sune.

ECJlH g (x, 8) - JIIiHei'IHaH

ep

yHK

UHH

H e = R

a

•

TO

ouenxa 8

6

sa-

X

O.n

HTCH B

Sl

BH

OM sane.

n pe.nnOJlQ)KHM,

'I T O B JIIiHeHHOH cx

ex

e

ep

yHK

UHH

gj

(x) JIO-

KaJIbHO uar erpnpyesn ; C

KBa

.npaTOM, T. e.

dJ

6 =

~

g

~

(x) dx <

00

,

6

i = 1, ..., q,

~

E

sm.

Torna MaTpHUbl V6 =

(

~

gj

(x)gr(x)dx) i.r=l .....a

6

r " I

HeBbl pom.neHbl II, eCJlH

~

= J g (x) £(x) dx,

TO

0

6

= V'6

~6'

6

O'l

eBH,ll,H

O. 8

6

-

nec

xem

enn

aa oueuxa 8: Me=

e,

a ee

KO

p-

peJISlUHOHH

aSl

MaTpHua IIMeeT BH.n

6

a~

= vr'

(~

)

B(x, y) g (x)

g'

(y) dxdy)

V;l

.

(3. 1.3)

6 6

Bsenea .nHarOHaJIbHY

IO

MaTpHuy d

6

= di ag

(

d

i

6

)

i

~

1

.....Q H MaTpH

'l

'

ny

io Mepy

fA.6

(d/')

aa (

R",

}8n) C MaTpHueH nJIO

THOCTI1

(fA.

rj

(~))

r

.I

=I

,

....Q' rne

l

2

~l

r

J

(

~

)

=

g~

)

(/,)

g~

)

(

A)

(

~

I

g

~

)

(

1.

)1

2

dJ.

JI

g

~

)

(/,W

dJ.

r

/

,

R

n

~

g~

)

(A

) = Ji <

'I.

·

x>

gJ (x)dx, r, i = 1,

...

, q.

6

3aMe

TH

M, 'ITO dJ6 = (2n)- n JI

g

~

)

(/')1

2

d

J.

. ECJIH n

OJ1

e e (x),

Rn

X ER" 06JIa.naeT CneKTpaJlbHOH

nJlOT

HO

CTblO f

(A).

I.E

R", T

O,

HC-

nOJIb

3YSl

(1.2.2), H3 (3.1.3). nonysa

ex

1

d

6

a

i

d

..,

=

(2n

)"

(~

fA.

6 (d/,)r

1

i f (A

)fA.

6

(d

J.

)(J

fA.

~

(d/,)t •

Rn

~

Rn

(3. 1.4)

11

8

c:D

V.H .

I.

fA.

6 -+-

fA.

npn

~

-+

00 H var fl = U' flrj (d/' )) . - He-

r

,j

= I ,....a

Rn

eupoxcneauaa MaTpHua.

M epa

fA.

na (R

a

•

m

a

).

cy

w,eC

TBOBa

H

He

x

orop

oa noc

ryaap

y

er

cs

B

YC

J1

0B

HH

I. Ha

3bJ

BaeTC

Sl

cneKTpa

J1b

HOH

Mep

OH

ep

yHK

UHH

pe

rp

eccun

g (x).

MaTpH'lHble Mepbl fl6 H fl, soo

oui

e roaops,

KO

MnJIeKCH03Ha'l-

a

ue

. Flpaxravecxoe onp eneneane Mepbl

fL

MomeT 6blTb

OC

HO

BaHO

aa

COO

TH

OllleHHHX

- I

-I

r r i

<'I.

h> V.H.

d

r6

d

j6

J gr (x + h)

gj

(x) dx -+- J e • fLrj(d/').

~

-

~

00.

6n(6

-h

) Rn

r, i = 1, .. .. q, xor opu e aH3

J1

0r

H'lHbl [152, 53].

E

CJl

H f

(A)

.

f..,

ER

n

- aenpepsianas

or

pauasenuas

ep

yHKUHH,

TO

V. H.

11

3 (3.1.4 ) II I Haxo.nHM npa

~

-

~

00

a

~

=

l

i

m

d

6

a~d

6

= (2n )n( J

fA.

(df..,)

t

1

j

f

(/')

~ t

(

d

f..,

)

X

R

n

Rn

X(j

f.l.

(d

J.

)t

l

.

Rn

Hecaeuieaa aa ouen xa 8

6

na pax

er

pa e Ha3bIBaeT

CH

COCT

O

Sl

-

T

eJ1

b

HOH

B cpennea

KBa.npa

TH

'leCKO

M,

eC

J1

H ee KOpp

eJ1H

U

HOH

H8H

MaTpHua CTpeMHT

CSl

K HyJIe

BO

H MaTpHue npn

~

-+-

00.

n Y

CT

b

9.'1

= {M - C

HC

TeM

a

J1

I!HeH

HO

ynopano

vean

stx

MHO-

meCTB C OT

HO

llleHHeM no

psn

xa «c », npH'leM

!:l.

-+-

00

, 3

cj:l

y HKUHH

f (

J.

) orpanavena H n0

4'

TH

BCIO

.ny n0

J1

0mHTe

J1

bHa. n OBTOp

5l

51

pac-

Cy

>K.ne

HHH A. C. X O

J1

e

BO

[127], 38K

J1I04'

aeM,

4'

TO

.nJ1

H

COC

T0

5lTeJJb

-

HOCTH B cpennea KBa.npaTH

'leCKOM

ouenxn Ha

ll

MeHbWHX xsan-

paT

OB

0

6

He06xo.nH

MO

H .n

OC

T3TO'lHO,

'l

T06bl

.nJ151

m060ro

BeKTOpa

Q

8 = (8

1

, .. .,

8

q

)'

ep

yHKUH

Sl

g (x, e) =

~

eJg

j

(x) y.nOB

J1

eTBOp

5l

J1

a

1=1

YC

J1

0BHlO

~

gZ

(x, 8) dx = 00.

R

n

Teopeaa 3.1.1. n pe.nn0

J1

0)l{HM.

4'

TO ee

xr

op-rpyaxuas g (x)

ynoaner aops

er

YCJlOBHIO

I H noxoopnan ar no Y

CJ10

BHIO

(1.8.10).

nY

CTb

r axxce onnoponaoe C

J1Y

4'

ai'lHoe nOJIe e (x), x E

Rn

ynos-

JIeTBOp

5l

eT YC

J1

0BHIO

CHJlbH

OfO nepeae

um

aaaas C K03qxpH

I..I.

HeHT

OM

a (p), npH

4'e

M

M le (0)!2+0< oo,

a

(

p

)~kI

P-

n

-

'I',

,\,

6> 2n,

6>

0. (3.1.5)

11

9

Torna y nons e (x ) cyuiecrsye r nenpepsrsuas

11

Orpa

HI1'1eHHaH

cnexr pans uaa

nJIOTHOCTb f CA),

'A

E

Rn

H.

eCJIJ1

M

aTpli

ua

r V.H.

J f (I.)

J..l

(

d'A)

HeBb

lp

O

il\)l,

eHa, TO npn

t'1

- " 00

Rn

A

ID

d6 (0

6

- 8)

--

Nq (0,

aT)

.

.&.

C YlJeTOM 3

<J

Me

'l

a1l

11

H 1.6.1 yraepacnenne cnenyer

11

3 Teo-

peMbi

1.7.5, ecrm EbI

6p

aTb B K

aT.{

eCTBe KOMnOHeHT BeCOBOH <pyHK·

unn g6 (x, 8)

<P

YHKIHlYI

g

~)

(x, 8) =

gj

(x) djt,l, i =

I,

.... q, KOTo-

psre CeHlJaC He 3aBJiCHT OT 8. _

YCJIOBHe

nep

eMeWI1BaH

Yl

51

B (3.1.5) MO)KHO

OC

JIa

611

Tb ,ll.JUI

ce Me HCTB

c.m, COCT0

51

IUI1X H3 np HMo

yr

OJJbHHKOB, uiapon (CM.

F

n. 1.7). B cny xae

/::..

-+

00 BMeCTO (3.1.5) MO)KHO I1CnOJIb3

0naTb

orp

aHHT.{eHI1

H, nana raea u e na Ko

3¢

epHUHeHT nep eMeWI1B8HIHI

a (p, d)

11

3 reopeuu 1.7.2 liJlI1 1.7.3.

E C,l l1 B (3.1.2) g (x) =

I,

f (0) > 0, TO y

rse

pscnenne

re

opevu

3.1 .1 osaasaer , 'I TO ouenxa 6

6

= j £(x) dxf\ LiI nap

aa

er

pa 8 npn

6

v.n. I 2 IDA

Li- 7 00 8C

Il

MnT

OTH'IeCKYI a opa

ansaa:

I/::..

/ / (8

6

-

8)

-+

N (0,

(2nr

f(

0»

.

H

cn

oJlb

3Y

51

cne

ncra

ne

1.8.1, r e

op

ev

y 3

.1

.1 MO)KHO 0

606

IUI1Tb

no

nJl8HY

.n.

OKa

3aTeJl

bCTBa reopevu 1.9.2

no

¢ YHKUl1

oHa

Jlbll

oH

u . n . T. ,UJJ

51

Y1

JJ

IOCTpaUYlH CK

a3

8HHo

ro

,

pa

c

cv

orpnv

n

OJJ

e (3.1.2)

Ii

np

enn

on

o

soes

, 'ITO 9JI

COC

T

OYIT

H3 paCWl1pHIOIUHXC

51

tua

po

a v(r).

A A 2

0 603H

a'l

HM 8 (r ) =

8v{

r ). O YCTb epyHKUH

51

a (r) = d v(r ) MOHOTOHHO

B03paCTaeT. EC

JJI1

38MeH

Yl

Tb yC

JJOB

He (3.1.5)

YCJJ

OBYIeM

(l .9.2).

TO Be

p0

51

THOCT

Hble

MepbI P, B C [0, 1],

co

oTB

eT

cTBy lOIUHc CJJy·

'1aHHbIM

np

ou

ecc

a»

l

2(

Xr(t) = ta

/

r) (e(a-

l

(ta (r)) - 8] , t E[O, 1], (3.1. 6)

np

n r

-+

00 CJIa60 C

XO,ll.51

TC

51

K Me

pe

Wb B C [0, 1], cOOTBeTcTBYIO-

IUeH np

ou

eccy 6POY

HoBcKo

ro

)l,BfDKeH

H51

Wb (r), t E[0, 1J C

JJO

-

1

K3

J1

b Hofl

,ll.H

cn

ep

cl1

eff

b = (2

nt

j f(A)!1(d'A) (B (3. 1.6) a-

( .) _

epYH KU

Yl51

,

06

pa THa

51

Q ( • ».

R"

T a K, e CJIH B (3.1 .2) g (x) = Ix IV, 'V>°(

pana

ansn

aa

n O

JJ

YlH

Q-

n

+

v

MIJaJIbH

a51

pe

rpecc

nn ), TO a (r) = (n +

v)-ll

s

(1)1

r

,

a

-I

(u) =

= « n +

'V)I

s ( 1)1-

1

u ) :/(n+ ,'l,

X,

(t) = (n + v)1/2 Is (l

)1-

1

/

2

r - (n+\I)/2

X

X f jx l" e<x) dx, tE

[O

,

IJ

.

<1rl

l

/(n+ \I))

OT

C IO

)l

a npa v = 0

lIo

.nY

lJae

M CJIY"l3

ilHbl

e npoueccu , nocrpoeu-

ssre 8 n. 1.9.

B cn

ysa

e, xorna

g)(

ecr

s C

HC

TeMa

p a

CWl1p

HIOIUIJX C

51

UI3 p

OU,

8 none e

(X)

, x E

Rn

OAH

OpO)l.HO H H3'OTp

Onf

IO,

nOHHTl1e cnexr pan

s-

lIo

li

MepbI <pyHKUllYl p

erpe

ccna M'OJKHO

MO.llYl¢IlUlip

OSaTb

raK,

lJTOObI ee SbI

T.{

I1CJJ

eHl1e CBO.ll

Yl

JJ

OCb JI{ aH a

JII13Y

O,ll.HOKpaTHbIX II

B-

T

erp

aJI OB.

P a CCMOTpl1M

)l,JJH

n

pO

CTOTbl JJHWb cxevy (3

.1

.2) npa

j,

= v (r ).

Tor na ouenxa HaYl

MeHb

IllH

X xnanpa-roa nap av erp a 8

6y

.:reT m leT b

BI1

11

8(r) = f

~

(X)

g (x)

dx

/d

~,

d; =

d

~>(r

l

= f g2(x) dx.

~(

rl

<'

(r

l

O

ep

eH)l,eM K nO

JIHpHb

lM xoopnnnaa (1. 1. 1). n YCTb

e; (r) =

=

f g (r, u)

S

~

(u) dm (u) -

1<

03

¢

¢H

Ul1

eHTb

I ¢

ypb

e

<PY

HI<UI

III

5(1)

g (r, u) no OpTOHO

pr.-

lYIpOBaHHoH CI1CTeMe

crpep

axecxax

rap

MOHIU<

{

S

~

(un , (m, I) ET - {

OJ

1,

...

} x {OJ 1, ..., li

('1,

m)} (CM. n . 1.2)

Torna

,

d~

=

5)

g2

(p, u) pn- 1dpdm (u) =

U 5(11

00

titn .rn, r

=

~ ~

j

[

g

~

(p)]2

pn-

'dp, (3. I. 7)

m=O 1

=1

a

np

Yl'leM B CYlJIy paSeHCTBa O

ap

ceBaJI

51

,ll.

JJ51

n

pe06

pa30

BaHlI

51

L

JJI

-

KeJIH

,

00

,

f

'A

[f

g;

1I

(p\ p

[g

~

(p) p(

n-

2JI2

J

2

dp = ) J

m+(

I1- 2)!2

('A

p) p"

i 2

d pt d

'Jv

,

~

u a

(3. 1.8)

C yse

ro

a (3.1.7), (3. 1.8) SBe)l.eM ceaencrno HOpM l1

po

Ba

HHb

lX

Mep aa

(

R~

,

~

(R~»

l:-l~

"(d'A)

= [f

r

g~

(

p)J

m+

(n-

2l

/

2('Ap)

pnl2dP

rAd

'A

X

a

,

X [J

g

~

(p)F pn-1dp r.

o

120

121

00

f.:

(n, m)

n YCTb A

na

,dr) = am

.dr

)1

~

am.1

(r),

i,ne

am.l(r) =

m-

O 1=1

= j

(

19~

(r»)2

plt-'d{l, (m, l) ET. O'leBI1)1,HO.

o

......

fJ.(

n . mJ

Am.,(T

)

~

0,

~ ~

Am,t{r) = I,

m= O 1=1

TaK

tlTO npu Ka )!{)I,OM t aanaao Sep05ITHOCTHoe pacnpenenenae ua

MHO)j{eCTBe T.

3a

MeTHM,

-r

ro rip» r

-+

00 '-lHCJla

Am

'! = lim

Am

,! (r), (m,

I)ET

r-.

oo

te CJlI1 npeaeau cyuiecrayicr) He ec

er

na 3a)l,aIOT B

epo

5ITH OcTH

oe

pacnpeneneaue H3 T, nOCKOJlbKY MO)j{eT CJl

Y'lHTb

C5I,

'lT

D A

m.

! =

= 0. (m, I) ET.

Ilpenn

o.n

o

xa

a, xro y nOJl5I e (X), X E R" cyuiecrayer

11

3

0Tp

on-

- 1

Ha H cnexrpansnaa n JlOT

lI

OCTb g

(1,),

AER

+.

Hcnonsays (1.2.11)

II

re

op

ex

y CJlO>KeHIUI )l,Jl5I epyH KUHH Beccens, n

onys

aea )l,JlH

Jl.HC-

nepcnu

0;

Ou eHKI1

e,

raxoe

npe

ncraaneauei

cr

;d; = j

~

B (I X - Y I)g (

X)

g (y) dxdyld ; =

L(r) tJ(r'

t (

00

h( • •m )

= j j j j g (PI' u

I

) g

(Pa'

u

2

)

~

cT(n)

~

S

~

(u

I

) X

o 0 $( 1) 5(1) 0

m=

O 1

=1

( ) J

~

)

~

)_

(n-2)

/2 J

(~

)

(~

)-

(n-2)(.l

X

S

m

'

u

2

m

+(

n-

2)!2

(fIo

PI (I\PI

m+(n-2

)/2 I\P2 "'P2 X

X

dG

(I,)

(

PIP2)

n- 1dp1dp2dm (u

I

)

dm (u

2)

ld; =

00

h(n ,m)

00

r

= q (n)

~

([~

g~

(p) J

m

+ (n_ 2)/2

(t..p)

(t..p)-(n-

2)

/2

pn

-

1

dp tX

m=

O !

=\

~

0

00

h(n,m )

00

X G(d

'f..

)/d; = (2ft)"

~

Am,1

(r)

~

g

(t..)

~~.

I

(d

t..

). (3.1.9)

rn=O 1=1 (I

BBe)l.eM Ta I<Oe y CJIOBue.

g)

II

np

H r

-+

00

11

m

. !

-~

IL

.

• r r

Im,l'

IimAm. !(r) = Am" E

[0,

00),

(m,!)

ET;

(3.1.10)

,_

co

122

B (3.1.9) paspe

un

eu

11

04J1

eHHh

IH nepexon K npene

.ny

n pl1 r

-

~

00

H

00

k(

n,

tn )

00

cr

~

= (2m)n

~

Am., Ig(

),

)f!m. ' (d

)')

E(0, 00). (3.

1.1

I)

m=

O

!=

I 0

CeM

eH

cT

BO Mep {f!m.

I},

(m, l) ET , CYW.eCTBO

BaHli

e xoroporo 110 -

c

ry

aapye

rc

s YCJlOBlIeM

II,

Ha3

0BeM

H30

Tp

onHOH c

nexr

pansno a

Mep

OH epy

HKIum

perpeccan,

Il

ycr

s g (r, u) = g (r) - pannansaan

epyHK UH5I. T

orn

a g6(r) = g

(r)1

s (l)l'n, oc'ransnue s; (r) paaau

HyJJIO. C JleJl.OBa1reJlbHO, AO,l (r) = A

o

.

l

=

I,

ocr ans usre A

m

•

1

(r) =

= A

m

.

1

= 0, H

1(

3.1.10) Bb

IIl

OJlH5IeTCH. B yCJIODHH II HaJl.O r pe-

ill

60BaT

b,

'lT

06bl

npa

r-+oo

JlHW b

f!

~.1

~f.1

0 .1'

ro

rna

BMeCTO

(3.1.11) Il OJlY

'l

H

:.'.

l

00

cr

~

= (2rt)n

~

g (

t..

)

f.l

o.1(d

t..

) E

(O

, 00).

(3.1.12)

o

Bbl

6Hpa51 B xauecrse g (r, u)

C

Y

~

I:vtY

pann a

ns

no a ¢ YH KU

li

H

H K

OHe'lH

Or

O

'lHCJla

p a3J1H'lHbIX c

rp

epnsecxax r

apMOHH

K, 6YJl.e"l

noayva

ri,

npn n = 2, 3 BeJJli

'l

lfH

bl (3.1.11), B KOTOp bIX Am.1 OT-

JIH'lHbl

. OT ay Jl5I )l,Jl5I K

OHe'lH

bIX

n

O)l.

~1HO)j{

e

CTB

MHO)j{ eCTBa T.

ECJIH epy HKU

li

5I g

(I,)

nenpepuea

a

11

or pana

sesa

H ssmonneno

YCJlOBHe II , TO H3 (3.1.9) n

OJlY'lHM

1i m

cr

;d

; =

cr

~,

rne

o

~

o npe -

r-+ oo

nensercn COOTHow eIlHeM (3.1.12). Ecnn <PYHKUH5I g pannansnas.

TO )l,Jl5I

cr

~

cnpaaennueo COOTH

OllleHHe

(3.1.12).

Teopena

3.1.2

. n YCTb npn

!:J.

= V (r) Ha6J1IOJl.aeTC5I CJlY 4a

HHo

e

none (3.1.2),

npH'le

M onnoponaoe asorponnoe none e (X), X E Rn

y)l. OBJleTB Op5IeT yCJIOBHIO (3.1.5), a ep

yHKU

H5I perpeccnn g (x) =

= g (r, u) - YCJlOBHlO (1.8.10) npn

!:J.

= v (r) .

nY

CTb BbIOOJIHe-

HO

raroxe ycnosne I

r.

Toraa cyuiecr ayer H

enpep

bIBH 25I

Or

p aHI1-

- I

tJeHH a5I

li3

0Tp

OnHa5I cnexrpansnas nJI OTHOCTb g (I.), I, ER+

11

np H

A

ID

2

r

-+

00 d, (8 (r) - 8)

-+

N (0,

02)

.

Y TBe

p)j{)l.

eHH= reopevsr cnenyer 1i3 reopexu 1.7.3 H 3

aMe

4a -

HHH

1.6.1.

P a

3yMee

TC5I, n

OHHTHe

H3

0TP

OllH

Oii

cnexr pam.noa

Mepb

I M O)f(-

HO BBO)l,HTb

aa OCH

OBaHHH

pa3J1

0iKeH

HH (1.2. 12), (1.2.13) npn

n =

2,3

.

Fl

yers

reneps

nOJIe

(3.1.2) H

z6JIIO

)I,a

er

CH na c

rpe

pe s (r) H

D; =

~

g2

(x) an (x) < 00 )l.Jl5I

JJI060r

o r > 0, 2 none e (X),

W)

x E

R,

n OJl.HOpO)l.IO U

H3OTp

OOHO.

01.leHKa

H

aHM

eHbWlf X KBaJl.paTOB

123

A r 2

napaaerpa

()

l1i.HeT Bl1jl

e*

(r ) = J g

(X)

~

(X) dm

(:<)10,

. 3lTO He-

'('

I

caeuieunaa o

ue

nxa, a ee nncnepcus

00

n tn.rn;

00

h(n.m)

l

0:

(r) =

ci

(n)

~

Am,d

r

)

bm

.

/l

(r)

(

~

[

g

~

(rW t ,

m=O 1= 1 m=O /= 1

(3 ,1.13)

00

ntn.m ,

rne

-

01

.1(r) =

[

g

~

(

r)

]

2

/(

~

[g;n (r) ]2] ,

m= O /= 1

b

ln

•

n

(r) =

00

j J;n+(n

-2

)/2(Ar)

(A

r)2- n G(dA), G (dA)

n

yxacrayer B npen

cr

aaaeaa u ( 1.2.4).

H anpHMep, nYCTb B (r) = Y

/l

+

2

(or), a > 0, G (

A)

= " na- n

npa

'A,

~ a; G

(A

) = 1 npu

A>

a. T or na b

m

.

n

(r) = 2

n

-

2

r (n/2) X

X

::1.

n

/

2

J

~

+

( 1 1

-

2

)

/2

(ar) (ar)2- n. E CJII1 B (r) = exp { - ar

2

} ,

G'

(A)

=

= (2a)

-n

/2

A

n- 1exp {- A

2

/4a}, TO b

m

•

n

(r) = 2 (2/a)

(n

- 2)/2

n

n/2 X

2

n

X r (n/2) r -

exp

{ - 2ar

2

}

J

m

+(n

- 2)/2(2ar

2

) ,

a>

0.

A P

E CJI" sup bm,n (r)

~

k, .nJl

51

scex r, TO e*(r)

-+

8 npn r

-+

00

.

m

Paccvorpua sonpoc 06 aCHMnTOTlf lJeCKOM nOBe.neHl

III

0:

(r)

npn r

-+

00

.

II I. LlJl

51

nexoroporo y E(- l, n - 2), n

~

2 0'

(A)

=

= ls (l

)IA

'\'h(A

), rne h (A) nenpeju aaa B aexoropoa OKpeCTHOCTI1

uyns, h (0)

=1=

°h (

A)

orpanaueaa H3

R

~.

n YCTb B

blD

OJlHeHO I II. F blopaB B JleMMe 1.5.4 K (z) =

= J;I+(

11

_2)/2z

,\,

+2- n C yueron

COO

THo

w eHHH (I .5. 16)

n Q

JlY

411

~

1

n

pH

r

-+

00

bm,n (r)

~

--=

Is (1)1h (0) c

2

in, m, y) r-

I

-

,\,

(I +0(1

»)

, (3. 1.14)

rne c

2

(n, m, ,\,) =

I'

(n - 2 -

1'

)

I'

«(2m

+I' + 1)/2) [2

n

-

2

-

V

X

X 1'2«n

--

J - 1')/2) r ((2m + 2n - 3 _ 1')/2))- 1.

E CJIH Pb

lfl

OJ

lH

ellO y CJIo

13

l1'

e III, cyuiecrayior

lim Am.

dr

) = Am,IE

10

,

00),

(m,

1)

ET, (3.1.1 5)

'

.....

00

np

HlJe M B (3. 1. 13)

pa

3

pelll

eH n

04Jl

eHHb

IH n

epex

o.n K npe,lteJ:y

np

ll

r

~

00

11'

00

It(n .

m)

(J'

5 =

(211

)n h (0)

~

AmA,dn,

m, y) E(0,

00),

(3.1.16)

m = O 'I

=1

124

TO

11

3 (3.1. 13), (3.1.14), (3

.I.J

6) n

orr

Y4

11M

IJpH r

-+

00

hln ,m I

a

2

(r)=a

Hr

v

.!-I

.L;

~

[g:n (rW f-

I(

J + 0 (1)).

m=

O /= 1

B lJ<lCTHOCH 1,

eCJH1

epy HKUHfI g (r, Il) = g (r) - pannansna»,

TO YCJlOBHe (3.1

.1

5) EbrnO

JIl

liJeTCfI II npn r

-+

00

0

2

(r) = C

3

(n, 'V) X

X h (0)

r-

,\,

-l

g

- 2 (r) (I + 0 (I)), rzie

c~

(n, 'V) =

~T

(n) r (n - 2 -

2

r

2

- }i) r ((I' + 1)/2)/[2

n

-

((n - y- 1)/2) r

«2n

- 3 - 1')/2].

3 3MeTIl M. lJTO

nJlH

P3.n.ll

3J1

bH0I1 qly H

KlH

H1 g (r) ouea xa

8*

(r)

napaverpa 8 II MeeT MHHJHvlaJIb

HYlO

znrcnepcmo B KJlaCCe scex

.'IHli eHHblx uecwemennsrx

ou

esox na

pav

erpa 0

11

39, rn. IV, § 3].

B

43C TIlOC TH , npa g (r) = I On n

1M3JIb

llO

H oyner ouenxa Ba H-

r-. I r

Me

HbWIIX

xaanparos 8*(r ) = Is (r) r \ £(x) dm (x) uenasecrnoro

sir )

cpenuero 8. B r eopeue 1.7.8 npu II

~

2 0

11l

1CaHbl YCJlOBHH acna n-

TOTlf lJeCKOH HOPM3

J1

bHOCTII ::

nO

H oueaxn.

P aCCMOTp HM reneps npanepu aCI-iM nTOTH

yeCK

HX pacnpenene-

I-Ill

ii oueuox HaHMe

HbWllX

K

B3n

paTO

D, xcropsre CTpOHTCH ua oc-

HOB'

am

ll-i

pe

3YJlbTaTOB rn. 2.

n YCTb B cxene (3.1.2)

11

0 Ha6JllOACII IHI

1\I

{£ (x), X E ti (fl)},

~l

-+

00

H3n O ouemrn, napanerp 8. Ilpenn

o.n

osor», 41'0 none

s (X) = G('I) (x)), X ERn, rne

CP

yH

KUH51

(]:

Rl

-+

Rl ynoaner -

aop

se

r YCJlOBHlO III n . 2.1 C K03cjxpHUH

ell

TOM Co = 0, m =

= r

an

g

G,

a

cJlY

4a HHo e none 'I) (x), X E

Rn

ynoaneraops er YC-

JlOBI1IO

XXI n. 2.10, T. e. HMeeT ueorpaan vennyio B nyne cnex-

rp ansnyio nJlO THOCTb , Torna

no

Jle MMe 2. 10.1 npn r

-+

00

K.

cp

.

B ( Ixl) = B (r) nOJIH 'I) (x) HMeeT B

Hn

B (r) = c 4(n, a) r: (I +

+

0 (I )), a E (0, (n + 1)/2),

c

~

(n, a) = h (0) C

21

(n, a) , rae

C

21

(n , a) onpeneneuo B n. 2.10. O THOCHT

eJlbH

O

CPy

HKU

HH

perpe

c-

CHB

g (x) = g ( Ix I) n

pe

jl

nOJl

o)j{HM

noxansnyio n

rr

rerpup y e

aocr

s

C xaanparov H nonvcrnaocrs

LJ.Jl51

K.

cp

. B (Ix I) nO

Jl51

'I) (x) (CM.

n .

2.5). T

aK

HM

o6pa

30M, AJIH m = r ang 0 > I c y

uie

cr s y

er

no

.n

o-

)j{IfT

eJlbHbIH

np

en

C

5

= cd n, m, a, g, ti) = m! Jim

J

r(

J

g (p,1x

I)

g (u ]y DX

11-+- 00

66

X «: (u) Ix - Y I

-am

dxdu,

ex

E(0, min tnlm, (n+ I)/2}. (3,1.

17)

nilfl

napaMe

rpa

e

C

Xe

~

1

b1

(3.1.2) paCCMOTpl1M ou e

!l

Ky

0

6

(1') =

d;

2 5 g (x)

~

(x) dx,

d~

= f g2(x) dx.

6.(

11

) 6

(11

)

125

- -

B YCJIO

Bl

Ul X

reope

x u 2.5.1

npn

f.L-+

00

H a E (0,

min

{n/m,

(n + l) /2}

d

tD6~(",)

=

A~

(u)

cd

I +0 ( I)), r ne

A~

(u) =

C;n

X

- 2 m

2n-ma

2.( ) n

X

(

m.

1)

C4

f.L

g

rt.

pl1

11·

-+

00

npe.aensnue pacnpenenenas

C.B.

~

l!

= (

et.([t)

-

B)/[

A

m

(r.l.)

d~2c~/2]

11

C.B. Y m,l!,g(I) =

m

=

sgn

{C

m}

c

:j

m/2

c5

1/2

f.L

(m

a -

lI

g

- 1(u)

~

g (x) Hm

(~(x))

dx COBna-

M",)

):(aIOT. B YCJIOBI1HX

reopevu

2.10.2. npa

f.L

-+

00

npene.u.nu e

pacnpeneneuas C.B. Y

rn,l!,

g( l)

(a

3HallHT,

11

C.B.

~~t)

3a~aIO

T

C

5I

K. c.

11.

sgn {Cm} Y m,g (

I)

(CM. (2.10.7)). B lIaCTHOCTI1, AJIH

nu e HKI1

e(r) HeH3BeC THOrO cpe

znrero

B nOJIH {t, (x), x Ev (rn B YC-

JIOB

I151

X reopeu u 2. 10.1

cnp

aBe):(JII1BO npeae.ns

noe

COOTfIOWe

ll

l1e

r.

'il

mu

2

~r

=m!

r

/

1 v (1) 1

1

/

21

c; ,- l

c

:j

rn

/2

c

2'I/2 (8 (r)- B)

-+

x,

(I

) sg n {Cm}'

rne c

2=c

2

(n, m, a,

v(l))

nae

rcs

epOpMyJIOH (2.1. 10), a

Xm(l

)-

COOTH OIUe

ll

l1eM (2. 10 I).

3.2.

COCT05lTEJlb

HOCTb OUE HKH

HAHM.EHbWHX

KBA,D,PATOB

nPH

HEJIHHEAHOR

nAPAMETPH3AUHH

R

q

np

e):(nOJIOIKHM,

'ITO

e c -

OTKp

bITOe

MHO)KeCTBO, a epy

ll

K-

u.mI g (x,

B)

B (3.1.1) HeJIHH

ei

'IHO 3aBHCHT OT napanerpa 8 E

e.

B

n.

3.2 - 3.4 6 y ):(YT paccaorpensr uexoropue a CHM n TOTH

'I

eCK l1e

CBoHcTBa

ouemor et.

rrpn

L\

--+

00.

nYCTb

d",

=

d",

(B) =

diag

(d

i

",

(O))i=I. ...q» 8E

e.

M aTpHu eH

d",

6y):(

eM HOpMl1pOBaTb

OUeHKy

en-

E CJII1 g (x, 8) ):(l1

epepe

peHul1p

y eMa

no

8 H ee nponsaonnue gi (x, 8) =

(iJ/iJB

i

)g

(x,

B)

JIOKaJlbHO C

yM

-

MHpy eMbI C KBa):(paTOM.

no

x, TO eCTeCTBeHHO

B3HTb

dZ

",

(B) =

= .\

gZ

(x,

B)

dx, i = I, ..., q. n YCTb

TE

e - uexoropu n KOMn

aKT

.

'"

nanee cnysaea Bbm OJIHeHH5I COOTHOWeHI1

51

O r paHHlI HMC5I Ba)KH

bIM

lim inf IL\

1-

1

/

2

d

i

",

(B)

> 0, i = I, ... , q. (3.2. I)

"'

~

oo

OET

(Jlesas qaCTb (3.2.I) MO)KeT oopaiuarscn B

6eCKOHe'l

HOcTb). Ce-

MeHcTBo c . B.

B ~

=

8

~

(Go (x), x E6) Ha3

bIBa

eTC5I pann

oa

epuo

COCT05lTe.

TJh

Hb

lM

cCMei

k TB

oM

oueaox napaa erpa 8 BO MHO

lK

eCTBe

T e e (e",- pasnoa epuo COC

T0

5lTeJIh Ha H ouenxa 8 B T), eCJIH

):(JI5I

JII06

0r o p> 0

supP{

1

8",

-

BI>p}

- 7 0.

(3.2.2)

8E T

t.~

oo

,

126

0603HallHM

CPt.

(B

1

,

B

2

)

= I

(g(x,

8

1

) -

g ix, 9

2

))2 dx ,

wt.

(8

1

, 8

2

)=

t.

= Se (x)(g (x, 8

1

) -

g (x, 8

2

))

dx, 8

1

,

8

2

EEl';

z~

(81' 8

2

) = <p; l

(81'

f,.

8

2

)

W", (81' 8

2

) ,

8

17"=8

2

,

Zd

8

1' 8

2)

= 0, 8

1

= 8

2

;

i' (

i'l

) = 1

L\

/- IJe

2

(x )dx.

no

onpenenemno

e",

n. H.

~

1 r-; 1 A

8 fl

B

~

1'(L\)

~

\ L\

1-

L",

(8t.) = y( L\) - 2 1

L\

1-

w

~

(8",, 8) +

1

r.

+ I

L\

1-

cP~

(8~,

8), (3.2.3)

HJII1

cP~

(8

""

8

)( z

~

(8

""

8) - 1/2)

~

O. (3.2.4)

nYCTh

B

~

(8) E B

C

-

60p

eJIeBCKOe

MHo

me

CTBO Ii (8). Ec.

TI

Il

e6 E

B

~

(

8

)

11

~JI51

JII06w

6>0

11

8

1

, 8

2E8

c

in!

qJ

",

(81' 8

2

» 0,

18,

- B

.I>b

TO

11

3 (3.2.4)

BbITeKaeT,

-rro

P

{

8

~

Ee, (8)}

~

P { su p

z

~

(1:,

8)>

17

2}.

(3.2.5)

t

E

B

~

(

8

)

Hepasencrao

(3.2.5)

5lBJI51eTC5J

HCX

O,LJ.HbIM

~,/IH

nO JI y q e HI1

51

)1,OC

-

TaTOllHhIX

YCJIOBI1H pasnouepnof COCT05lTeJIhHOCTI1 8 t.. B xacr ao-

CTH, eCJII1 Bt. (8) = e

c

n{r :

I.

- 8 1

~

p}, TO CXO):(HMOCTb npn

L\

--+

00

npasof

lIa

CTI1 (3.2.5) K Hy JIIO paanovepxo n o 8 ET nne-

tIeT

):(JI5I

e

~

BhIDOJIHeH He (3.2 .2).

nO

JIO

)l{

HM

CD

t. (up u

2

)

=

CP

t. (8 + I

~

j112

dt:

1

(8) up

(-J

+1

L\

1

1/2

X

X dt: ! (8) u

2

) ·

Ilp n epHKCHpOBaII HOM 8 E

El

epy HKUH5I

CV

t. (up u

2

)

on -

peneneaa na M

HO)Ke

CTBe

U~

(8) X

U~

(8), U

~

(8

)=

1

L\

/- I

12

dt. (8) X

X (

El

-8).

npe):(IlOJI

O)K HM c

nen

yiouiee.

IV.

Il

Jl

51

JIIo

6b1x e > 0

11

R> °c YIUecTBy eT raxoe 6 = 6 (8,

R),

qTO

sup

I

~

,-I

CD

d U1, U

2

)

~

8.

(3.2.6)

OET

tJ

i

.

u

2

E

U~(8)nvC(R);ltJf-u21"';1\

V.

Il

JIH

aex

crop

oro R

o

> 0 H JIl0 6o

ro

P E(0, R

o

)

oYIUeCTB y·

lOT qHCJI8

a = a (R

o

) H b = b (p, R

o

)

TaKHe,

1J

TO

in! in! I

L\

,

-I

cD

",

(u,

O

)

~

b;

(3 .2.7)

BET

UEU ~

(8 )O(V

c

(R )

".

~p

»

o

12

1

in! inf I

IJ.

1-

1

(] )t; (u, 0) > 4B (0) + Q . (3.2.8)

RET uE U' (or

....

<,

VIR

0

)

6

JleMMa

3.2.1.

I'l yc-n, n

one

E(x) ynoanereopser YCJIOBIHIM

Jlem 1LI 1.8. 1 npn In = 25,

s>

1

J1

nJI51

JIl06bIX

el'

O

2

Ee

c

I

IJ.

II/2 su p Ig (x,

ell

- g (x, e

2

)

IX

rp

:f2(81' e

2

)

npn

~

00

. Tor.aa

\ E6

4S

M I w "\ (61' (

2

) 1

~

klrp~

(

8l'

(

2

);

(3.2.9)

25

M I

'Y

(~)

- B (0) 1

~

k

2

1

~

,-

5. (3.2.10)

A H ep aBeHCTBo

(3.

2.9) cos nanaer C (1.8.11). H

epa

aeucrso

(3.2.10)

BhITCK

aeT

113

roro, 4TO K03

cj)(

flH UHeHT

nepe

MCWHBaHH

Sl

a (p)

2

n OJl

51

t: (x)

lie

IJp eBOCXO)

l}

1T a (p). •

B

npHB

O,ll.HMOH HB)Ke r eopexe BbIBO,ll.bl JleMMbI 3.2.1 HCnOJIb'

3YIOTC

fJ

KaK nOCblJlKH.

nO

JlO)KHM

~

6.

(e) = I

~

1-

1

/2

d

~

(e) (86. - e) .

T

eopeva

3.2.1 . ECJlH B

bm

OJlHCHbI y

CJlOBH51

IV, V, (3.2.9) Ii

(3.2.10), TO

LJ}lSl

JlI06

0ro

p > 0

sup

P

{

I~6.

(

e

)

l>p

}=

O

(\

~

I

-

S

)

npa

~~OO.

(3.2.11)

SET

• n YCTb p E(0, R

o

)

epH

KCHpOBaHO, R

o

•

b

11

a -

411CJla,

CY-

uiecruonanne KOTOp bl X otec nenneaerca V.

nO

JlO)!{HM Z

e:,.

(u)

==

=z:.

(El

, e +I

IJ./-

1

/2 d

6

(e) u).

no

H

epaBeHcTB

Y (3.2.5) M il JlIO'

6

0ro

0 ET

p

{I

~

_

\

(0) I> p}

~

P { sup

Z6

(u) > 112} +

U

E

U~

(S

>'-..VC

R

o

'

+ P

~

s

up

Z

e:,.

(u) > 1/2} = P

1

+ P

2

•

UEU

~

(f)lnI Vc(Ro

'"

"l Oll

(,-lcnoJlb3Y5I HepaBeHCTBO I<.OWH - 5 YH

51

KOBCKor o. YCJlOBHe (3.2.8),

HCpaneJlCTBO 4 e

6blw

eBa II (3.2.10), Hax o

,n,

li M PI

~

P {v

(~)

>

> 1/4 in! I

~

/-1

eD

t:.

iu,

O)}

~

P {V

(~)

- B (0) > a/4} =

"

EU

~

(e

)"

u<Rn

l

= 0

(1

~

,-

5)

.

C

l l

ycr

s

F'

,1)

, ....

pm

)C V

(Ro)

"",-V

(p) -

saMKHyTble

MHO)KeCTBa,

m

U p(

i>

= V' (Ro)"'..v (p).

,lI.I1

aMeTP Ka)!{,n,

oro

p I

ll

MCHb

we

6, OTBe·

; =1

{taI

0we

ro no YC

J1

0BHIO IV 4HCJlaM e H R

o

(BeJlH4HHa e oyner

(0 F (O U

C

(e) .

8bJ6paHa Hl1)f{e

)

, U E n 6 • t = 1, ...• m. Toraa

m

P2

~

P{

su p

z

~

(a)

~

1L2};

I=oJ

uE

P(

I>nu~

(e)

128

P{

sup

z~(u)~l!2}

~

:P{

1

Z

l\

(

u

(i)I~

1!4} +

u

E

F(i)n

u

'~(

e)

+ P {

sup

' IZ

6.

(u

1

) -

Z

e:,.

,(u

2

)

I

~

J/4} = p¥)+

P~"

.

"

1

·

u

2

E

P-(

{)

m

U ~

(

0

)

no

YCJlOBHIO (3.2.7) H aepa

seacra

y (3.2. 9)

P

~

)

~k

9(1)t;U

(u(i),

O)~

~

k

a

b

-2

S

I

~

I

-2'

.

3aM

CTHM,

'IT

O

I

Z6(U,

)

-

Z

e:,.(U2)1~l

w6(e+

+I

IJ.

(

tld

t;l

1

U

l'

e) II

CD;I

(ul'

0) -

CD

:t;1

(tl

2

•

0) I +

<!>t;

1(u

t

,

0) I

WA

X

X (e + I

~

1

/2

c:

1 (e)

Ill'

8 + I

~

,1

/

2d

t:

1

(e) U

2

)

I; I

(1)

t;1

(Up

0) -

I ( 0) _ <n

1

/2

(1)

1

/2

( )

(£1'">-

1/ 2 ( 0)

,.,...,

-1

/ 2

-I

-

cD

-

6 U

t

•

I::::::,,,,,

6

Up

u

2

'*'

t>

U

l'

w6

(U

t

,

0

)+

<1

) 6 (Ul'

0)

<D

t;1

/2

(u

t

,

0» .

CJleJl

OBaTeJlbHO, .

,lI.

Jl

SI

U

l

'

~

E

pi)

n

o:

(e )

no

(3.2.6) H (3.2.7)

IZ6(Ul)

-

Z6(U\l)I~'Y1

/2(~)<!>

:f2(Ul'

ut)((l

+

+2

1/2

)

(1);1

(u

t

,

0) + 2

1/

'2

(Dt; '

12

(ul'

0)

<D6'I

/2

(u

2

'

0» I

~

1

/

2

~

(I +

+2

312

) e

l

/2

b-

1

1'1/2

(~).

n O.3TOMY

p~1)

= 0

(I

~

,-

').

ecna e B

bI6p

aHO TaK,

'IT

O b

2/16

(9+

+ 4 1

/2)

e> B (0). C

06

Hpa H ouesxa

,n,JlSl

PI' Pt.

p

~1)

11

p

~

{),

n o-

JlY4

aeM

yrsepacneaue TC

OpeMb

\. C JI)'l.Jafl . xorna

,n,JlSl

nexoropux i

p

I)

n

u

~

(e) =

0,

paCCMa TPIl Bae TCfl TaK )KC. •

Teopeaa

3.2.2

.

nYCTb

e (x), x E R" - on

ao

pozi

ao

e CJlY 4aH-

Hoe

rroae, o6J1a,ll.alOmee or p aHH

4eH

HOH cnexr pansaof nJlOTHO-

CTb IO

f

(A),

II V

(IJ.)

.L;

B (0).

Tor

na eCJlH PblnOJlHeHbl yCJlOBHil.

~_

oo

IV II V. TO

J.l.JlSI

JlI0

60

ro

p > 0

sup P {I

~

6

(e)I> p}-

;;.

O. (3.2.12)

AET

6.

->

00

...

'u

oKa3aTeJlbCTBO n OBTOpHeT npen

un

yuiee. D o H

epa

BeHCTBy

LJ

~6blLUeB

a

p

~1)

~

16M I

z~

(U

(I

»

f2

~

16

sup

f

(A

)

Q'6

1

(uul, 0) = 0

(I

~

1- 1).

).,fR

n

C XO

,1l,

HMOCTb P u

p~

i)

K

ayzno

04eBH,lI.Ha. •

ECJlH I

~

1-~12

i

l

;',

(8) - ;,.

00

JlJI51

4a

CTI1

I1JIH scex

HH,n,eK

COB

A~OO A

j = 1, ..., q, TO

(3.2.12)-

60Jlee CHJIbHoe CDOHCTBO oueaxn

eA.

4~M

pasaoaepnas COCTOflTeJlbH

OC

Tb

(3.2.2).

YCIlJl

HB YCJI0BHe IV H ycnonre p a3J1H4IlM OCTIl napaaerp os

(YCJlOBHe

K

OHTpa

'

~Ta)

V, MO)!{ HO

nOJJyqlf

Tb

oone

e oouiee,

qeM

TeO-

peua 3.2.1, yraepacneane 0 aeposrnocrsx

60JlbwHX

YKJloHeHIIH

ou eaxa 86..

O~03HallHM

fJ

h (8) = as (

8)

(8 -

e),

iD

6

(u

l

' u!) =

=

rp6

(e

+

dt;

l

(e

; u, 8 +c: (S)u

lI

) , u

l

'

~

EU

6

(e)

.

VI.

!.LIm

HeK>TOpOrO a E(0, I] cYllI.eCTBYIOT TaKHe KOHCTaHTbl

9

-i

·IIO

Itg

k = k(T

»O

H

a=a

(

T

)~O

,

sr

o

SUp SUp

cD

~

(

Up

U

2)

I U

l

-

U

~

I

-

a~

k(I + Qa),

ElF

- C c(Q).

~

u

1

,U

2

EU

~(

El)nu

,u

j

u

1

Q>O;

(3.2. 13)

VII. inf inf

cD

~

2(U,

0)

qr;l(/

u D;;;' 1,

(3.2.14)

Of

:!

U

E

U

~

(O)

rne

ep

YHKll,fUl qr

~

(x), x

~

0 MOHOTOHHO He

y6

bIBaeT

no

t1

H X,

qrto (0) = 0, x

G

+

a

qr; 2(x) +0 npa

t1

-+

00

, x

-+

00

.

C

ep

O

pM

yJ

lH p

yeM

O.ll.HO

yrsepxcnea ae 0 M

O.n.

YJle uenpepu saocr n

CJlY4a HHOr O

nOJl~

.

Teopeva

3.2.3

[54, 1391. n YCTb

II

(u) - ce

rra

paoensaoe H

H3Me

pllMoe

CJlY4 aHH oe none, onpenenenaoe na

3aMK

H

yToM

MHO-

q

»cecree F

~

R

,

II

)l.JI~

JII06blX

U,

~

,

U +

~

EF

M

ill

(u + u)- 11(u) r"

~

1(u) IUIII

(3.2.15)

npn nexoropux m

~

p ;» q 1I JIOKaJlb HO orpanme nnos

epY

HKll.IIH

1(U) 1 R

q

.+

R

~

.

Torna

)J.JI

~

JIl

00

blX Q, h H

C>

0

P{ SUp

111

(

U

'

)-11(U

")

I

>

c}

~

u'

,

u"E

Fnu

c

(Q

)

;

l

u

'

-

u

"

l

~

ll

~

k

o

(

SUp I (u)) Qqh

ll

- qC- m,

(3.2. 16)

uEF n uc(Q)

rne KOHCTaHTa k

o

3aB

ll

CHT OT p, m H q H He saBHCHT or Q, h, C 1I

MHO)l{eCTBa F. B

lIaC

THOCTIf , npa Bb

InOJlHe

HHIf nepevncnennu x

YCJlOBHH

P { sup In(u') - n(u") I>

C}

~

flo

( sup 1

(u»

Qllc-

m

,

u' ,u" E

Fn

uc(Q) UEF nuC(Q)

(3.2. 17)

k

o

He

3a

B

lf

CHT

or

Q, C If F.

Teopeaa 3.2.3 u

cn

on

ss

y

erc

s B

,l],O

K8S

8Te

JlbCTBe cn en y

io

iu

er o

yr s ep )I{.n.eHH

51

.

Teopexa 3.2.4. Flyers s

un

o

aae

nu YCJlOBH5I VI, VII H

.n.Jl5I

m raxoro, -rro a

m>

q

M IW ... (8

1

,

aa) l"

~

k

4

lJ'

:1

/· (8

11

8

2

), 8

1

,

8

2

Ee'. (3.2.18)

Tor

zi

a

.ll.Jl

5I nexoroporo k

5>

0

If

H>

0

sup P

{!a:(8) (8"- 8) I

~

H}

~

k

o

H(a+a)mqr

-;-

2m(H) +

ElET u

00

+ (k

o

/

H) ) x

(G+Q

lmqr;

2m

(x) dx.

(3.2.19)

H

...

060

3Ha

llHM

V IP) = (v

c

(H (p + 2))"'. v (H (p +

1»)

n

U

~

(8),

p = 0,

1,

LLJlH moooro 8 ET I

P {Id" (8) (e

-8

) I> H}

:;:;;;

i P { sup

z~

(8 + dt; l (8) U,

8

)~

1/2}.

D= O uEU(P) .

0 0 YCJlOBHIO (3.2. 14)

P { sup

z

~

(8 + c: (8) U, 8)

~

1/2}

~

P { sup I

w~

(8 +

. uEU(P)

u

EU

~

(

e

)nuC(

H(p

+

2

)

+ c: (8) u, 8)

~

2

-I

qr

~

(H (p + I»}.

O pHMeHHM K CJI

yq

aHHoMy nOJlIO

11

(u) = .\' e (x) g (x, 8 +

~

+

d;

l (8) u) dx, u E

tr

: (8) reopea y 3.2.3.

3a

Me

Tlf

M, xro OJI

ar

o

.n.

a-

ps YCJlOBH5IM (3,2. 18)

11

(3.2.13)

aepase

ncrso (3.2.15) I!bInOJlH

51

eT-

C

51

.n.JIH

1(u):=::

(1

+1

u lat' H p=

a.m

. n 03ToMy

.n.

Jl5I up u

E

U

~

(8)n

C

nu (H (p +

2»

2

m

M I

w~

(8 +

dA

1

(8) u

1

' e+

c:

(8) u

2

) 1

~

0ln

:;:;;;

kmk

4

(l

+ (H (p + 2)t

)m

IU

l

- U

2

/

H,

CJle.n.O

B

aTe

JlbHO,

no

(3.2.17)

P{ sup

z

~

(8 +

dA

l

(8) U, 8)

~

1/2}

~

2

m

k

o

k

m

k (I +

4

IIEU(P)

+ (H (p +2» Q)m(H (p + 2»amqrt;

2m

(H

(p + I

».

T aK

H~

06

pa30M,

00

sup P

{J

d" (8)

(

~

- 8) I

~

H}

:;:;;;

k

5

~

(H

(p

+ I »(G+a)m

qr;

2m

X

eET p= O

X (H (p +

1»

:;:;;;

k

o

H(G+almqrt;

2m

(H) + k

H-

1

X

5

x Ix(

Q+

a)m

1Jl

t;

21n

(x) dx. •

9·

131

180

Cneocmeue 3.2.1.

Ily

cr

s qr

.dx

) =

k

x

~,

0 <

~

«,

2~

-

ex.-

_ a> 0. T Or,A3 np

as

as xaer» (3.2

.1

9) ec

rs

BeJII14HHa nc

pnnxa

H

-

(

2

~

-

a

-

a

)

m

B

RIO

. 'iaCTHOCTIl, eCJIH

cc.

= i"' = , a = , TO

m

su p ?

{!«. (8)

(e

t.

.-

8) I> H }

~

k"'H-

.

(3.2.20)

SET

MbI

IlOJI

Y4HM

60JI

ee C

HJIb

HbIH peay

ns

ra

r , 4eM reopea a 3.2.1,

nOJIO)l

mB

B (3.2.20) H = \ 6. 1

1

/

2

p, p > 0, m = 45: Jl.JIH

rno

ooro

p > O

2

sup?

{\~t.

(8) I> p}

.,.,

0 (Ifl l-

m

/

)

n

pa

6. -+

00

. (3.2.21)

SET

Ce

MeHCTBO O. B. 8t. = 8t. (to (x) , x EM Ha3bIBaeTCH CHJI

bK

O

COC

TOHTeJIbHbl M

ceMeA

cT

BOM ouenox napaae

rp

a 8Ee

(8

6

-

CI1J1

b'

HO COCTOHTeJIbHa

51

ouexxa 8 Ee), eCJlH 8

t;

L\

--->~

8 o. K.

Y Ka

)l{

eM npoc

ru

e

,AO

C

TaT

0 4H

ble

ycnosa s Cli JIbHOH COC

TOHreJl

b-

HOCTH

A

8 ,AMI C

JIy4a

5l, KOf,Aa e

,

- KO

Mll

aKT Ii MaT

pHua

dt;

{OJ

0

1

np

on

opn HOH3JIbHa MaTpHue 1£'\ 1

/

2

s;

VIII. m- JIHHei'IHO

ynop

H,A0

4e

HHOe MH

O)j{e

CTBO C or nouie-

HHeM

nOp

HJI,Ka «

'--

'I>

Ii

cyuiecrnyer TaK

afl

nOCJle,AOBa

reJIbH

OCTIo

1

MHO

)l{

eCTB £'\t

Em

, xro: 1)

6./c6.

,+

I

;

2)

lim!

At+!\ jLltl- = 1;

1--->lX>

3)

~

Is,

1-

1

/

2

<

00.

1

YCJIOBlI HM 2) H 3) y,AOBJIeTBOpHeT, uanpaxep, n OCJle,AOBaT

em.-

HOCTb MHO)l{eCTB 6.

t

C

06

0e

MBMH l

£'\

t I = r": &> O.

----

-I

IX. I) lim

sup

(\6.1

q>

A

(

e

l

,

8

3

)

-

<p

(

8

1'

8

2

)

~

0

,

(3.2.22)

0--->00 a" StEa e

r ne

¢Y

HKUHH

<p(8

1'8

2

);>0

nerr

pepueaa na

e'

x8'

H

qJ

(81' 8

2

)= Q

rorna H TOJlb KO T

or

na,

KOf,Aa

8I = 8

3

;

2) Iim su p

16.1

-

1

Jg! (x, 8) dx <

00.

0--->00 SE

e'

A

X.

AJIH

JlI060r

o p > °cyme

cr

a

yer

'I HCJlO k

e

= k6 (p) :> Q

-raxoe,

qTO

,AJI5I

JII

o50ro 8E8

inf \6.!-l<pt. (8 + u, 8) >

!le

. (

3.2

.23)

uEl8

C

-{l

)

'\.

~

p

)

YCJIOBHe

KOHTpa

c

T8

X -

<>T

O

ynpOlue

HHhlA

B

3pH

aHT V.

Teop

eva

3.2.5 .'

nYCTl>

Bhln

OJ

lHeHbI

yCJIOBHfI

VIII

-X,

a OJl.-

uoponuoe none e (x), x E R"

o6J1B

,u.aeT

cJIe,u.ylOUJ.HMH

cnOOCTBaMI'l:

132

I)

'\'

{6.

)----~B(O)

n.

H.; 2) iBr. (x

)dx<oo

, T or.n.a 8 6

" 4

00

.J

~"

OlJl

bHO COCTOHTeJtbHaH ouesxa napaxerpa aEe,

..

nOJl

O)KRM no ('") = 1

6.1

-

1

Se (

x)

g (x, r) dx,

't

Eee, n OKa -

)K€M,

4TO

"

sup

In

(r

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24

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t Ea C 0 '\

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eas

xo, B COOTBeT

CnmR

C (3.2.3)

- I "-

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8)

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(3,2.25)

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u

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n YCTb L Eee ¢JJKCHp oBaH

o.

Torna

M

11~

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re

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3.2.5

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li

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c R

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d~l

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O)

...

li

a C)i i

COIi

i 0) a: 0 a:

(O)

l' . °e

,•.dqg

(O

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yJI

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O

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1

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e.

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li

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+

e

,

(8)-1

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a

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6

134

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+

el

+

e

i'

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e (x)

s~i)

(x, u) dx,

li

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~

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Lll

l

/

2d~1

(8) u).

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JI

OrH4HO

06,o3Ha 4HM

a

~

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k

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nJl>l

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0+60ro

R > 0

cYULecT

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1

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U

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l

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2

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8

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sup sup ILlI

1a l

-

J

(

d~

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CD

a

li

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z

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ll

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l

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2

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~

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S)n

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,

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XII.

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~

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(3.3. 3)

SET

XIII. n Jl>l nexoroporo m

;;'

2 H JlI

060Ji

JlOKaJIbHO HHTe

rp

H'

pyexon C KB8,ll,pa TOM

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n

M Ife (x) s (x) dx [2m

~

k

1

L\'

SZ(

X) dx

t;

(3.3.4)

... t.

m

M I

'\'

(Ll) - B (0) 1

~

k

s

1

6

/-m

12

• (3.3.5)

'uOC

TaT

O'lHble

yCJIOBH>I

BbInO

JIHeHHH (3. 3.4 ) H

(3.3.5

)

np

asenenbl

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fl.

1.8.

HH

>Ke MbI C

'l

HT8eM xopp

exr

usna

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aHll

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peM

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M

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,

't

q

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lp

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L~

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orO

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Hoe

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e6

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e K cxe-

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Ilpe

)!.

Il

OJIO>KHM,

sro

't'li

-

~

00,

T

t:.tr

_

~

0, t = ILl

I-

l

/2 .

6.---t00 6.

-"

00

Teopeaa 3.3,1.

nYCTb

Bbin

OJIJ-eHbI YCJlOBH

>I

XI - XIII, a

135

oue nx a 6

06JUIJl.aer

CBOHCTBOM (3.2.21).

Tor-na

6

r-1

sup?

{j

d

....

(8)(

.e~

- 8) -

~

r:»: (8) I>

TAr-I}

=

96

v-I

= 0

(I

s

l-

m

/2) + 0

('t6"2m

/,) npn

!J.

-+

00.

(3.3.6)

h'liA

(8)

-

O,!J,H

Op

0Jl.Hble sexropasre nOJlHHOMbl

CTeneHH

v OT

CJIy·

'1aHHbIX nepeaenasrx bt. (a, 0), Ia I= 1

....

,v

C

paBH

OMepHO

n o

A H 8ET

or

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K

o3<f$HUHeHTaMH

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alK

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O

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MM.

JIeMMa

3.3.1. Ycnosne (3.3.2)

,!J,JlH

BeKTOpa

a, 0

~

Ia I

~

! - 1 ssrrexaer

11

3 coor nourerraa (3.3.1),

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1lJ1fl

a +

ii

' i = 1, .. , • q.

..

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COC

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1

Hbl X

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ll

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l'

u

2

E

U

~

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(R). •

0

60

3H

a'IH

M At. (a. 8) =

(a~

(a, 0))/= 1... .

4'

B

li

(a, 8) =

(b~

(a ,

(J

)

j

~

I

....,Q.

B~

)

(6) =

(b~

(0, 0))

I,H

,...,Q H

3anHWeM

pasn

oxeaae

M

aKJl

OpeHa

no

llepe

M

eHHoH

U

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-

1

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1

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2

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6

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a

-

2tB

~

)

(8)u + ~ (

lIal)

(At. (a.

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A

(a, 8)) u +

~ 6

(u);

'

~

l

a

l

:S;;

r

-

1

(3,3.7)

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(~

16

(U))I_I .....Q·

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(u) =

~

((l

lal

) t

B

\

(G

~

a

)

(x, 8 +

t-

1d

'6

1

(8)

un

-

v

IClI-

,-1

Ii

(ttl

a'

- GI

(X

. 8)) dx)

u,

IUi I

~

Iu

I.

(3.3.8)

AHa

JlOrHtJH

oe

p33J10 ;KeHHe

,!J,JHl

qJYHKUflH

taLI

/A

(8 + t-

1

d

L;

1(8) u)

JlMe e T BH)].

(J.L

ijt>

(0+

rc:

(0) u) =

2:5~

(8) -

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(0, 0) +

+

~

(

1/

a

l

)(

a

~

'

(ex

, 0) -

2tb

~

(a, 0)) u

a

+

~

/j

t>

(ul;

(3.3.9)

J

~

lal<!;;,-

2

t (u) "'"

~

((I

lal

) S

(Gil)

(x, 8 +

t-ld~1

(8) u;,) -

il A

~_,

-

2

II

- G;,'

lX,

e)) ua, Iuj/I

~

Iu

I.

(3.3.10'

ECJIH

!=

2, TO CY).1MbI B (3.3.7) H

(3.

3.!:i) OTCyTCTByJOT

Ii

OCTa -

1d'6J

T

04HblH

tJJIeH B (3.3.10) 1]//6 (ill) = t'l.L/

iA

(8 +

t-

(8) u) -

- (JL /

j

/}, (8).

J1eMMa

3.3.2.

nYCTb

Iu I

~

6 < 1 H ocy m eCTBJl

51e

TC

51

coosrrae

{,\,

(

~

)

~

B (0) + J}. ECJIH Bhl nOJ1He HO

xi,

TO

sup/

I;,t>(u)

l~ks6'

,

supl~

u

.(u)l

<k46

'-

),

i,

j=

I

•..

. ,q.

B

ET

BET J

..

Floxaacen,

no

npn

qJHKC

Ii

p OBaH HOM 0: , Ia I= !

I

C,1

(a. u) I=

t'J,

II(G(a)(x. H+ t- )d;:' (8) u) -

e(

a.) (x. 8)) dx I

~

klJ ··

3 I1Me THM, qTO

~A

(ex, u) =

~~

)

(a, u) +

~

)

(a, u).

r,lle

C~

'

\.

a , u) =

21

IJ.j(Jal

/2H

(eft.

(8))- 1

~

Ie(x) (S(

CX

I (x, 0) - S(

CL

) (

x,

u

))+

A

+

S(U

) (x,

U)

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X,

0))] dx;

~

)

(a, u) = ~ k (a,

~)

IIJ.l

lI

Ct

I

/2

H

(d

~

(e))

-1

r(S(a

-il

l (x,

1~1~1,Il

<

o

A

u) S(/l) (z, u) - Sl

a-

1l

1 (X. 0)

Sl~

)

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~

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- uen

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KOH

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TbI.

Y1

CX

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H3

ycnoBHH

HaCTOH-

mefJ

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H pes

yxsrara

neMMbl

3.3.1, noaysaea

I

lI

Ct

I

~~

)

(a , u) I~ 21

~

I-l

I

12

(

~(8)

)

-

1

[(D~

(0, u) "( 1/ 2

(!J.)

+

d"....

(a, 8 +I

~

1

'

~dL;

)

(e) u) lli

l-l

/2(D

~

(0. u)J

~

2

(k~

/

2

(a, I) (B (0) + 1)1

/2

+ k (a. 1)k

e

)

6, (3.3.11)

q

npHQeM

MOIKHO B351Tb k

e

= 2

(~

k'l.

(et, 1)

r/2.

AHaJlOrl1'-IHo

npa

1-= 1

qJHKCHpO

BaHHblX

a H

~

I

~

I

(l

Ct

I/ 2

1

- 1

(eft.

(8))-

1I

~

.

(

S

l

a.-

~)

(x, u) s

(Jl

) (x. u) -

6

_ s(

Ct-

Il) (x. 0) sllli

(x,

0)) dx I

~

(I

~

/

li

a

- tlH I/'l (dr -

Il

)

(8))-

J<D

;J

-=-

Il,

/},

X

X (0,

u»)

(\

~

1(

1

~I-

ll

/

2

(d

~

(8) - 'dt.

(~

,

8 + I

IJ.1

1

/2dL;

1(8) u) +

+

(i

~

jCltlH

1

12

(~(e))-I(j)

~

l

( 0, u)) (i

~

I(l

a-tll-ll

/'l X

X

(d

~-

p(e»-

'at.(a-p,t:l»

~(k,k(~,

I) + kgk (a - I\ 1))6.

(3. 3.12)

l3IJ

137

Леоненко Н.Н., Иванов А.В. Статистический анализ случайных полей

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