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

q

)tn

- S

(x,

u

2

)

Idx

~

'f

a

~

( ILl

,

1/

2

d

~1

(8) sup ' SI (x, u) I)2 X

(

1= 1

<E

ll.

uEG

UI

X a

(j

)

qll2

~

k

5

a (j ). (3.6. 19)

O ueHKI1 (3.6.15) - (3.6.19) rrOKS3bIB3fOT, qTO Gyw.eCTBYfOT raxae

KOHCTa HTbl k

o

H k" qTO

P { su p

W

~

)

(8, u, v) (I + k

o

ILl I

I

/

2

P

(j»

- I> e/2} ~

" EC(j)

1

~

P {k

e

ILl

I-

~

(X(x) - MX (x» dx > (eI2) p (

j)

- k,a

(j

)}

.

(3. 6.20)

k

7

P) ILl

/-

vm 1m

B eJIHq ylH a (e/2) p (j) - k

7

a (j) = «e/2) (1 - p) - , >

>

0, ecnn Bbl 6pSTb P

,UOC

TaTOqI

W

MaJl

bIM.

n pHMeHHM K

c. B. X(x) H X(y) nepasencrao JIeM Mbl \. 6.2.

Torna C yxe

roa

(3.6. 19) .uJIIJ 1> 2

l-

2/f

(I x -I(WV (X(x), X (

y»

I

~

10

(1\1

X(x)

t'

(MX (y»1/

'a

Y I)

~

2

i 2ft

~

k

s

a

/

t

(j ) a

-

( I x - y I)· (3.6.21)

C JIe

.n.O

BaTeJIbHO, no aepaaencrsy LIe6blWeBa BepOHTHOGTb (3.6.20)

Ma

)l

w

pHp

yeTcH BeJIHIUlHoi'I

_

00

kga(t) It1j- l+v

m

(2- 2/t)/

l7'

o,

a(f)

=

~

a

l

-

2

/,

(p) pn-1dp. (3.6.22)

u

T aK KaK a (p) = 0 (p-

n-

e) .uJI5I nex

or

oporo e > 0, TO aCt) <

<

00

, eCJII1

I>

2 + (2nle).

LI.JI5I roro qTo 6 bl BeJIH

qHHa

(3.6.22) CXO,UHJIaCb K

HyJI

fO npa

I

Ll

I-+

00

, .uOCTaTOqHO BS5ITb y < 1/2 + 1/(21- 2). 0 60SHaqHM

III

(x) = (s, (x, ) - S/ (x, 0»

(2X

{S

a (x) < S (x, vn- 1), 1

2i

(x) =

= 2s

1

(x, O)( X

{~e

(x) < S (x, vn- X{e(x)*O}), i = 1, ... ,q. Torna

P

{Y~)

(8, v)(1 + k

o

ILl I

I

12

p

(j

»- I > e/2}

~

8 (ek

o

)- 21Ll/-

J

p

-2(j

)X

q

2

2 ( )2

>(

~

di6 (8)

~

M I(tn (x) - M

in

(x) dx . (3.6.23)

OpHM

eH HM K o. B. t

l l

(x) H il l (y) He

paB

eHCTBO JIeMMbl 1.6.21

i

2f t

(Ix -

2f t

D

(~

t

u

(x)

dX)

~

10

~~

a

-

y I) (M I t

li

(x) n dx

~

1l 6 6

~

1

0a

(t)

<D

itl

(v, 0); (3.6.24)

D

(S

t

2i

(x ) d

:C

)

~

40

a(t)

.\ 57(x, 0) IF (s

(x,

) - S (x, 0)) -

tl L\

2

t

- F (0) 1

/

dx

~

40a (t)' odt (8)

(~

( I

Lll

l

/2d

~1

(8) X

11

1.

X sup Isd x,

v)I)

2

(aU)

+ p

(j»2

1

I

q

ll

l.

(3.6.25)

<E

tl..DEC.

)

VIs

yC JIOBHi'I reopeu u, (3.6.24) fi (3.6.25) BbITe

KaeT,

qTO npa-

BaB qaCTb (3.6.23)

o

r

p

aH

H~

e

H

a

Be JII1QHHoi'I

k

io

(I) It1/-

l

p - 2

(j) «a (j) + P U))2+ (a

(j)

+ p

(j»

2It) =

= 0 ( I Ll I-

1

+

2

'1'

mn - 1It)f,;1,).

DO

JIYQeHHbJe ouenxu rrOKa3b1BafOT, qTO rrpH i = \, ... , Nn - 1

sup P { sup zt (8, u) > 8} = 0 ( I Ll/-

1

+

2m

"O- 1It)lm

o

) (3.6.26)

8ET uEG(/)

AJlrI aexoroporo m = m (j) <

mo.

Paccxor paa c Jly

qaH

i = Nn. Osesnaao,

P {

sup

zt

(8, u) > e}

~

P {

sup _ IRt(8 + I

t1

/1/

2

d; 1 X

uEC'N

e

)

lulo

":;

11l1-

vmo

/

m

,

X (8) u) - Rt (8) -

MRt

(8 + I

t1

fl

12

d

;;:

1(8) u) I> e}, (3.6.27)

3 a rrH

weM

su pa

xen

ne, c

rosmee

nozt SHaKOM HOpMbl B (3.6. 27),

B BH,Ue CYM

Mb

I BeK

To

pOB

~

I

A

(8, u) +

~2

A

(8, u) +

113

A (8, u), rne

~1

1l

(8, u) =

c:

(8) I(v

s

(x, u) - v

s

(x, 0» (2x

ge

(x )*s (x,

un

-I

),

~

2

1l

(8, u) = 2

cfi

l (9)

~

VS (x, 0) (X

{G

e (x)*s (x, u)} - x{e (x)*

O})

dx,

A

~

31l

(8, u) =

c:

(8) SVS (x, u) (2F (s (x, u) - s (x, 0» - 1) dx, Jler-

A

KO nOKa33Tb , 'lTO .uJIrI Iu 10

:os;;;

I

Ll/-"

m

ofr

n.

Q

)1

/2

_

I

~

ltl

(8, u) I~ I

11/

112

~

d

it,

2

<D

itl

(u, 0)

~

k

ll

l

t1

/1

12

- vmolmo:

(

i- I

(3.6.28)

159

158

Q )

112

- 2 " 112 - I

IB

3

1\

(0, u) I

~

(

i

~ a; (0) diD(8 + I

i'1

/ d" (8) u) x

x (2't lDr («, 0))

~

k I

L1/

112

- 'I'm

o

/m,

(3.6.29)

o

12

E CJIII V> 1/2, TO )l.JlH n OCTaTO

'l

HO

60

.TJ

b

lllH

X

I/),I

nOK

a3

3Te

JlII

cr eneun A (3.6.28) H (3.6.29)

or

p

au

ar

ens

au

.

TaKHM

oopa-

30

M, OCT3 CTCH OUC

lIIJTb

l3ep051THOCTb

P{

sup

1[

\6. (8, u) I >

e

}~

IU

l

o

~I

I\

\

-'I'

mo

/

m

,

.

~

P {2

(±

«:

(8)

~u

p

Igi (x , 8) \

)2)

1

/2

~

; (X) dx > e

ll

~

i= 1 Ell. .\

l12

~

P

{k

13

1

i'1

,- iX(X) dx >

e}

, (3.6

.30

)

Z (x ) = X{ in! s (x, u) - g (x, 8)

~

e (x)

~

I

U

.I

~II\I

-

'I'

m

o

/

m

,

sup

s

(x, u) - g (x , O)}.

~

l

UI.

~

I ,

\I-

'I'

m

.

/

m

.

HHH

np

eJlb

I.nyrn.eH JIeMMbI BbITeKaeT, 4TO

V.H.

inf P

{A

l~

(8)}

-+

I,

i'1

-

~

00,

i = I, .. . , q. (3.6.32)

AET

1l.TJH C06blTHH Ie

n

- 8

1<

p,

p<

P,

R,

~

(0

",»

0,

OO:3TOMY COOTHow e.

± + y

HI1H(3.6.32) sepnsr H .nJIH COOblT

HI

I Bi

c-

(8)

= {

R

i

~

(8) + MR

r..

(0

6.)

~

> - e (I +IMRl (e

",

)

I)}:::l

A

i~

(8), C npyron CTOp OHbI, Rt,

(8)+

+

Ri,;(8)=

2 Sl g, (x, 8) IX{f,(

x)=O

}d

x=

O n. H. , H C06blTf:l51

-,

Bt;. (8)

paBH

OBepOHTHbI C

06

b

IT

flHM

cit

(8) =

{

R

~

(8)+MR

1t,

(

e",)~

+ v 1

~

e(

1 + I

MR",

(8,.)

Il}

. LLaJlee, ll.JI

51

8 <

q-

B

~

(8) nc:(8) = D

1;"

(8) =

{/

M

R

~

{(

)

+

«t:

(8) I

~

6

+ v

~ e (I + I

MR

L\

(81\)

I)},

(=

1, ... , q; (3.6.33)

.q v v

n

Dt.

(0) {I MRt (8",) + r:(8) I

~

qe (I + IMRt (8J Il} =

1=1

~

{I M

Rt

(8",)I< (I - qe)- I (qe +I «: (8) I)} = et (8),

I

+

V~

+ v

n OCKO.

TJb

KY no YCJlOBHHM

reopesiu

Mx

(x)

~

k

14

I

L1

r-

'I'm.

l

m.,

T. e. in

fP

{E

L\(8

n-+

!'

/),-

~oo

.

T aK

K

a

K

P

{

I

M

R

d

8

1\

)

I

>

p

}

~

BMeCTO (3.6.30) nOCT3T04HO oueuirn, llJlH

JlI060ro

B> 0 BepOHT-

8ET

~

P {E

f(

8)}

+ P {I Rt (8) I> p (I

-q

e) - qe}, TO C. B. IMRt (

86.)

I

HOCTb P { Ii'1/-

112

I(x(x ) - Mx (x))dx > e}.

~

e

p

a

B

eIIC

T

B

O

4e-

or

pa u ns

eaa

no

Bep0

5lTHOCTH paBH OMepHO

11

0 8ET.

COrJIa

CHO

(3.6.33)

osnn

esa

11

oue

nx

a

CO

V

(X

(x), x(y)), a

lIaJ

IOrH 4HaH (3

.6.2

1),

noxa-

v + v

VH

3bI BaIOT, 4TO n

OCJlC

ll

H

51

BepoHTHoCTb

orpaanvena

BeJl1l411HOH

supP {[ Rt. (8) + MRt. (8(\) I> e(1 +IMR

",

(OJ

/)}-+

0,

i'1....:...

~00,

AET .

2

kin I

L1/

-

'1'

m

o

/

lm. . T aK KaK BCe Ou.eHKH paBHOMepHbJ

no

8 ET,

n03ToMy

cnp

an

en

nn

no (3.6.31). 3 aMeTHM, 4TO

Orp

aHf:l4eHHOCTb

no

JleMMa 3.6.1 no x

as

ana. I!I

± v ±

BepOHTHOCTH c . B. I

MR

t (8",) I r a

rox

e

Mom

HO

n0JI)'4

f:lTb neno cpe n-

n O.

TJ

O>KHM IHR" (0,,) = (MR

",

('t)) 1

1:

=8

",

'

CTBeHHO H3 (3.6.8)

Ii

yCJIOBHH reopeau. •

Jle

MMa :J.6.2. B YCJIOBll5lX reope

uu

3.6.1 nJiH Jl

I0

60

ro

e> 0

JIe

MMa

3.6.3. B yCJIOBHHX reopena 3.6.1 .nJIH J!I060r o

e>

0

v V.H.

v v

VH

sup P {I

Rt

(0) + MRt (8,,) I > e}

-+

0,

/),

-~

00. (3.6.31)

supP {1

MRt

(8J

- 2't (0)

~6

(8)

d6.

(0)

(86.-

8) I> e}

-+

0,

/),

....:...~oo

8ET

OET

(3.6. 34)

±

± ± v ± v

. BBe.neM C0

6b

ITHH AI'" (8) = {RI'" (0) + MR

I

",

(

8",

)- Rt

",(

8

",

)

~

• E

CJm

BeJIHtIHHa I

~I\

(8) I MaJIa, TO 1f3 HepaBeHCTBa (3.6.14)

± v •

> - q I + I MR,d 8",) I}, t = 1, ..., q. Hs (3.5.4) H yrse

psc

ne-

H Or paHf:l4eHHOCTH n o Bep

051TH

OCTH o. B. IM

Rt

(8

Y

",

) I CJIe.ny eT

or

-

160

I I - 6· 110

161

paHllqeHHO

CT

b no

BeP

0

51

TH

O

CT

Ii HOpMbl sexr opa d

t,

(8)

{o

A - 8).

YTB

ep:>K,ll,e

HH

e JleMMbl asrrexaer H3 (3.5.4) H nepasea

cr

s (3.6. 1

1)-

13.6. 13)

.•

...

ll

OKa)KeM reopevy 3.6. 1.

CooTHO

We

ll

H

51

(3.6.31) H (3.6.34)

noxa

ausa

ror, 4TO

.ll.Jl

51

moooro s > 0

sup P

{I

(2't (On- IAt, (0) Rt (0) + d

t,

(0) (8t,- 0

I»

e}

~o

,

6.

~

00

.

seT

(3.6 .35)

TaKHM 06pa30M,

npene.nsnue pacnpe

,ll,e

JleH

H

51

(ec.

TJ

J.1

rax

os

ue

v 1 +

cym

eCTBy

10

T) BeKTop

OB

d~

(0) (0

~

- 0) H - (21'(0))- At. (0) R

6

(0)

COBna

,ll,

a

IO

T. D O

CK

OJlbKY

R

7;,

(0) - dit,l (0) Jg i (x, 0)

11

(x) dx =

\

= 2d

it,1

(0)

~

gi (x, 0) X{e (x) = O} dx = 0

n. H.,

'"n( R/(X.S» Oj

V.H. v

TO npa

6.

-

;,.

00

Be

KTOp d

t>

(0) (0A - 0) pacnpeneaeu TaK )Ke, KaK

H B

eKTOp

- (2r (0))- 1A

t>

(0)

rn

t>

(0),

rne

rn

t>

(0

)=

(d;61

(0).\ gi (x, 0) X

.\

X

11

(x) dX)/= I.....q'

O,ll,

HaKO

no p. U. n, T. 1.7.5 sekr opu +

rnA

(0)

paB

HoM

epHo B T aCH

MllTOTH

4eC

l<H

H

Op

MaJlbHbl C

HYJleB

blM cpe

,ll,H

HM

H KoppeJlHuHoHHofi MaT

pH

ueo (2n)n

~

t;

(A.

)

f.I.

(d

')."

0).

V1

3

store

R

n

¢ aKTa

J.1

(

:3

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,ll,

eHHe re

ope

xsr 3.6.1 noayxae» T

;:lK

)Ke,

KaK yTBep

)K,ll,

eHHe re

opexu

3.4.2. •

r JJABA 4

OUEHHBAHHE

KOPPEJl5lUMOHJi

O A

II>Y

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4.1.

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O

UE

HO K

D Y

CT

b £ (X), x E R" - onaoponaoe cJlYlJaihlOe rro

.ne

C HY-

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HM O

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H,ll,

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3

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cp

. B (h);

6.

E

~n

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na

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se O

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h.)

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(x) 6Y

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an

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B~)

(h) = 16./-

1

S

~

(x)

~

(x + h) dx,

A

1

B

~

)

'

(h) = 1

6.

1- S £(xH (x + h)dx,

K01

o

pu

e

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C51

He.

" n (A - h)

llapaMeTpH4eCK

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MH

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cp.

B (h). aHaJIOn l

lJ

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r paMMe CJlY4aHHor o npouecca.

Bb[60p04Ha

51

cne

xrp

ansuas nJIOTH

OCTb

(nepaonorpa

aa

a) no

Jl

51

~

(x) , aaoznonaevo ro na

MH

O

:>K

e

CT

Be

6.

, onpeneneaa pasea-

CT

BOM

I

~

(A) = (2n)- n

16.

1-[ ISel (

'J..

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~

(x )dx r

.\

OTBe4alOwa

51

eii Bb16

opOlJHa

51

K.

cp

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B~

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T. e. oue

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a

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:>KeC

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6.

+ K. T. e. pacnona

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er

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.

3,ll,eCb

o

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MH

B (h) MoryT CJlY)lUITb ycpennen

nue

no

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51

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(x) npa.

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163

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00

a

no

m

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au

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(x)

np

a

cjJ

aK CHpOBaHHOM 6.

H

m-+

00

.

Ha

np

OT51)l{eHHH BCe!! rJJaBbl npenno

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rcs .

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TO i; (X),

x E R" - ona oponnoe B Y3K OM CMbICJIe C. K

.-

Hen

pe

p bIBHoe

H3-

M

epHM

o e cenapaoensuoe none. 113 HS

Mep

ll MOCTH I; (x) H reope

au

cI>y6HHH- T OHeJIJlII ssrrekaer .

'lTO

H

HTer

paJI J

1;

2(x) e- 1x1dx n. H.

Rn

KOHe4eH.

nY

CTb Q - C06bITHe, SaKJJ

IO'l

aIOlueeC5I B TOM,

'l

TO I;(x),

x

E

R"

- HSM

ep

aMa51 ¢ YHKUH5I a

~2

(x) e-

1x1

- Cy MM

Hpy

eMa

51

na R"

¢YH

KUHH. T aKH M

06pa

30M, P

{Q}

= I.

Fl

y

ers

DE )5 n

-n

pOH3BOJIb-

Hoe

orpaaasennoe MHO

)l{

eCTBO;

lz

ER" fl

co

EQ q;JiKCa pOBaHbI ,

S(x) I; (x + h) = U

h

(x). Torna IUh (x) IX

D

(x)

~

+

1;

2(x) X

D

(x)

+

1

++

~

2

(x +h) X

D

(x)

~

.

2-

1

e

xp

{r (D) - I x

J}

1;

2 (x) + 2- X

X ex p {r (D

1hl

)

-I

x + h !}

~2

(x + h), rne r (D), r (D\hl) - pa

.D.

If

YCbI uiapoa, cO

.D.

ep

)I{a

lU

HX MHO)J{eCTBa D H D\hl'

Cnenosarens

-

HO,

~

I Uh (x) Idx

~

ex

p

{r

(D1

h1n

~

1;

2(x) e- ,x'dx <

00.

D R"

n OJlO)l{ HM

B

~

)

(h) = I

~

,-I

~

Uh (x) dx,

B

~

)

(h) = I

~

,_I

X

d

X

~

Uh (x) dx, eCJIH

ro

EQ,

B

~

)

(h) =

B

~

)

(h) = 0, eCJIH

CJ

) E

d

n(d-

h)

E

Q"'-.

Q.

cI>

y HKUHH

B~

)

(h),

hER", i = 1, 2

51B

JI51IOTC51

C

JIY4aH

HbI-

MH nOJJHMH.

B

ca

MOM nene, npn

¢HKCH

pOBaHHbIX

.1.

H h ER"

no

r eo

pea

e

cI>

Y

OI1

HH- T OHeJIJIH C.B .

1

B~)

(h) = 1

.1.1

-

1

~

UIl ' x ) dx,

B

~)

(h) = 1

/),

1- j Uh (x) dx

., 6 n (d - 1ll

np

HHHMalOT KOHe

'lH

ble 3HalJeHH5I

JI,JI

H

co

E

~

l

A

.

h

,

P

{Q

d ,h} = I . C JJe-

B~

)

(h) =

B

~

)

(h),

AOBaTeJIbHO,

B

~

)

(h) H

BJI51IOTC

Sl

c. B., r ax KaK

i = 1, 2

.D.

JI

Sl

roEQ nQ d ,h' a Mepa P rIOJIHa.

OIyq

aHHOe nOJIe

B

~

)

(h), hE K - n. H.

aenpepu aso.

3T

OT

¢

aKT

esrre xaer us uenpepu

snocr

n

mrrer

pa na

Jleoera fUh (x) dx

s

no h.

Bonpoc 0

n.

H. nenpepsrsaoc

ru

nOJI

51

B

~

)

(h), hE K He-

164

CKOJIbKO

CJIO)J{

Hee. n Y

CTb

h, It +

"C

E K, h -

¢HKcHpoBaHo

. Torna

I

B~

)

(h +

.)-

B

~)

(h) I

~

/.1I-

l

JI

Ub

+

~

(x) - Uh (x) Idx +

A

+ 1

.1

I-I I i u«(x) dx - j Uh (x) dx I= /

(1)

+ / (2).

d nCd - h- '

[j

tin ed- h)

(4. 1.1)

Hnr

erpan 1(1) CX

O,D,H

T

C5l

K H yJUO npn t

-+

0.

060

s Ha

l

IHM R e Q = (R U

Q)

"'-.(

R n

Q)

cH

MMe

TpH

'Iec

K

ylO

pas-

(2)

HOCTb

MIl

O)l{ eCTB R

Ii

Q.

Be

JlH4HHa 1

MaiKOpHp

y eTc5I HHTer pa-

JIOM

1

.1

1- J I Uh (x) 1dx = 1.11-

1

X

ti n

«ti

- h- TJ6 (d

-h)i

X J I

U_

h

(x) Idx

~

I

.1

/- '

5 I

u_

h

(x) Idx.

(d +

hln

(d

6(

d- T

»)

AE

)d-T

)

(4. 1.2)

rJOCK OJIbK Y

.D.

JI5I

JJ

I0

60ro

Or paH H

'IeH

HOr O

~

E)5 " I

~

e

(.1

- . ) IHr

O

,

TO II 1(2)H t O.

TaKHM

c6

paSOM, nO

JJ

5I

B

~)

(h ), h EK, i =

1,2

no-

p0

)l{ ,ll.aIOT B npo

crp

ancrse

C (K )

Bep0

5lTHOCTHble Mepbl

",,

~

l).

i =

= 1, 2 aa )5 (C (K) ).

4.2. COCT05lTEJI

bHOCTb

H aH,ll.eM y C

JJ

OBH

51

, npn KOTOp blX oueaxa 81°(h), hEK, l =

= 1, 2

COC

T05lTeJlbHO B nopae C (K ) ouenasa tor K.

ell

. B (h), h E1

<'

n O

Jl51

S(x) npa .1

~

00

. ClJ

llT

aeM,

'lT

O 9n - JJHHeHHO ynopanouen.

Hoe MHO)J{eCTBO C orao

ure

nuea nopaaxa «S

lJ.

rJ pe

.n.

nOJlO)J{HM cae ny iomee.

I. LLJJ5I .110601'0

ct

> °cyuie

crs

y

er

TaK

a51

nOC

JJ

e

.n.OB

<lTeJIbHOCTb

.1

IE9n

, 'I TO:

I)

I1

l

c

11/

+

1

;

2)

~

(1.1

1

+11

1.1

1

I-

l

-

1)2 <

00

;

3)

~

1.1/

I-a

<

00

; 4)

.D.JI51

t:» to H aex

orop

oro

¢Hl<cHpoBaHHor

o

t

k

<00

MHO)J{eCTBa .1

1

c V (k I .1

1

Il/n) HBJl5IIOTC5I E

bl

nY

KJIblMH.

II. Ai II;(0)

I"

<

00

H

.D.JJ51

HeKOT oporo

~

E[0, I)

11

l im sup

j.1

I-

S

(Is

~

~(x.

x+h,h,

0)1+

d~

oo

hEK

6-

1

+IB (x - h) B(x + h)

IJ

dx <

00

.

(4.2. 1)

165

ECJ1H npa 8.

~

00

I8. 1

-+

00

nenpepusno B TOM CMbl

CJJe

,

4T

O

I

b.

\

npH

HH Ma

er

Be

e npoa eacyr oxnu e 3Ha4eHH5I H3

urrrepe

an a

la

o'

00),

a

o

> 0, TO 1) - 3) yCJIOB

I151

I s

cern

a

BbIn

OJIH5IIOTC5I,

llO

Cl<OJlbKY B K

a4

eCTBe MHO>KeCTB 8.

1

MO

IKH

O

B351T

b MIIO>KeCTBa C

a-

1

+ 6

06be

MOM

111/

I= t , 0>

O.

Tp

e6 0BaHI1e 4) y

CJIOBI151

I K no-

( JIe,nOBaTeJIbHOCTH

~

I

B

03

HH

Kae

T B CB5I3

11

C H

3Y4

eHI1eM ouenx n

" (2 F

B

Ii

)

H

BbIn

OJIH5IeTCH, eCJlH au nyrot

ae

MHO>KeCTBa 8.

1

-+

00

(CM . n. 1.1).

ECJII1 8 (x) EL

2

(R

n

),

s

up

~

I

5

~

i;,

)

(x, X + ti , h, 0) Idx <

00,

TO,

hEK Rn

OIJeBH.ll.HO, YCJIOBHe (4.2. 1) BbTnOJIH

5IeTC5I

n p l1

r

~

= O.

0 603Ha 4HM Wh (x) = Uh (x) - 8 (h)

11

.ll.Jl5I i = 1, 2

z

~

)

(h) =

= I8.1-

1

I W h (x) dx, 8.(\) =

11,

11(

2) =

11

n

(11-

h).

lI (1)

Teopeaa 4.2.1. E CJII1 asmonnenu YCJIOBI1

51

1)- 3) 1

11

II, TO

LlJJ5I

JII060 r o h EK

B~

)

(h)

rz:

8 (h) n . H

To

IKe yrsepacneaae

nepno H

)J..JI51

B~

)

(h), eCJIH K nepe4HCJIeHH biM YCJIOBI1

51

M 1l.06a -

BHTb

4) 1.

A Il oxaacea . 4TO Ll.'1

51

zno oo r o

hE

K

z~)

(h) - - ;,.0 n . H n o

1I~

00

Y

CJJ

OBI1IO II

M

(z

~

l)

(h))2

~

1111-

2Ir(ISf ) (X(I) - X (

2),

X(I) -

X(

2) + h, h, 0) I +

1I11

+ 8

2

(x(1

) _ X(2») + 18 (x

(1)

-

x(2)

+ h) B (x

(1

) - X(2) - h) I) X

X dx

(1)d

x(2) = 0 (I

111

-

IH

).

nYCT

b

11/

- nOCJIe.nOBaT

eJIb

HOC

Tb

MHOIKeCTB

H3 YCJl

OBH51

1.

o

rs

e

va

10

ui

as

a = 1 - B

::::;

1. Torna

.nJl51

K

aIK.n

or o h EK

Z

~

l

(h)

_~

0 n . H. Pac

ca

orpn« B

eJlHllHH

Y

I

l

~oo

qo

(h) = sup I

z<~)

(h) -

z<

~)

(h) I

lI

tSlI

ClIt+1

t

II nOKa>KeM, 4TO

~

i)

(h)

-

~

0 n. H. Jlerxo 3aMeTI1Tb,

'IT

O

~j

t)

(h)

::(

t

~

oo

~

(i

l1t+

l

l l

l1

tl

-

l

-

l

)

IZ~

:(

h

)1

+ nj

ll(h),

i=

1,2,

rn

e tljll(h) =

= 16

t1

-

1

~

IWh (x) Idx,

11l

2)

(h)=1

I1

t

I-

I I 1Wh (x) Idx,

6r(h)

=

lI

t+I

"-.

li

: 1I/ (h )

=

(.6.

t

+1n(I1t+l - h»

"(I1/

n

(11

/ - h)).

nOCKOJIbKY

M (tljll

(h))2

~

M I

~

(0) 1

4(1.6.+1/

I 6/ ,

-I

- 1)

2,

TO

nil) (h)

t~

:

0 n. H. Cnpa-

106

Be)lJI I1BO BKJII04eHHe

11/

(h) =

(11

1

+1

"'-

11

/) U

((I1

I

+I

"l1

t

) -

h) (4.2.2)

11, CJIe1l0Ba TeJIbHO, M

(n

1

2J

(h))2

~

4M I

~

(0) I"

(I

11

/

+

1

II

I1

t

1-1- 1)2,

T. e.

11

!2) (h) - ;,.0

n.

H.

TaK

KaK

z

~

)

(h) =

B~

)

(h) - 8 (h), TO

t~

oo

LlJI5I ouenx a

B

~

)

(h) r eopeva

Il

O

Ka3a

Ha.

3

aM

eTHM,

zianee,

4TO

1

z}?

J(h) =

B

~

)

(h) - 8 (h) + (1

-1

nn(n

-h)

II

11

1-

)

8 (h).

(4.2.3)

n OK

aIK

cM,

4TO Is, ,

-I

1

11

/ n

(8.

t

-

h) \-

;,.

1.

J!

eHc TBHTeJIbHO,

{

~

oo

1

-111/

1-

1

I

I1

t

n

(I1

t

-

h) I = I

I1

t

I

-I

I

11

/

"(8.

t

-

h) 1

~

::(Is, I- I IM

,,

(l1

t

)- lhl l. (4.2.4)

Bocnonsayeuca

reopeaon

3.4.1

11

n OJIO>RHM g (p) = ex p { _ p2 /

2I n

{I

11/

} ,

q = n.

Tor-na

C Y4 e ToM 4) 1

111

/

,,

(I1 /) - lh

l

e-

/t.

·

~

1

. r -lxl'/lllt/

2In

I- li n

~

J e

.u«;

b I5

n

_

1

( 1) : Ih

I,

b = b118./ I , b,=

t'>t"-.(lI

/Ll

hl

00

= 2

~.

e-P 'pndp . CJIe.nOBaTeJIbHO,

pa

snoaepno

no

hE

K

o

1

11/

"'-(l1d- lbl II

I1

t

,- I ::::; b

l

I5

n

- 1(1) Ie

'"

Ih II

11

/

1-1/11.

(4.2.5)

J!JI5I

KOHKp eTHblx MHOIKeCTB n/ B ouenxe (4.2 ,5) KOH

cTaHTy

MOIKHO YMeHb w I1Tb.

3 aM

eT11

M, 4TO

1

s,

(h) = sup

1/

11/

-

111

n

(11

- h) I

-I

11

/1

-

1

I1

t

n

(11/-h)

11

=

t'>tS

D

£

lI

t+ 1

2

= om)V 01

)

(h); (4.2.6)

61

1l

(h) =

111/1-

11

I1

t

(h) I

::::;

2

(I

I1

t

+

1

11

I1

t

I

-I

- 1)

-;,.

0; (4.2.7)

(

~

oo

6

~

2

)

(h) = I 6/ n

(I1

t

-

h) I

(I

I1

t

,

-I

- 1

11

1+

11-

1

)

=

/)

=

111/

1- 1

11

/ n

(~t

- h) I

(l

-I

I1

t

/111/

+11

- -

;,.

O. (4.2.8)

1-> 00

r-; 2

COOTHOllleHI1R

(4.2.3) - (4.2.8) nOKa3b1Ba1OT,

'I

TO n . H.

8~)

(h)-..

-

;,.8(h)

.•

~->oo

167

I'

3aMe

~aHu

e

4.2. 1. Y'CJIOBl1e 4) 1 B ,ll.OKa3a TeJIbCTBe reopea u

aepase

ncrsy

I'ensnepa, (4.2.9)

11

(4.2. 10)

4.2.1

HcnOnb30BaJIOCb He B nOJIHOH Mepe . YTBep)K,ll.eHl1e reopeau

t

AJUI Ou.e HKI1

B

~)

(h) OCTaeTC5I B CHJIe, eCJlI1 lim

sup

I s,I- I Is, -,

ag')

(h, r )

~

2

p

-

1

(M(5IU

Il

+ (x) - U

h

(x) Idxy+

o

'"

(~t

- h)I =

O.

(

-->-00

hEK

Il

UJIH

)l.OK333TeJlbCTBa 60JIee CHJIbHO

rO

yT

Bep)K,ll.eHI1H 0

COC

TOH·

+ I D I

P

I B (h + T)- B (h) I

P

)

~ 2

P

-

'

(k

1

+

k

~

) I D I

PT

+

t

\ •

Te

JlbH

OCTIt Ol.(eHOK

B

~

)

H

B

~

)

HaM nOHa,ll.0

651T

CH

60JI

ee )KeCTKl1e

(4.2.14)

Tp

e60BaH

HH K noma S(x).

C

npyroa C

TOp

OHbl ,

no

rpopxy

ne J1eoHOBa-

Wl1pHeB

a (1.3.2) H

II

1.

.llJIH HeKOTo

pbI

X p ;» n, 0 <

~

1,

(\,

(\ > 0

11

JII06bI X

(4.2.11)

p

m

h, h

-I-

T

EK

dt} (h, T)

~

(a~)

ih,

.)t

/

m

~

21'

s

up

(MI

~

Wh (x) dx n

/

~

hE

I<.

D

1) M Iu

h

+ (0) - Uh (0) I

P

~

k

1

I't

jrl+o,;

(4.2. 9)

t

~

k

5

I

tJ.

I

P

/2. (4.2. 15)

2)

IB (h +

T)

- B (h) I

~

k

2

1

~

,

fl,

p

~

= n + 6

2

,

(4.2.10)

n YCTb Y < 6/(n + 8). Torna, I1CnOJIb3YH

Oue

HKI1 (4.2.14) H

(3aMeTHM,

'ITO 2)

cnenyer

113

1)).

(4.2.15), nonyvae» pas noxepao no hEK

JlJIH

P

11

3 YCJlOBHH

III

n OJIO)KI1M m = [pJ + 1, eCJlI1 [

pl-

2

n

es

er

aoe '1I1CJIO; m = [p1+ 2, ecnn p - aeuenoe, a [p

]-

ag')th, T) =

(a

~

)

(h, 't))

1'

(ag')(h, 't»)I-

1'

~

k

4

1

tJ.I

P

(l- W )

IT jfl+ O',

sern oe lJHCJ10 ; m = p, eCJlH p -

lJeTHoe

,

r ,ll.e

k

4

= (

i'-

l (k

1

+ kg))J-

1'

k~

,

6' = 6- Y(n + 6). •

IV. 1)

Mi

s(0) ,2m <

00;

2)

,ll.JIH

r = 2, ... I m npn

tJ.

-+

00

I'lonyr no

113

(4.2.14) ,ll.JlH JIlo 6 0 r o MJIO>KeCTBa D E

~

fl

H fl. H.

su

p

~

IS;U'h(' »

(X

(I ) - X(f>, ' " , X

(f-I

) _ x'", 0)

ctx(l

, ... dx'" =

nenpepu anoro

rJO

JIH

~

IWh (x) Idx c

nenyer

ouenxa

IIEI<.

c/

D

=0

(!

tJ.If

/\

(4.2.11)

b

Yj

)(h, T) = M I

~

(I

w h+'t' (x) 1- I

Wh

(x)l) dx I

P

~

Il

4

ID jP 1T (

TO

.

D

O'leBIl,ll.

1I0, (4.2.9) I3bInOJl HHeTCH, eCJIH M

1

~

(

0)1

2

P

<

00

H

2P

(4.2 .16)

Mis

("C)

- s (0) 1

~

ksl T 1

2fl

-tO

.ll.JI5I

nexor opor o

6>

O.

Jloc

TaTO q -

n OCKOJlb KY

aupaaa

rs

CeMHI1

HBapl1a

HTb

l sucox oro 1I0p H,ll.Ka

HblM AJIH BbInOJlHeHHH yCJIOBHH

(4.2. 11) HBJlHeTC5I npennoncx enae

n OJI H

Wh (x)

'lepe3

CeMI11lHBap l1a

llT

bI I1CXO,ll.HOro nOJlH S(x),

Bo06

·

06

uarerpupyea ocr u CeMHH HBapHaHTOB S;WIl(' »): eCJlI1

uie rosop

a,

He np

OC

TO, nposep xa YCJIOBI1H (4.2.11) HJlH (4.2.12)

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060HTH,

eCJlH npenno-

(' (Wh(

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(1)

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(I)

d (f - l ) 2

sup J ISf (x , ... I X , d

x...

X <

00,

(4. .12)

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lJTO C. B. S (0)

oonanaer

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ro

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4

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00

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113

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ell

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aa

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tJ.

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tJ.

-+

00

11

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W2m

a~)

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or,

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6

1

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- )

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Ifl

-tO

'. (4.2. 17)

IV. T or

na

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1),

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n, n + T EK

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ne

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tJ.

• JleHCTBI1TeJlbHO, ag')(h,

.)~

(

a

g)

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-

2

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~

k( I

tJ.

l

iJlI

n o yCJIOBl11O

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g) (h,

.)

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2

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~

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t

4

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lm 2

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s

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I I1I

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I- I, 2m). (4. 2.18)

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6

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7

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I, III

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12 - 6· 110

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176

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

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