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

q

0 6 03Ha4HM

~r-

I

(u , t)

pa3JI

O

>l<

elH He (3.3.7)

6e3

OCTa

T0

4Ho

ro

r-n e MO)l{H O norroso

rn,

k

7

= 2

(

~

k

2

(a - B+ cit I)r .n, k =

s

4JIeHa ,

~

oo

(u, t) -

pHA

, nOJlY4 eHH

bl

H

11

3

Br

-I

(u, t) ep

opM

aJlbHblM

{

~

I

np0,ll.OJI}KeH

ll

eM

CyM

MHp

oB

aHllH

no 6ieCI<OHe4HOCTH:

= 2

(

~

1,2

(B

+ e., I)

r

/

~ .

,

=1

Y TB

ep

)l{,ll.eHl1 e Jle M

Mbl

surexaer as (3.3.8), (3.3. 10), (3.3.11),

(3.3.12)

11

04e BI1)J,Ho

ro

HepaBeH CTBa IU Ii

al

,> Iu

a

I

.•

0

603H

a411M

L

~)

= (Liit)/.i=I...: » reCCHaH epYHKUl1l1 L/;.

J1eMMa 3.3

.3.

TIY CTb 8 ET, OCYW.eCTBJlSlIOTCSl C0

6bITHH

{y

(~)

~

~B(O

)

+

I}

11

{tlb

/;(a, 0)1<

<5

},

la!=2

, ...

,r

,ll.Jl

Sl

,ll.OCTaTOq.

1-10

MaJlOr O

<5

> 0 Ii nuncnne

nu

YCJlOBHH XI

11

XII. T

or-n

a

HaH'

nercs raxoe

411

CJIO X

o

= 'X

o

(T) > 0, 'ITO

inl "'m in

(t

2L

~

)

(A +

t-

l

cf6

J(8) u) ,> "'•.

U

E

~

ro)(\

v c(X

o

'

..

Bocnonsayeaca CO

OTHow

eHHeM

11

20, c. 104J

/"'mln

(t

2L

~

)

(8 +

r'a:

(8) u)) - A

ml

n

(236 (8))

I

~

q m

ax

It

2

L.

./;

(8 + r '«: (8) u) -

23i

(8) I. (3.3.13)

.

I

J

i

.J

nO

CKOJl

bK

Y B npasof 4aCTH pa

3J1

0)KeHHH (3.3.9) no YCJlOBHIO JleM.

Mbl

t I

b~

(0, 0) I< 6, a CYMMa ,ll.aJlbHeHW

ll

x cnaraesiu x

no

YCJlO-

13

HHM

ll

aCToHJUeH JleMMbJ

Ii

pe3YJlbTaTY JleMMbI 3.3.2 He npeBOCX

O,ll.

I1T

k

g

<5

, kg = kg (T), TO npasas qaCTb (3.3.13) He 60Jl

bW

e BeJl1l4

11

HbI

q (2 +

kg)

b. C Jle

.n.O

BaTeJlbHO, JleMMa cnpaee

zi

nasa npn X

o

~

<5

<

J.*

(2 +kg

)-

Iq-

l. R!

ECJlH ocyutecranserca C0 6bITHe

{I

'

~6

(0) I

~

x

o}

11

Bb

In

OJlHeHbI

y CJlOBlH1

Jl

e

M

~1

bl

3.3.3, TO oroopaacenae u --+ t

Z

Lt,(8 +

t-Id-;:l

(8) u) Bbrn YKJIO B w ape tJ (x

o

),

11

CI1CTeMa

ypaBHeHI1H

t2VL

", (8) = 0 (3.3.14)

11Me eT e.ll.Hf! CTBeHHOe

peurenae, cosnana

iotnee

C e6 '

TaK

I1M 06 -

pa3 0M,

0=

t

2v

Lt,

(06

) = -

2t8

6

(0, 8) +

t

ZL

~)

(8 + r

1d-;:1

(8) u*)

;';6

(8),

Iu* I

~

I

~

6

(8) I, (3.3. 15)

11

n o

Jle

~

nv

le

3.3.3

I

~6

(0)

1=

21I

(t

2L

~

)

(8 +

r's;

:(8) U,,))-I

Ba

(0, 8) I

~

zo;:

18

",

(0,

l:J)

I. (3.3.16)

138

00

B

oo

(u, t) =

~

(I

fal)

(A", (a ,

lB

) - 2

tB

6

(a,

l:J))

u

a

.

(3. 3. 17)

l

al=

8

n YCTb u (t) = th ; + ,.. + t'

-lh

r_

1

+ .

00

-

rp

opaans uoe

pa3

J10,Ke-

Hl1e

B

psn

pe

We

HII

5I

ypaBHeHI1H

09

""

(u, t) = O. Ilpa n O,ll.CTaH OBKe

er o Ha4 aJI

bHOrO

OTpe3K3

(r -

I)

I

tr-

II

u (t) = t

II

+

...

+ lr

-I

(3.3

,1

8)

i

B

S!

""

(

U,

't)

4Jleff

bl, conepx auure t

,

i = 1, ... I' - 1, oopauiaior-

CH B Hy JIb. n

03T

OMY MO

)l{H

O sanaca

rs

09

00

(u

(r-I

) (t ), t) =

.s

t'h

"r_l

. (3.3.19)

i

~

1

JleMMa

3.3.4. Bee hi' i = 1, ... , ,

-1;

her_ I' i

,>r

11

3 npen-

CTaSJIeHI1H (3.3. 18) H (3.3.19) -

aexr

opu , KO

Op.ll.I1H

aT

bI KOTOpbIX

5lBJIHIOTCH O,ll.HOP0.ll.HbIMI1

rJ

O

Jl

Ii

H

O

~

l

m

1l

1

CTeneHI1 i OTHQCllTe.% HO

b

6

(a, 0), Ia 1= 1,

00

. , i.

..

'u

oKa3

areJJbcTB

o JleMMbJ

np

OBell.eM HH.ll.YKu

lleii

no r. ECJl I1

r = 2, TO hI = A.\ (8)

B/;

(0, 8), a yTBep)l{,ll.eHl1e OTHOCI1TeJlbHO

h i,l'

i,>

2 nposepaercs Hen OCpe,ll.CTB('HHO. n YCTb yrse

pac

nenne

sepao .ll.JIH aexcr oporo , ,> 2 Torn a h, HaH,ll.eTCH H3 YCJlOBll5J

oopaineuu» B UYJlb K03epep

llull

eHTa npJl t' B

B

oo

(u

(r - I) (t) + t'h"t) = £

00

(u

(

r-I

) (t), t) - 2t'36 (8) h, +

..

(3.3 .20)

B (3.3.20)

oropouieaaue

cnaraeuue co

zte

p

aca

r creneuu t", m > r.

VI3

(3.3.19) HaX

D.ll.I1M

h

r

=2-

1

i\ 6 (8) n:,r- l.

11

yrsep

ac

neuue OTHO-

CI1TeJIb HO

h, cnen

yer

11

3

np

e,ll.nOJIO)KeHI1Sl

IlH

,ll.YKUI1 H. B eJIl1lJHHa

b

6

(a, 0) C Ia I

~

r + 1 npa c

yr

cr s yer B B

""

(u

(r ) (t;, t)

JlIl

Wb B c.na

rae

xa

x

san

a ts;

(~,

0)

(u

(r )

(t ))

t3

/

~!

,

I

~I

= Ia 1- 1, xoropsre co-

A epm aT t B CTereH SlX He MeHbW I1X, lJeM Ia I. n 03TOMY b

6

(a, 0)

lie

sxonnr B hi,,, eCJIIl Ia I> i

3a

MefH M, 'ITO

npl1

nO}!.CTaHOBKe B

B

oo

(u

(

r-I)

(t). t)

seJIWIHH

r'»

: (a, 0) BMe CTO b

6

(a, 0) p H,ll.

B

oo

(u

(

r-

I) (I), t) i.e 3aBHCHT or t. El10 Bb

lT

eKaeT

11

3 (3.3.17)

11

3

Y CTaH OBJleHHOrO

ssnue CBoi1cTBa U

(r-I

l(t) . n

03T

OMY h,.

r-I

- O.n.HO-

ponaue nO

JJ

HHO

>1

bJ crenenn i no ney

eM

eHHblM

b

6

(a, 0), I

ce

] =

=

1, ..., i. •

Osesnnno,

cjyHKUHH

Br-1 (u, t) n

orys

aer

ca

H3 B

oo

(u, t),

xor-

zia a", (a. 0) =

tA

(a, 0) = 0, Ia I>r. Coxpauna

0603

HaQeH

Kft

139

u

(r-I)

(t), hi' h

,.r-l

B

np

llMeH

eHHIi

K

~r-

I

(U,

t).

T

orna

BMeCTO

(3.3.19)

nOJIyqaeM

coor ao

ur

ea ne

~r-

I

(ul

r

-

I

)

(t), i ) =

~

t

'hl

.r-h

(3.3.21)

;

~

r

B KOTOpOM CYMM8

conep

xcn

KOHeqHOe qJ.lCJlO

enaraevux

.

..

D.oKa>l<eM

reopeay

3.3.1. nOK8>l<eM, 'ITO ecnn

Ut:,

(8) Ev (x

o)'

V(d)

~

B (0) + J. Ib

...

(a, 0) I

~

k"t~r,

lal = I, ...

,',

(3.

3.22)

TO

I

~t:,

(8) - UC;:- I )

(I)

I

~

k

1

ok

r

t'T

t:,

.

(3.3.23)

Orciona

asr

rexaer yTsep>l< )l.eHH e

reopexu

.

D.eikTBHTeJIbHO,

no.n

o-

>l<HM

B (3.3.22) k = 1

?10

1

/,

Torna

11

3 (3.3.23) n0J1Y4aeM

supp

{!

;t:, (0) -

U

~-

I I

(t)

I>

Ttl }

~

sup P

{I

~t:,

(8)

I>

Xo}

+

BE

T

81!'T

+P {y

(~)

- B (0)

~

II

+

~

sur

P {Ib

6

(a, 0) I>

k"tt}

=

e

"

/al _ l

=- n

t

+ n

2

+ n

a

. (3.3.24)

BeJJllqHHa

nt=O(ldl-m

/ 2) [} CHJJy (3.2.21);

n2

=O(

I~I

-

m/

2)1l

0

(3.3.5).

I1cno.nb3yH

II

cpaBeHcTBO

4e6bIWeBa,

(3.3.4)

11

(3.3.1),

nonyuaev

supP {I

bt:,

(a, 0) I>

k't

~

r}

= 0

('t6"

2m/r), laI=

I,

..., r .

BET

003T

OMY

11:3

= 0

(T;

2m/r).

Hepas

escrs

o (3.3.22) Ii yeJJO

BllH

re

opeau

n0

3BOJJ5IIOT BOC-

nOJJb

3

0BaTb

C~

JJeMMOH 3.3 .3

11

3aKJlIOIJHTb,

'ITO

SA

- peurenne CH-

cr ea u

ypaBH

eHHH

(3.

3.14),

np

H4 e M

BbIn

OJlHeHO (3.3.16) . n OJIO-

)l(HM y6 =

kt't

~r

.

n o npcnnon

cos

ea

lllO

Y

...

-r

0, KpOMe roro, li 3

6--'>00

(3.3.16) H .(3.3.22) czcn

yer

I

~

6.

(0 ) I

:::;;;

k

l1

!h . (3.3.25)

nO

CKOJIbKY B (3.3.18)

11

(3.3.21) enaraeasie t'h,

11

l

h,

.

r-'

. , . I

0AH

OP0)lHhl

no

tb; (a, 0). TO

til

hi I

~

k

l 2

YA, t' Ih

l

•

r

-

I

1< li

12

y

t.

.

Orciona

!tp

-l

' (t) I

~

k

H

Y

t>.

'

12r-1

(lfr-I)

(t), t)

~

klrJY~

'

(3.3.26)

VI3 JJeMMbl 3.3..2, (3.3.25) Ii

l-

ro

li3

Hep2

8eHCTR (3.3.26)

CJJeJly

eT

I' a

(~a

(0))I

~

kl6Y~,

I

~A

(Ul

,-

I)

(I»)

I

~

k

17Y~

'

(3.3.27)

VI3 2-r o

Hep

aBe HC

'TBa

(3.3.26) Ii 2'-ro nepaeeucrsa (3.3.27) nO

JJ

y-

'laeM

f! I

'YLa

(0 +

t-Id~'

(8) u

(r-I

) (t)) I

~

1£'-1

(U(

r-

I (t), t) 1+

+I

~

,

\

(U

(,-I

) (t)) I

~

(k

lb

+ knl

Y6

(3.3,28)

n OJIh3Y5lCb (3.3.7), (3.3.27)

l-J

H

epa

BeHCTB OM Iur -

u

n~

~

k (a) Iu

1

-

UsI(i ul l V Iu

21)I

Ct

I-l,

u

1

' u

2

Etr. sarnuueu oueuxy

t! IVL 6 (8,) -

VLf::,

(8 + ,

-I

dA' (8) u(

r-

I) (t)) I> I

~'-

I

(

~

t:,

(0),

t) - Br

-I

(u

(r-

I: (r), t) 1-

(

I

~

....

([;

....

(8))

I

+

l~

.\

(U

(,-

I)(t))

I»

>I

~t::.

(8) - u (

r-

I) (t) I(2",* -

2qt

max

I

b

~

(0, 0)

1-

i,1

r

-

~

-

~

flU) (I

~

.1

(8)II V Iu (r - IJ (t)

I')

- (k

u

+ k

17

)

!It::.

~

;c: l

~

'}...,.I;t::.

(8)-U('-I

) (t)

1-

(k}&

+ k

t 7

) s;

(3.3.29)

nO

CKOJIbKY V

LtJ,

(8:) =

O.

aa (3.3.28) H (3.3.29) nOJl

Y4a

eM

~

.3

.2

3)

.•

TOqHbIH

BI1

J1.

nOJJHHOMOB hi

t:,

M

O>l<H

O HaH

TH.

nO)l.CTa

B.Tl

HH

U(,-l

l (t) B (3.3.7) H

npHpaBHH

Ba51

K HyJlIO

1<

03qxpHUHeHTbI npu

~TeneHHX

t

l

(ex. JI6MMY 3.3.4) . VICn

OJJ

b3

Y51

3TOT

npH

eM, H,lXO,lUIM

q

hll~

(6) =-

(~

A~l

(0)

b~

(0, 0)

),

=I..

..

.

q;

1

,-1

Q

~6

(8) =-(

1}

A~'

(8)

A~

I.

(b

~"

(0. 0)

b~

(0, 0) -

1

....

.

/-1

-

ClA~I.

(8)

a~I.I.

(0, 0)

b~

(0. 0)

b~

(0, 0)

)I

~\'

..

,Q'

HeTpy)l.H

O YOO,l;HTbCH B TOM, Q10 B 8blpa)j{eHHtl

)l.J151

h

2t

, (8)

a~~

(0, 0) = 2

([~

I)(II

)

(8) +

n~II)

(j

)

(9)

+

n~k

)(1

1

(8)),

fI~f)

(

k

)

(0) --

= I

!1

jl/2d

u

1

(8)

'ift.l

(8)

dkl

(8)l

gil

'

:x,

8)

gh

(x,

0)

dx,

140

141

Cneo

cm

eu

e 3.3. 1. ECJIH B r eope

ae

3.3

.1 yCJIOBHe XI BbIOOJ]'

HCHO

.ll)

IH

r = 2, TO M51

JIlo60

ro e> 0

n

",

=

supP

{Idt>

(8) (8", - 8) -

hit>

(8) I> e} = 0 (I

~

j- m

/2)

eET

(3 .3 .3

0)

npa

~

-+

00

.

,U

Jl51

JlOKa3

aTe

JlhCTBa

¥Tsep

>K,Il.

eHli51

,Il.OCl'a

T0 4HO nOJlOm HTb B

(3.3.6) r = 2, "

t>

= 81

~

,1/2,

e>

O.

Cneiicmeue 3.3. 2.

Fly

ers

BbIOOJlHeHbl YCJlOBH5I XI npn r = 2,

XII,

(3.3

.4)

,Il.

Jl

51

m=l,

(3.2.1

2) H

)'

(

~)

~

~B

(

O

)

.

Tor na

,Il.JH

l

"'

<Xl

JlI060ro

e>

0

nt>

-+

0,

~

-+

00.

(3

.3

.3 1)

3TO

T rpaxr B

hITeKaeT

11

3

(3.3

.2

4) II

CJle,ll.C

TBlH

I 3

.3.

1.

Ecrnr e (x), x E R" - rayccoscxoe onaoponaoe none, TO

r

ecpevy

3

.3

. 1 MO}KHO y TOlIHH Tb ,

Il

peosne scero

,Il.Jl

51

ra

yCCOBC

KO-

ro

nOJl

51

e (x) H epyHKUHfJ g (x, 8), Y

,Il.OB

JleTBOp 5l101ueH YCJlOB

H51M

IV, V reopeau

3.

2 .1 ,

COO

THoweH

He (

3.2

.2

1) cnpasen.mao npu

JIl0

60M

m>

2. Bonee ro ro, B YCJlOBH

51

X VI H

VII

re op

enu

3.

2

.4

B~

l

ec

T

O

c

re

nennon o

ueax

n (3 .2 .2 1) MO}KHO nOlIYlI HTb sxcnoaen-

1

UHaJlb

HylO

OUeH

K¥

~KO

PO

C

TH

y6

b

I&

1H1U1

no

Is 1 /2 BeJlH

lIll

HbI

su p

P

{I

~

(El)

I> p}.

fl

OJlO

lKHM

"t>

=

logr

~

Ifj, I.

I<1crrO

Jlb3Y51 Hep

a·

eET

BCH

CTBO

00

Je-

x

'

12

dx

~

e-J

e-e'

lZ,

(3 .3 .32)

e

YCJlOBIl

51

(3

.3

.1) H CIIHTa5I, 'ITO none e (x) oonana

er

orpaanseu-

HOJi cnexr

paas

aof

nJlOTHOCTblO 1(I.), AE R

n,

ne

rx

o nOJI

YlJlfTb

ouenxy

sup P {Ib

t>

(a , 0)

I;;;'

k l o

gl

/21

~

I}

~

(2

/n)

I/

2/

J/

2

k

o

k-

l

lo

g-

1/

2

1

~

I X

OET

X ex p

{-

k

Z

log I

~

1

/2

1

o

k~

}

,

rJI.l

to =

sup

f (

A.

), k

o

= m ax k (a , 0), k (a, 0) - KOHCTaHTbI

11

3

l.ERn ICtI= I.....'

(3 .

3.1).

B03bM

eM k = IJ

/2

kom

l/

2

.

Torna

m ax supP {:b

t>

(a, 0) I

;;;'

k

log

l/2 1

~!}

= 0

(I

~

,-

m/

2).

la

l

~

I

....., aET

(3. 3 .33)

H epaBeHCTBa

(3.3.24) H

(3.

3.

3

3)

npHBO

,ll.51

T K CJle.AYIOw.eMY YTBep-

1K.1eH

1110.

Teopeaa

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"I' E(0,1) - nexoro

poe

4IiCJlO.

OYC

Tb

r»

,.

..

,

pS

) Ev

C

(

Po

) - 3aMKHYTble MH

om

eCTB3, n

li

aMeT-

psr KOTOp

bIX

He

np

eBOCXOn 5lT BeJlH4HHbi 6, OTBe4a lOIlJ,eli no YCJlO-

BHIO

XV 'lHCJ13M R = Po H e =

~

a

(p) "I'/2,

~

E(0,I) - uexoropoe

4HCJlO. Buoepea TO

T.J

KH u (1) Ep I) n

U

~

(0), i = 1, ..., s . Torna

p (3) = P{ sup

16.

1-

1

1b

<l.

(8, u) I;;. y'a (p)} ~

u

Eu

~(O

)

nu

c(po

)

~ ~

P { sup

11.'1

1-

1

I

bt>

(0, U

t

) -

b

<l.

(0, U

2

) I+

i= 1 u..

u

.

E

U~(

e

)

(\

F

(

1)

+ 1

6.

,- 11b

t>

(0. ur

i»

) I;;' y'a (p

n·

3aM

eTHM,

4TO

I

b~

(0, U

1)

-

bt>

(0, u

2

)

I

~

I

Rt>

(U

1

) - R

t>

(u

2

)

I+

- - (I )

+ M Ie, (U

1

) - u,(U

2

) I

~

2<D

t>

(U

l

,

u

2

)·

D03T

OMY no YCJlOBHlO XV

S

p(3

)

~ ~

P {I

Ii

I- I I

bt>

(0, uri)) I;;.

(l

-

~)

y'a (p)}. (3.5.7)

i-=l

OueHHM

KmK

)l.oe cnaraexoe nocnenaen CYMMbI OT

lleJlb

HO. 0 603Ha-

'lHM

Q(x, u) = Q(x, °+ I

6.

I

/2d :;-1 (0) u). Do JleMMaM 1.6.2, 1.8.2

H yCJIOBHlO XVI

2

I

Ii

1-

2

Db

A

(B, u(i») = I

6.

1-

~

JCOy

(Q

(x, U(I)) , QW,u

(l»)

) dxdy

~

<l.

A

~

J0

16.1-

2

jj (k IQ2+

"'

(x, U(i)) I/I2+"') (M

Q2

+

'"

(y, U(I))) I/(2

H

)

a6/(

2+1'll

X

Mi

x (I x - y I) dxdy ~ 20 ·

2

il

/

(

2+

~

)

I

6.

1-

2

~ ~

a

M 2

+

6

)

(I

x-y

I)

(

j..t

~

+

~

!)

+

<l.<l.

+ I s (x, 0) - s (x . u(i

»)

1

2

) dxdy ~ 20 .2

M2

+ "')a

l

(Il~~t"'

>

+

1

+I

6.1-l (D

~

)

(u(i), 0)) I

6.

1-

~

20· 2

M2

+

i»

a

t

( fl

~

4-

2

...

+ "') +k(Po)) Itl l-

I

,

a

J

=

00

j p

n

-

1

a

M

2+

6

) (p) dp, i = 1, ..., s.

(3.5.8)

o

T aKHM

o6

pa30M, H3 (3.5.7), nepanencrsa 4 e

6b

I

we

aa H (3.5.8)

no

-

1

JlY

4ae

M

p(

3) = 0 ( 1

6.

1-

) .

C JlY4aH, [<0

rn

a nexor opue

M

H

Om

e

CTI~a

F(i)n

U

~

(8) ny

cru

, pacC

MaT

pHBaeTC5I

Cl

HaJl

OrIl

4HO. •

Ilp nse

nea

On HO n OCTaT

OT.J

HOe YCJlO

BHe

c np aBen JlHBOCTH XVII.

Ilpennono

aoo

a,

4TO

c.

B. E (0)

c

I

M

~

l

e

Tp

H

4 H

a

H ee

<p

.

p.

F -

a6

co-

JlIOTHO

nenpepsis na. F' (x) = • (x) . Flyer s B

bIn

O

JlIl

5l10T

C51

yCJIO-

BH51.:

sup sup Ig (x,

01)

- g (x, 8

2

)

I = go< OOl

(3.5.9)

' ERn e,.e,Ee

c

inf T (x) = .*> O.

(3.5. 10)

I

x

l

~

g

o

OO

CKOJlbKY

nJlfl

moooro g E[- go, go]

Igi

M Ic (0) + g I- III =

~

(Ig I- Ix i) F (dx)

;;.

T*g2,

- igi

nOCTOJlbl<Y

ll

JllJ u E

U

~

(8)

I

6.

j-

1

Mk

..

(u)

;;.

)-ll + r" I

6.

1

-

l<D~

)

(u, 0).

n

03

TOMY XVII f1BJl5leT CfI

C

Jle

nC

TB

He

~

1

cnenyiouiero YCJlOBH lJ.

XVIII ' 1I.

JlfI

moooro p > 0 HaHn eTc fI raxoe

T.J

HCJlO a (p) >

> 0,

'IT

O

inf

I

6.1

-J()J

~

)

(U, 0)

;;'

a (p).

eET

uE

U

~(

e

)

"'-

lJjp)

Ii cY IlJ,eCTBy eT

rax

oe Po> 0, 'I TO .*a (Po) = qolll + ao; qo> 2,

ao>

0 - aexoropue 4H CJla. Ycaonue XVII aHaJlGf

HT.J

HO V aa

n .

3. 2. O

'l

eBH)l.HO, (3.5.9) o

oec

ne

ua

saer I3bInOJlHeHl1e y CJl0BH5I

XVI.

149

14

8

Teop

eaa

3.5.2. n YCTb e (x), x ERn- onnoponuoe rayccos-

c

xoe

nOJIe C K.

<p

. B (x) EL

1

tR

";

ECJ1H

BbIn

OJ1I

IeHbl

YCJ1

0BIHI

XV -

XV

II, TO oueuxa a

,..

y,noBJIeTB0p

51

eT (3.5.4).

....

)l.

oK3SaTeJIbCTBO 3TOfl r eop

enu

OT.T

IHlJaeTC5I or ,nOKa3aTeJIb·

CT!l3

npenuayuiea .rJllW b B oueaxe Be

J1

HlJUHbI I

~

1-

2

Db

A

(e,

U),

C

u Eu; (e) nV

(P

o)'

nO

CKO

J1

bKY B = i I8 (x) Idx <

00

, TO e (x)

Rn

oonanaer cnexrpansaca nJlOTHOCTblO. CJIeJl,OBaTeJIbHO,

no

reope-

Me

PUMaH

3

-JIe6er

a 11m 8 (x) =

O.

n YCTb B > 0 - raxoe

Ixl--7

CX)

'lUCIl

O, "ITO k

o

= sup (I B (x)

1/

B (0)) < I. Tor na

Ix

l>

r

l

ti

l

-

2Db,..(

O,u)

~

l

til-

2

H (B (O) + IS(X,O) -

6

'

n

il

x

-VI

~

el

- S (x, u) /2)dxdy +

Iti,

- 2 JS cov

(Q

(x ,

U)

,

A' n llx- J/I> el

Q(y, u)) dxdy = D

1

+D

g

.

no

YCJIOBUIO XVI

D

1

~ (B (0)+

lti

,

-

I

<J

)

~

)

(u. 0)) Iv (I) l en1

til-

1

~

(8 (0) + k

(Po))

Iv (1) l

en

I

~

,-I. (3.5.1 I)

)l.

JlH

oue

HIlBa

HIi

51

D

2

socnonssyeaca pa3

J1

0

)l{

eHHeM (2.1.5) n

ay

-

MepH

oA

raycco

ncx oa

11

/10 TII OCTH:

00

2 2

D =

\ti

/-

2

~

\

(l/

ml) Hm(t

l)H

m(ig

)(2n

)-

le

-<I

I+lz)

/2

X

2

m= 1

R'

IB

I

2

x (

~

/2 (0) t

1

+ S (x, 0) - S (x, u) I IBl/ (0) 1

2

+

Atr

Hlx-v

l> el

+ s (y. 0) - s (y, u) II B (x - y) 1

m

B (o)- mdXdY) dt.dt;

~

00

</

2+

(

2)

2

E;;;

I

/).

1-

2

~

.\ ( 11m!) I Hm (tI) H m (t

2

) I

(2n)-l

e

-

1 2 / X

m= 1 R'

2

x H (8 (0) 1/

1

/

2

1+ 2B

1

/

(0) lt

1

1Is(y,O) - s(y, u)l +

6

'n

ll x- 1,1> e l

+Is (x, 0) - s (x, U) )2)I B (x - y) ImB-

m

(0) dxdy dt

1d

t

2

•

(3.5. 12)

2

Oue HHM p5l,n, COlOmeTCTBYlOlUHH

CJ113r

aeMolVlY 28

1

/

(0) H

11

Is (p,

0) - s (y , u) I BO BH

yT

peHHeM nnre

rp

ane (3.5.12). 3 aMeT IIM, "ITO

no H

epaBe

HCTBy K OWH- OYHHKOBCKIO

ro

(2

n)-

1/2 j"It

,a

I Hm (t) I e

-t

'ntdt

~

(fn J)I

/2

, a = 0, 1.

R:.

'

rI

03TOMY

000 ?

+~

)

2

28

1

/

2

(0) Iti ,- 2

~

I (

11

m!) I H m (t

1

)

Hm (/2) II I (

2.rt

)-

l

e

-<I 2 / X

m= I R'

X dt]dt

2

II

Is(y

, O) - s(y, u) I IB (

x-

y) \mB-m

(O

) X

A'

nilx

-YI

>el

1

x dxdy

~

28 -

/

2

(0)

(1

- ko)- II ti

l-

2

JSts (y, 0)- S (y,U)!I

B(x-

,

AA

- y) Idxdy

~

28 -

lt

2

(0) B

(1

- kO)- 1k

l

/

2

(Po)

I

~

,

-I

.

T

Ol.JH

O T ax )l{ e P

51

,n. COOTBeTCTBYIO

Ill

Hii CJlar aeMOMY 8 (0) It

1

/

2

1,

Ma>

kOp

u

py

eTC51

BeJ1H

l.J

HHOH B (I - ko)- I I

~

,-

1,

a p5I,n, COOTBeTC'r-

BYIO

Ill

IIH CJlar aeMOMY Is(y, 0)- 5 (y, u) 1

2

, BeJIUl{UHOH

B-\O)

B(I -

- .s: k (Po) Iti l-

1

. C Jle,nOBaTeJlbHO,

D

~

B

(1

-

kO

)

-I

(2B-

1

/

2

(0) k

1

/

2

(

Po

) + 8-

1

(0) k

(P

o)

+ 1) 1

60

I

-I

.

2

(3,5.13)

O u eHKH (3.5.11) II (3.5.13) nOKa3

b11

3aIOT.

srro

1

~

1

-

2

Db

A

(

e

,

U

)

=

0 (I

til

- I). •

n OCKOJlb KY )l,JlH r ay cco

sc

x

on

nJIOTHOCTU cnp asenn a no (3.5.10),

TO, aanpnxep.

J],JI5I

Or p aHHl{eHHOIf epy

HKU

HIi

g (x, 8) YCJlOBHe

XVII Bb

I11

0JlH5IerC5I, eCJIH au no

nn

en o XVIII'

3.6. AC

HMn

TOH flI ECKASl HOPMAJl bHOCTb

OU

E HKH

HAHM EH b WH X M

OAY

JlE

A

O CHO

BHa

51

TPYJI.HOCTb, KOTOPY

JO

n

pax

o

mr

r cs np e

ono

n eea

rs

npa

,lLoKa3 aTeJIhCTBe a

ClfM

nT

OTHl{(:CKOH HOPM8JIbHOCT

lI

ou

eu

x a

e

A

,

3aKJIIO

ll

ae

TCli B TOM, sr o lpYHKUHOH8

J1

RtJ.

('t) u

en

udxpe

pe

u -

unp

ye

n,

Ii

Mbl He MO)l{eM BOC

11

0Jlb30 BaTb

C5I

CTaH,napTH bIM cn

oc

o-

60

M paccy)l{,neHHH,

npHp

aBHHBaH rpa

nn

enr Rs

('"C)

B TOIJKe

St>

K

HyJlIO

(cp, C n 3.3 - 3.4). C ,nP}

ro

H C

TO

pOHbl, Y

<P

YHKUHOHaJla

Rt>

(1")

CY

lUe

CTBYIOT

np

on

aaon uu e n o scea a

an

pasn

e naa» , e

cna

TOJIbKO q>YHKUDH g (

x,

8)

ll

llq. cj:ep

eIl

Ul1pyeMa B 8, H onCpaUH5I

151

150

npenensaoro nepexona non 3HaKOM nnrerpana paspeuieaa.

De

HCT-

BIl TeJIbHO, eCJIIl

'E

e, a

1-

np Oll 3BOJIbH Oe nanpasnesue B R

q

,

TO

«

(J

liJl)

RlJ.

(

rr)

=

~

(Vg (x, r ), I)

(2

X {£o (

x)*

g (x, 'f)} - 1) dx, (3.6.1)

lJ.

rne

«*»

03

H

atIa

eT

~

,

eCJIB (Vg (x, c), I) > 0 H

<',

rr), I>< O.

nY

CTb p> 0 - paCCT05lHlI e Me

:>K

,lJ.

y KOM

na

KTOM

ecnn {Vg (x,

T Il MHO

:>K

e-

CTBOM

«<;e. E CJIH P< P Il oc

ym

eCTBJl5IeTC5I C0

6b

IT

lI

e I

elJ.

-

- G1< p, TO

,lJ.JI5I

moooro H

anp

aBJIeHHH I (

(J

liJl)R

lJ.

(8lJ.)

> O.

3 TO npocroe

3aMe

tIa H

il

e n03BOJIIlT nOJIYtI

llT

b aCHMlI TOTHtIecKYIO

HOpMaJIbHOCTb

oueHKH

e

lJ.

.

BB

e

,lJ.

eM

PH,lJ.

rpeo oeanua K cxe

ae

Ha6J1IO,lJ.eHHIt (3.1.1), HC-

nOJIb3

Y51

06

03 HatIeHH5I npenu

zr

yumx n y HKToB Il coxpanns rpe-

60Ba HH5I

n.

3.1 K cpy HKUHH g (x, 'f) .

e

C q HTa H e Bhm YKJIblM MHO

:>K

eCTBOM, 6Y

,lJ.e

M

npezm

ona

rar

i.,

c

tITO y <py HKUHH g (x, e)

cy

meC

TI3YIOT H nenpepu aa u B ace

npO

H3BO

,lJ.

Hble

no

nep

e

~

leHH

b

l

M

8 = (8

1

, .

..

, e

q

)

,lJ.

0

2-ro

n Op 51

,lJ.

Ka

BKJIlOtIl1TeJIbHO

.n.JI

H K

a::>K,lJ.

O

rO

x ER", np" qeM ¢ YHKUHH gi (x, 8) H

go (x, 8), i,

i>

1, ... , q JIOKa JIb HO mrre

rpa

pye

as

; C KBa

,lJ.

paTOM

no

x

,lJ.JI

H JII0

60r

o 8 Eec.

XV

III. .lI.JI5I JI

I0

6 0 r o p > 0

s

up

sup

I

!J.1

1

/ 2

d

~

1

(8) sup ISi (x, u) I

~

k(i) (p).

eET c c

xE

lJ.

uEU

~

(

e

)

nv

(p)

i = 1, ... ,q;

(3.6.2)

su p sup \

!J.

1'

/2

di6! (8) dj61(8) dii

lJ.

(8 +

eET

uEU~

(

e

)

n

v

c

(

P

J

+I

!J.

1

/2d

6"!

(8) u) ~ k

W

)

(p), i, j =

I,

... , q. (3.6.3)

Hs (3.6 .2) H (3.6.3) B COOTBeTCTBIIH C JIeMMOH 3.3.1 su re -

Kal

OT acnonssyeaue IUi

::>K

e HepaBeH CTBa

2

su

p

Slip

I!J.

'-'

CD

lJ.

(U

l

,

u

2

)

lUI - U

2

/-

~

k (p); (3.6.4)

BET UE

ui

(B)n Vc(p)

SUf s

up

d~~

(e)

CDilJ.

(UI' U

2

)

l UI - U

2

I

-~

~

k

i

(P).

BE

u E

U

~

(e

)

n

tl

c

(

P

I

i = 1,

...

,q.

(3.6.5)

•

C a MO y CJIOBHe (3.6.3) HBJI5IeT CH yCJIOBHeM (3.3. 1) n p n la 1= 2,

a (3.6.2)

,lJ.

OCTaTOtIHO

,lJ.

JI

51

cn paaennnaocr u (3.3. 1) npu Ia I= I.

XIX .

C.

B.

Eo

(0) cH MMeTp lJl

lll

a.

Ee

cp. p . F (x) a 6cOJJIOTIIO ne-

npepu sna: F' (x) = rr (x), H I) 'to= su p T (x) < 00 ; 2) T (0) > 0.

xER

n

3)

l'f

(x) - T (0) I

~

H \ x I. H - HeKOT

Jp

aH KOHCT

aHT

a.

Bnecre C O

,lJ.

HOPO

,lJ.

HbIM B Y3I<OM CMblCJIe CJIYlJaH

ll

bg l nones

Eo

(x), x ER" paccsror paa O

,lJ.H

Op0

,lJ.

Hoe B Y3KOM csiucae cnyxa

n-

Hoe

n OJIe T](x) = 2X {

Eo

(x) < O} - I, x ERn.

OqeB

H

,lJ.

HO, Ml1(x)=

= 0

,lJ.

JISI

CHMMeT

pHq

HOIt c. B. e (x); 8

n

(x) = M'!) (x) 11 (0) =

= 4P {e (x) < 0, e (0) < O} - I.

EC JIIl e (x) - rayccoscxoe none, TO no diopa yne

CT

IWTbeca -

Illennapna [75, c. 321J

o U

8

1

) (x) =

4~

'

~

<p

(I; s, 8 (x)18 (0) dtds - 1 =

-

00

-

00

= (2/n ) a

rcs

in (8 (x)

IB

(

0»

).

n YCTb a' (p) -

K0

3

cj:Jep

Hlu

le

ll

l nepeMeUHIB3Hl1H (CM. n. 1.6) n OJI5I

n(x). Tor

-n

a

CG

' (p)

~

a (p),

CG

(p) -

K0

3epep

ll

Ul1eHT nepea eumsaa na

nOJIH ,,(x). E CJIH

,lJ.JI

H

CG

(p) uuno.nneno, nanpnvep. rpe

ri

oeanae

IX

n . 1.7, TO B'l (x) 11HTe

rpH

py eM3 aa R" (CM. 3aMetIaHHe

1.6.I), n, CJIe

,lJ.

OB

aT

eJlbHO, none 11(x) 06

J1

a

,lJ.a

eT nenpepuanoit II

O

rp

a Hl1tIe HHOIt cnexrpa

nsa

of n JlOTHOCTblO III

(A.)

,

A.

ER

".

XX. B eKTop

-cj:J

y HKUHH Vg (x, 8) paBHOM

epHO

B T

o6

!Ja

,lJ.a

eT

CIJeKTpaJI

bllO

H MepoH

f.l.

(d

A.,

0), U MaTpHu a

~

.

tII

(f...

)

f.l.

(dA, 0) pas-

R"

HOMepHo no 8 ET n OJI

O::>K

HTe.TJb

Il

O onpe

zi

enena.

Teopea a 3. 6. 1. n p

e,lJ.

TI

OJIO>KIlM, tITO

K0

3

qxpHU

IleHT

CG

(pi ynos-

JIeTBOpHeT y CJIOBHIO IX n . 1.7, ¢ YHKU HH g (x, 'f) -

YC.l0

BIl

51M

XVII I, XX

11

XII H3 n. 3.3, c. B e (0) - ycno

nm

o XIX.

V.H. v

!J.

- ".

00.

nY

CTb OUeHKa 8

6

06JIa

,lJ.

aeT CBOliCTBOM (3.5.4). T 0f11

Cl

pasaoaepno B T

'.t>

.H.

2F

lJ.

(., 8)

-4-

(

N

~

)

K

.(

e

)

( . ),

!J.

- '"

00,

(3.6.6)

rne 2F

lJ.

(y, e) - cp. p. aexropa

dlJ.

(0) (8:, - e),

K

2

(8) = (2n)" (4.

2

(0))- 1(t

fA.

(d,

A.

, 8)f'

0/

II

(1))

f! (d

l>

,

e»

) x

x

(t

fL

(dA,

e

)f

'. (3.6.7)

152

15

~

- -

nOK

a3aTeJlbCTBO pa306beM aa HeCKOJlbKO

sran

os.

n YCTb ll

....

... ,

lq

- nonoscarensuue aanpaaaeans KOOp

,lJ.

I1HaTHbl X ocea. Pac-

CMOTpHM BeKTOpW Rt (r ) C KOOp

,lJ.

HHaTaMH

R

i~

('

L)

=dit.

I

(O)

(010 X

X (± l

))

R6(r) , i = 1, ." • q H sexropa MR1

(r

) G KOOp

,lJ.

HHaTaMH

MRi~

('T:

) = ± d

i6,l

(0) j gd x,

'T:

) (2F (g (x,

'T:

) - g (x, 0») - 1) dx.

/:;

1=

1, ... , q.

(3.6.8)

OtIeB

II,lJ.H

O. M R; (8) = 0 B CHJlY CI1MMeTpHtIHOCTH c, B. s (0).

0

60

3

llatIHM

zl (0, u) = I Rf (0 +1

~

11

/2d; 1(8) u) - R1 (8) -

MR~

(0 +

+

I

~

11

/2d; 1(0) u) I

(l

+IMR1 (0 + 1

~

,1

/2d

;l

(0) u) I)- 1.

JI

eMMa 3.6. 1. B YCJlOBHHX reopeasi 3.6.1

,lJ.

JlH JlKJ6o r o e > 0

11

,lJ.OCTaTOtI

HO MaJlblX P> 0

v.a .

sup P { sup

zt

(0, u) > e}

--

O.

~

- 7

00

. (3.6.9)

. l

OET

U E

U

~

(8

)n~

c (p

)

A

,LloK

a3aT

eJlb

CTBO nponenea

,lJ.

JlH Be

.'lli

tI

li

Hbl

l'

(0, u). ,LlJlH

npocr

oru

6Y

,lJ.e

M ClJHTaTb. tITO p = 1, a cynpea yx B (3.6.9) non

3H

a

K O

~

1

BepOHTH OCTH onpenena

er

ca B K

y6e

C o = {u : Iu

10

=

ma

x IU I I

~

1} :::l V (I ). Ky6 Co noxpoea No = 0

(I

n I

~

I)

xy-

l

~i

~

q

6aMll

C Olo . .. ,C(N.) CJle

,lJ.Y

KJIlI,HM

06

pa30M . n YCTb

xacn

o p E(0. 1).

OO

CTpOHM x o

uu

eu r p

nx

ecxyio

CliCTeMY MHO

>l<

eCTB C(m) = {U:1u 1

0

E

E[

(1_

p)m+l, (1 - p)

tn

]}

. m = 0. ... ,

rn

o

-I,

cr » = {u : I

Ul

o

~

(l -

p)m.}.

Ka

xc

noe H3 MH

O>l<eC

TB

C(m)

no

xpo

ea O

,lJ.H

HaKOBbIMH

l

Ky6

aMH co CTOpOllOi'J am = (1 - p)m- (1 - pt + = p (1 - pm

11

nepea y x epy ea 3TH K

y6b

\.

OH

H H 0

6p

a3

ylO

T noxpu rue C O),

...

df

... ,C(N

.-I).

C(N.) =

C(mo

). BbI6e peM

rn

o

=

rn

o

(

~

)

HCXO

,lJ.

H H3

ycno-

BI1H

(l

-

p)m

. = I

~

\-

"1.

rn

o

=

[rn

o]. y E(1/2. 1).

3aMeTHM.

4TO

m

1·l

o·paccToRHHe OT C

W

,lJ.0

0 pa BHO p (j) =

(1

- p) I

~

\-v

/';;.

H

I' 10·

,lJ.

H

aMeTp

C(f) pasea a

(j

) = pi

f!.

,-

'I'm/m. npn HeKOTopOM m =

= m (j), j = 1, ... , No- 1. .LleHcTBHTeJlbHO,

nycrs

Ky6

C(/)

HBJlHeTCH 3

J1

eMeHTOM nOKpblTH5I MHO)KeCTBa cr: Torna a(j

)=a

m

,

p (j) = p

(1

- p

)m+

l + .

..

+ P

(1

_ p)m.

-l

+

(1

_ p

)m

..

ll

Hc Jlo

Ky60B

C

U

).

nO

KpblB

aKJlUHX Ka

>l<

,lJ.

Oe MHO

>l<

eCTBO c (m),

MO>l<H

O cne-

JlaTb He 3aBIIC5IIlI,IIM OT m H,

CJ1e,lJ.OB

aTeJlbHO,

OT

~

.

ll

T06bl y6e-

154

R

q

,lJ.

HTbC

Sl

B 3TOM. pac

cao

rpaa B JI106oi{ OKTaHT.

0

6

b

e~

1

nonas-

w e

i'J

B Her o tIaCTH MHO

>l<

eCTBa C

<ml

e

crs

(l - pt

q

- (l- pi

m

+ I) Q.

a I C {fl 1= a

q

(j) = pq(1 - p)mq. T aKHM 0 6p a30 M, B .n.aHHbIH OK,

Ta

ll

r «noa euiaer ca» He

60

Jlee ((l _ p

)m

Q

- (I - p)(m+l)q) P- q(l -

- p

)-

tn

q = (1 - (1 - p)q) P- q Ky60B. n OCKO.ijbKY

mo

= O(ln 1

6.

1),

TO II

N

o

=O

(l

n

l

~

I

)

.

B bI6e

pe

M

OE

T . Torzia

No

p { sup zt (8, u) > e}

~

1} P { sup

zt

(8. u) > e}, (3.6.10)

uEG

n

;= 1 "EGu)

Oue

HHM Ka

>l<

,lJ.

oe

cnaraexoe B (3.6 .10).

0611\l

\l1 3J1eMeHT MaT

pll'

Ubl-n pOH9BO,l,HOH D

6

(u) o

T06p

a /Ke HHH u

-+

MRt (8 +I

~

jl

/2

d;: ' X

X (0) u) HMeeT BII

,lJ.

D~

(u) = (OI(JUj )

M

R~

(8 + I

~

1

1/

2

d;: 1

(8) u) = I

~

,

1

12

d

iA,

J (0)dM

1(0

) X

X

~

SlJ (x , u) (2F (s (

x,

u) - S (x, 0)) - 1) dx +

n

+

21

~

I

l

/2d

iA,

1(0) d

if,1

(0) JSI (x. u) 5J (x, u)

l'

(5 (x. u) -

.l

- 5 (X. 0)) dx

=

l

D

~

(U

)

+

2

D~

(U

).

C Y

4e

TOM (3.6.3), (3.6.4) H Or pa

ll

114

eHHOCTI1

't

(x)

,lJ.

J1H

IU 1< P

nOJl

ytIaeM

1 2

I

~

I

-

I

/ 2

1

D

~

(u)

~

2 /

f'j.

1l/

2

d

iA

l

(0) dj61(8)

dill>

(8 +I

~

1

/

d; 1(8) u) X

..

X (I

~

j- I i(F (5 (x, U) - S(x, 0) - F (OW dX)1

12

~

2k(l

J)

(p) k

l

/2

(p) 'Co Iu I. (3.6.11)

C

npyroa

CTOpOHbl .

1T

1

\

~

l

-l

/2a

D

~

(u) -

'T:

(0)

3~

(8)I

~

'to [di6

1

(0) d

i6

(0 +

+I

~

I

l

/

2

d

:;-

l

(O)

u)di6

l

(0)

<I>

}

~

(u, 0) + d

iA

l (0)

CD

)

~

(u, 0)] +

1

+ d

iA

l (0) di6 (0) IigI (x, 0)gJ (x, 0) (T (5(x. U) -

- 5(x, 0» - 't (0» dx I' (3.6. 12)

no

(3. 6.2) H (3.6.5)

cnarae

a

ue

B xsa npa

rn

u x

cK06Kax

OueHHBa-

IOTcn

BeJl

ll

tIHHO

i%

'C

o (k(i)(p) k

l

/

2

(p) + k)f2(p) lUi. ,UJl5I nOCJIe

.n.

He·

155

r o c

na

rae

n o

ro

, ncn

ons

aya YCJlOBHC XIX H (3.e.2J npa U = 0.

uaxonua Ma

)KOpaHT

y

I

,1

11

I2d

~1

(8) sup I e.(x , 0) I

(I

~

r:r

(""

(s (x , u) - s (x , 0)) -

<EA

~

-

""

(O»)ZdX)1/ 2

:s;;

Il

) (0) H I

~

1-1

/

2CD

~

t

(u , 0)

:::;;;

k

(i

)(0).

11

'!l

JI2

(P)IU\.

(3.6 . 13)

n OCKOJlbKY no YCJlOBI110 XII

MaTpHQCJ

1~1-,

nD

6

(0)

=

2T(0

)

S

A

(e

)

nonosor

ren

sa

o onpenenena, npoaen

ennue

pa

cc

y)K,ll.eHH

Sl

nOKa3bI'

Ba101, 4TO }!JlH

,ll.O

CT

aT

041J0 M

JJlbl

X U (,ll.Jlf! npOCTOTbJ C4HTaeM.

4TO

.llJlH u EC

o)

H nexorop

or

o k

o

> °

I

I12

inf

IMR! (8 + \

~

d

-;-

' (8) u) \ > k

o

I

~

1

/'2

1u

10

' (3.6 .14)

fl E7

j lycr s J

=/=

No' u EC

lf)

- nponaeonsnas T04Ka . Tor na C vue-

10M

(3.6. 14) MO)K !l O

san

HC31 b

sup

zt

(8, u)

:::;;;

( sup

\

\I'

~)

(8, u,

u)

+

Y~

)

(8,

u»

(1

+ko

l~

1

/

2

p(i»- 1;

"Eew uEC

o

,

1

W

~

)

(8, u, u) =

~

W

~

(8. u, 0),

'-=1

wF

l (8, u, 0) = 21s; (8) JV

S(x,

u) (1.

{;

, (x)

",s

(x, un-

'"

-

'X

g o (x) < s (x, o

J})

dx [;

w

\!1

(8, U, 0) = Id

6"

1(0)

~

(V s (x, u) -

vs

(x, v» x

'"

x

(

2

x(s

~

(

x

)

<S(

x,v)}-1)d

x

l;

\

V

~

(8, u, 0) =

21

d-;I (8) j VS (x, u) (F (s (x, u) - S (x, 0» -

/j

-

F(s(x

,v)-s(x

, O»)dx !;

W4

U)

~

(8,

U,

v) =

I

d

;;

1

(8) j

('

(vs

(x, z) -

vs

(x,

v»

X

b

x (

2F

(s( x , v)

-slx,

0»

-I)d

x I;

Y

~

)

(0, 0) = Ia;' (tl) J

(Vs

(x, u) (2x g

fl

(x) < s (x, vJ) - 1)-

A

- vs(x, 0') (2z {e (x)*O} - I) - v

s(x,

u) (

2F

(S(x,

u) -

- s (x,

0»

- I)) dx I.

no (3.6.5) LlJll'l u, v EC(j) no

.n

yuaeM

Q ) 1/2

I

~

I- In \f/Yl (8, Ul, u)

:::;;;

~

d~

2

(8)

lC

D,

,,,

(u, 0)

:::;;;

k

1

a(j ). (J.6.I,'1)

(

1= 1

3aMeTHM,

na

.n

ee, 4TO

A

-1

/ 2

W

(j)

e - 1/ 2(

,,

1/2 )

ILl I :1A ( , u, v) = 2

""

0 I

~

I

\.,1

-'6 (U, 0 X

a

)1

/2

X

(

,~dit,

2

( 8) d;6 (8 +I

~

1

1/ 2

d;; 1(8) u)

~

k

2

a (j ) (3.6.16)

B COOTBeTCT

BHI1

C (3.6.2), (3.6.4)

11

X

IX

.

AH

aJlOrl14HO

q

)1/2

IQ

1-

1/2

W<

/1 (8, u,

0)

:s;;;

21:

0

~

d

it:.

2

(8)

CD

i

.

1

(rz, v) X

(

1= 1

X I

~

1-1

/2Q>

~

(v, 0)

:::;;;

k

3

a (j) . (3.6. 17)

OQ

eHI1M

w\

ll (8, u, v). ,UJlH JlK)6bIX U, v EC(j) n. H.

IXgA(x )*s (x, un - xg

fl

(x) < S (x, vJ) I

:s;;;

x{ inf s(x, u) -

/IEC( {l

- g (x , 8)

:::;;;

e (

x):::;;;

sup s (x, u) - g (x, O)} = x(x).

uEC(f1

C JIe)10 8aTeJlb HO, no (3.6.2)

q

)1

/2

I

~

I-

l

(

~

w\il (0, u, v)

~

,

~

/

-1/2

~

(dt;,1(8) sup j 51(x , u) I

)2

X

(

1- 1 'E6

X

~

X (x)

dx

:::;;;

k

4

/

A

,-I

~

X (x) ax (3.6.18)

6 A

Hcnoxssys rpopv

y.ny

KOHelJHblX n

pspauienu

a, HaXO.lll1M

I

~

r'

~

MX (t) dx = I

~

I-I J(F I sup S (x, u) - g (x, 8l) -

. 6 t. '"EC(iJ

-F

( inf

S

(x

,

u-

g

(

x

,

8

)

dx

~

't'

,

r

~

I

-

'

)

sup

/S(x, u

1

) -

uEC(j} A

ui'

uzEO(j)

157

156

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

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