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

HU

5J

a (r) orpamrsennoro n

OJl51

'1(x) = X

(~

(x) > a)

y,ll.OBJleTBO-

V.H

pser YC

JlO

B

I1

IO

: a (r) = 0

(r-P

), p

;»

n. Torna npa

~

-;0.

00

1)

P

,,

(~

)-

MI..

(~)}

la

(~

)-+

N(

O

,

I). OTM

eTH

M,

4TO

,ll.JI5J

rayccoscxo-

1'

0

npoue

cca

~

(x) (n = I)

K03epQlu

Ul

1e

HT nepeMe

WHBa

HI1

51

11

(x)

M

O)KH

O oue

mrns

xepea

K.

cp

. B (x) [105] .

Ilpn Re (fl -

A.

+ 1/2) > 0, Rez > °BBe

,ll.e

M

cp

yHKUHH

;-Y

T

Ii

T-

rexepa [5]

W

,

,,~

(z) =

z~+1/2e

-

z

/2r-

l

(f.L

-

A.

+ 112)

00

j

e

-ZII

a~

-'·

-

l

/2

d

u

.

o

Teopeaa 2.4 .8. Flye

rs

1;

(t), t ERl -

cra

unonapnu f

ray

c-

COBCK

IIH npouecc C ae npepsma

ux

n TpaeK

TOp

H

51

MH,

M~

(t) =

= 0, M

1;2

(t) = I

Ii

,ll.Ba

m

,ll.bI

~

l1cpcpe

p

e

H

U

H

pyeM

o

H

4e

TH

oH K.

cp.

B (t) = L (t)

lf

rx,

a > 0, L E£ TaKoH, 4TO

cpy

HKU

HH

B(t) =

=

- 8" (t) ynoanersopaer

YC

JlOBH

5I

M:

0'

2 =

i3

(0)

==1=

0,

B(f)

=

a(

a +

I)L(t)lf

a

+

2

•

(2.4.7)

7

TOr Jlil npn T

-+

00

AJI51

CP

YHKul10

Ha

Jla

L (T) =

~

VI + [s'(t)j2 dt,

-7

5IBJl

51

10

merOC5I

«

,ll.

Jl

I1

HO

H

peaJI

l13a

UI1

I1

» cJlY4a

HH

oro npouecca

1;

(t),

- T

~

t

~

T, np

l1M

eHHMa

HOpMaJlbHa51

annpoxc

aa

auna: [L (T) -

1 1)

- 2

V2a

T W1/2,1/2 0 /(20'2)) exp {II(4a

2

)}

] {C

19

T} - /2

-+

N (0, I), r ne

00 00

C

19

= ( 16/

0'

2)

~

[(2v)I(B"( 0))

2V]

-I[jYI

+ x

2<D

(

2,,

+1)(xla)dx

tx

,,=1

0

00

X .\ [B" (c)]2Vde.

o

•

Il

pouecc Sf (f)

5lBJI

51eT

C51

rayccosc

xna

C K.

cp.

B (t).

O,ll.

HO-

MepHble

pacnpenenenr s npouecca

~

'

(t)

HM

e

lOT

nJIOTHOCT

b

<p

o.oJ (u)

(CM. (1 .1. 3)).

113Y4

HM

nep

su

e rpn K0

3ep

cp

HuHeHT

a

pa3J10lKemm

CP

YH

KUI1H

G(a) = VI + u

2

B

p5l,ll.

<I>

ypbe B npocrpancree

i,

(R1,

ffJ

o

.o>

(u) da) i

Co

=

V2aW

1

/

2

•

1

/

2

(II

(20'

2))

exp {II (4a

2

)}

==1=

0.

C

1

= 0, C;

==1=

0.

f

Hc

n

OJl

b3

Y5J

AJI51

c}i

YIIKUH

I1

(2.1.3)

COOTH

ow

eHH

e (2.1.5), M

OIK-

HO sarmcart,

ra

xoe B"IpalK

eH

lle

,ll.Jl

51

,ll.HCnep

CI1H

:

00

DL

(T) = (4/

0'2

) ~ [(2v)!

(8

" (0))2"1-

1

X

\,= 1

78

00

T 7

X [j Y I + u

21l(2v+I)

(ala) dut j j [B" (I t - S

D]

2V

dids.

o _

7-7

I1CXO

,ll.5I

H3 JleM

rMbI

1.4.2 npa n = 1 (CM.

3aMe4aH

He 1.'1.1),

00-

Jlyq

aeM

r 7 27

j j [B"

(I

t - s I)]2\'dfds = 4T 5(1 -

al(

2T)) [B" (u)f "du,

-7-

7 )

OTK

Y,ll.a

npn T

-+

00

DL

(T) = C19T (I +

00)).

Teneps yrsep

ac

-

neane

reopexu

surexaer

11

3 reop

ea

si 2.3.1

.•

Ilpaaenea

,ll.

OCTaT04Hble

yCJI

OB

Il

51

BbmOJIHeHHH (2.4.7). D y

CTb

K.

cp

. B (f) = L (f)

/t"

L E£, 0 < a < I

,ll.Bam

,ll.bl

,ll.

H

cp

¢epcHIHl-

p

yen

a. Hac uHTepecy

lO

T

YC

Jl

OBH

51

aa

B'

(f), npa

KO

TO

pbIX

B' (f) = - al: (f)/ f

rx

+

1

•

(2.4 .8)

CJI

e,ll.

yeM paccy

>K,ll.e

H

H51

M B.

¢e

JIJlepa [122, c. 503). DPH °<

<a

< b H t

-+

00

5.

b

[B' (fy) fiB (f)] dy = [B (fb) - B (fa)]/B (f) = [L (fb)/(tb)a -

•

(2.4 .9)

- L (ta)/(t

at]

{L (f)/far

1

-+

b-

rx

_

a-

a.

n YCTb npn

cp

HK

cHpoBaHHoM y > °

sup

aacenae

tjJ

(t, y) =

= B' (fy) tIB (i) orpanase no. Torna no re opeae XeJlJlH 0 s

uo

ope

[122, c

.30

7] cyuiecr ayer nOCJle

,ll.

OB

aT

eJlbH

O

CTb

{f

r

},

r = 1, 2, ...

raxaa, 4TO lim

'!J

(fr' y) =

tjJ

(y) B T04Kax

Henpep

bIB

H

OC

TH

tjJ

(y).

t

,~oo

Hs (2.4. 9)

c.n

enyer,

4TO

tjJ

(y) = -

ay-rx-l.

Ilocnennee COO

THO-

tneaae He s

aBH

CHT

OT

BbI60pa

nOCJl

e,ll.OBaTeJlbHOCTH

{/rl, n0

3TOMY

lim

tjJ

(f, y) = -

ay-rx-I.

DPH y = I

nOJlY4aeM

(2.4 .8). O

OB

TO-

1---'>

00

p5151

3TH

paCC

Y*

AeHH

5I,

3aKJl

10

4aeM,

4TO

(2.4.7) cnpasennnso,

eCJI

H

npa JlI060M y > °

cp

yH

KUH

H

tjJ

tt, y) II

tjJ

(I, y) = B" (ty)f/B' (f)

orpannaensr. 370 Bbm

OJlH

5I

eTC

51,

uanpnvep, xorna ¢YHKU

I1

II

B (f), - B' (t), B" (f) neorp nuar

e.n

sn u H MO

HO

TO

HHbI

npa i > fo'

rne

fo>

O. .

2.5. YCJJOBHSI PE.ll.YKUIHf

B3BEIlJEHHbiX

If>YHKUI10HAJlOB

yb

Y4

HM

ep

YHKUl1

0HaJlbI

sana

~

g (x) G

(~

(x)) dx, r ne g (x) -

toO,)

BeC

OB

aH

epY

H

K

U

H~.

~

(A.

) - pacumpsnouteecn cenea

cre

o 06

J1a

CTei'

l,

~

(x), x ER

n

-

cnnopoztaoe H30Tp

ODII

oe rayccoscxoe none.

79

Ilpn

YCJIOBH5JX

I, III, IV paccvorpaa CJIY4

aHHble

npoue

c

cu

V

,.{t)=

I

g

(x

){

G

(

~

(x

)

-

Co

}

d

x

,

t El

O,

I],

~

(

),.t1

1

1L

1

r

ne

g (x) - HeCJI

y4

aHHaH

<P

YHKUII

51

.

IX .

nY

CTb g (x) - g

(I

xl) - pana

ansaas

nenpepuauaa, n

o-

.

TJ

O)l<HTeJIbHa

51

npn

lxl

> a

<P

YHK UH5I raxaa, 4TO AJI

51

neic

oropo-

ro uenoro m = r ang G

>=

1 cyuiecrsyer nOJIO

>K

ltTeJI

bHb

lH

rr

pe-

lleJI

= m, g, M =

fill

lim

~

r [g

(A

I x \) XC

20

C

20

in,

,

~~

OO

J

'"

D.

x g

(A

IY

I)

/

g

2

(A)J

[B

m

(A

Ix - y D/B

m

(I,)Jdxdy,

rne B (lx

l)

B

351T

3 H3 y CJIOBH5I IV npn a E (0, nlm) .

ECJII1 B

bIn

OJIHeHO y CJIOBHe IX, TO

<PY

HKUHIO g (x) 6 YAeM na

su-

B3Tb nonycru voa

,a.JI51

K .

<p

. B (x). Hanpnuep ,

<P

YH KUIl51

gl

(\xl) =

= I X I

13

L2( Ix I),

~

>=

0, L

2

EB nonycr naa

nJI51

K.

<p.

B ( Ix I) =

= Ix /-

O:L

(I x I), a E(0, nlm), L E

53,

(j

m,

gl'

6) = ml

~

~lxl

I3IYll3lx

_YI-

o:

mdXdy

.

c

2

0(n,

D.

Ilp

n

~

= °c

20

(n, m, g

l'

6) = C

2

(n, m, a, 6) (CM. (2.1.8)).

B'l

aCTHOCT

ll

, (l , m, g l' [0,

1])=

c

2

(l , m, a, [0,

1])=2

/

[(1-

c

20

-

mal

(2 - ma) ].

Teopeva

2.5.1.

Ecnu aunonu

enu

y CJIOBH

51

I, III, IV, IX,

TO npa A

-+

00

DV

d

l)

=A

~,(A)c2

o(n,

m, g,

6)(

1+

0(

1)),

(2.5.1)

r

ne

A;n

(A)

=

C

~~

,

2n

8

m

(A)

g2(A)/(ml)2,

npH4e

M

npe

AeJIbHble

pa

cnpe-

neJIeHII5I c. B. Y

t..,

g (l) = V

dl)

/[Am

(A)

Vc

20

J II C

.B

. YmJ..g(I) =

= A-

"B-

111

/2 (I,)

«:

1.

)

~C

2

0}

-1

/2 s

gn

~C

m}

j g (x)

If

m

(

~(

x))

dx COB'

Ml.

l

na

n alOT.

..

Co

orn

ouienue (2.5.

1)

rtpn G(u) = H

m

(u), In

>=

I cnenyer

113

IV

npa

a E(0, nlm) H IX. B o 6

111

e M CJlY 4ae

D[

j

g

(

X

)

G

(

~

(

x

)

)dX

I

=

5\(1.)+5

2(1.

),

D.

(AI

r ne

np

n

,,-+

00

B

cun

y JIeMMbI 2.1. 1

11

y CJIOBIl51 IX

51

(I,) = C

;n

(m

l)-

2,V (

~

g (x) Hm

(~

(x)) dx l2=

A

~

"

(A) (I -;- 0 (I )).

1'.(/.1

(2.5.2)

so

Onapasci, na (2 .1.2), aaxo

znu«

5

2

(

1.

) = :

2:

C

~(q

l)-

2

A1

[

r

g

(x

)

H

q

(

~

(X

)

d

x

f~

Q=m+

I .D.(t..l

00

~

(

~

C

~

(q

!)-

2).f

.r g (x) ,g (y) B

m

+

1

(

Ix - y I) dxdy

~

o

=m+1

<1

0.) D.(A)

~

k I (n, m,

(X,

g ,

t1)

A;

,,

+I

(A).

(2.5.3)

K aK H npn nO

Ka3

a TeJl bCTBe

.n

e

M

~

lb

rr

2.1.4, MO:iKHO fJOKa33

Tb,

4TO

np

l1 A

-+

00

Ml.\ g (x) H

m

+d t;(x))dx j2 = 0

([

r

g(x

)

Hm(~

(

x

))

dx

]

2

).

D.

(A' MAl

Tor

na

H3 (2.5.2), (2.5.3) c.nenyer (2.5.1), or xyna uerpynno no-

JlY41H

b COOTfWWeH

lle

lim M I Y

),.

, (1) - Y

m

.

A

,

1?

( I) 1

2

= O. •

A

-->o:>

Teopesta 2,5 .2. ECJl Il BblnOJl

He

Hbl yCJIOBH

5J

I,

III,

IV npn

a. E (0, n/m), g (x) = g (Ixl) E

53

- pannansuas . nenpepuanaa ,

noJIO>KI1Te,1bH a5J npn [x] > 0

<PY

Hl<U Il5J, TO npn A

-

~

00

DUd

I) =

A

~

(

A

)cz

(

n

,

m, a, t1)(1 + 0 (1)).

(2.5.4)

Ilpenenu «ouesnovepnux p

ac

npe,!!,eJleHI1H CJIY4aHHblx npoueccos

Y

t..

.g (t) = VA(t)/[A

m

(A) V C

z

],

t E[0, I ] H CJI

Y4aHHbIX

npoueccos

nB-

m

Ym.dt)

=

A-

12

(A){C

2

}- I12sg

n{C

m

} r

H

m

(

~

(

x

))

d

x

COBna·

M A/l l

n

,

ll.a fOT.

..

l-tcnonsays reopea y 2.2.2, aHaJlO

rl

14HO nO

K333

TeJlbCTBY

JIeM

Mb

l 2.

1.

4 nOJI

y4

ae M (2.5.4). n YCTh

tj

E[

O,

1J, a

jE

R I,

Rj(tj)

=

p 2

=Y

l",

~(ti)-Y

'.1

.

dt

i)'

/ = 1, ... , p, Tor na

M(2:

ajR..(tj»)

~

1= 1

r,

~

a;

~

M R( (t

j

) .

KaK

Ii npn /I.0Ka3aT eJlbCTBe Jle

MMbI

2. 1.4,

1= 1 /=1

HCI10JIb3

Y"

r eop

eay

2.1.I, n

erpynn

o n

ox

as

an.

, 4TO

,

~

r

~

M

R

~,

(

t

j

)

~

,,

~~

{

k

z

r

j S Ig (Al

xl

)l

g(

A) -

'\ (tl/

")

A«

li

n

,

I /

-l

l lg (Al

vl

)lg (A

)-I

Il Bm(A' x - y !)IBm(A)JdxdVF

12

+

+ k

s

[A;n.

t-!

(A

t

Y")

IA

~

11.)]

liT = o

.•

6

-6

-110

81

Ilpnxepa npene.nsnux pacnpenenennf C. B. Ym

.)

... g (I) npa

i..~+oo

,llaHbl B n . 2.10.

z.s. YCJlOBH5I

PE;I,YKUHH

<I>~

IHKUHOH

AJlOB,

3A BH

C5lJJ..lJiX

aT

nAPAMETPA

11s

YIlH

M <PYHKUHOH

aJlb

l sana \ G

(

~(

x),

t. (

A))

dx, \ G

(

~(

x),

",(,,) "'(

,..I

x) ax, rne t. (

A)

- pac

uinpmo

uieecs

ceMe

ik

TBo

06

JlaCTeH, S(x),

x EIt -

onHOp

O,D,H

Oe nscrp

ona

oe rayccoscxoe

none

Pacca or pux nsa THf1a

<PY

HKulI

OHaJlOB, SaBHC5Iw,HX OT n ap a-

MeTpa .

Ilepsu f

nw

yn

OBJleTB Op

51

eT yCJIOBlflO '

X. G (u, t.) , u E

R)

, t. E

lS

n-

,lJ,e

i1cTB

lf

TeJlbHa

51

HeCJIY

'IaH

Ha

51

<PYHKUH5I

raxas

,

'IT

O MG(s

(O)

, t.) = Co(t.)< oo, M

G2

(S

(0). t.)< oo.

DPH

YCJlOBHH X

<PY

HKU HIO G(u, M MO)j{HO paano

sorr

s B

P5l,lJ,

00

G (u, t.) =

~

c,

(11

) H q (u)/q!, c, (t. ) = I G (a,

~)

cp

(u)H q(u)du,

Q= O R'

q = 0, 1, ... , CXOn

51w,lH1C5I

B npo

cr

pan crne L

2

(R\ qJ (u) du).

(;'l

IlT

an Bb

mo

nHHMbl

Ml1 yCJlOBH5I I, X, n OJlO)j{HM T,..(t) =

=~'

G(s(x) ,

I1(A

)

dx-C

o(I1(A))O

,nl~

l,

t

ErO,

1]. Hac

6y

-

6(1..

t

1l n

)

n

yr

nnr

epecoaar s yCJlOBH5I, npn KOTOp blX on

aox

epnsre (

Hl

lH

KO'

He'1HOMe

pll

ble) npenensnsie pacnpeneneuus npoueccoa X

1..

(t) =

= T

dt)

[IC

(

I1

(A

))ja

(A

)(ml)-l

j

- l ,

tEl

O, I] Ii npoueccos

m

X

m

.",

(t) = sgn {Cm (

11

(A

m a; ! (A) I n; (s(x))dx, t

Er

O,

1] COB-

6(1..

t

1/n

)

nanaior A

nH

nexor oporo uenoro m ;» 1.

XI. Cyuiec

raye

r uenoe m ;;;' 1 raxoe,

sr

o

lim

DT

1..

(I)

/[C

~

(t.

(A))

a

~

(A) (ml)-2] < I.

~~

OO

JleMMa

2.6.1. Il pn YCJlO

BH51

X I. X, XI H A

--

00 npenensnsre

pa

cn

penen eu n s c . B. X

1..

(I) H

c.

B. Xm, l. (1) C

ClB

n

ana

lOT.

II.

fI

0 nO)KI1M R

1..

= X

1..

(I) - X

m

,

1..

(l).

B C!l

JIy

JleMMbl 2.1.\

DX",(l )

=I

+DR

1..

' 113 H

ep

aBeHCTBa

DT

,,(l)

;;;'C

~(I1(J

,,

))

a

~

n(i..)(ml)-

2

II YCJlOBH

51

XI c

ne

ayer.

tIT

O limDX,..(I)

=\'

n03ToMY lim DR

1..=

~

....

-

s-

cc

",-).

00

= 0, OTK y,lJ,a H

su

rexaer rpeoyea oe.

II!!

XiI. Cyuiecray

icr

ue

noe

m > 1 Ii a E(0, n/m) T

Cl

J<He, 'ITO

.lI.

Jlfi

K.a

il<.lI.or

~)

I E[0,

11

!i m

DT

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f

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12

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i

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cnenyer

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2.7. 1.

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}/qpz

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Z

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1

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a

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~.

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1

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~.

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f)

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r

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2

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X

J

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2r

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8+1l)

.

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hl

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s

r

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Il

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rt

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1

_

Z

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112

) B (Z) d

zJ

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o

2r

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rt

6

~.

z

rt-I

I

I-

z'/(2'"

«n

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o

Flocaennee

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e

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le CTpeMHTC51 K

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VII

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m = I.

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ou

e HHM,

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11

51

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o

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0). T or

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a

2, 8( zl

f "

~.

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11_z' /(

2r)'

«

11

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2

(r) X

~

P

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l r

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I

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+

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2

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z

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.

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p

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a

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r

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II

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6

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2.)'((n + 1)/2,

112)

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o

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r),

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o

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e L (a, b,

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ep

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- lp (a, b, - Y)]I + r

ep

(b,

b, y) -

q>

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o Jim 51(f )/A

hf)

=0

.

'->

00

P aCCMOTpHM

52 (f ) = C

t

(n, 2) f"

2r

f Z

,,-I/I-

z' /(2,)' «n + 1)/2, 1/2) dz X

(2,)

13

~

X I L (a, b, y) dy = C

1

(n, 2) f " 5 Z,,

-I

f \

-z

' /C2, )' «n +

o (2r) B

B

(l

)

+ 1)/2, 1/2) dz S

[{

cp

2(a)

"'

~)

(a) +

2<p

(a)

rp

(b

) X

o

X exp { - (a

2

+ b

2

) y2/(2

(I _ y

2))}

'\jJ

~

2

)

(a, b) +

+

cp

2(b)

1\J

~

)

(b)}/V I -

y2

] dy,

r

.n.

e

'\jJ

~)

(a) = e

xp

{a

2

y/( I + y)} - exp{- a

2

y/(1- y)} N

. . .

N 2ya

2

-

a

2

ys

(

a

~

- 6a

2

+ 6)/3; (2.7.13)

'\jJ

~

)

(a, b) =

ex

p {aby/(l - y2)} - ex p

{-

aby/( I - y2)}

C')

N 2aby + ab

y

3« ab

J2

+ 6)/3 (2.7.14)

npn

a'~

'

Y

~

0, aby

-+

0, b

2

y

--

0 (cnpaaezi

naa

ocrs

nocnemrero CJlEt-

~

zi

y

er

H3 VCJl

OBliq

XX). Tor na

2r

S

o.

(f l/A! I(f )

:(;jC

] (n, 2)

f"

rz" - I" - " Mr)l ({n +

12r

\

1

'~

8( z,

+ 1)/2" 1/2) dz r

2(Q<jJ

(a

:)-

b<p(b

))2 ydy

lAi

(f},

o

n 03TOMY

-

lim 52(,f)/A

2

1

(r )

~

1.

' -> 00

T

al

<HM 0

6pa

30M

, reopena c

.neny

e r

113

JleMMbJ 2.6.1 npa m =

= 2.

t'!1

Teop

ena

2.7.7.

Flyer s B

bJITO

JlHeH bl YCJlOBl1

51

I, VII npu

m=

= 4

11

yC JlOBI1e XX, B I<OTOpOM H8 jl,O

B351

Tb 8 H3 YCJlOBHH VII

l1 SClMeHHTb (2.7 .11)

COO

TH

OlIle

HI1eM 0 < b (f) < 1 < a (r) < 00 ,

a (2,7.12) - COOTH

oweHHeM

a (r) lp (a (r))

,...."

b (f)

<p

(b (r)) npn

r

-

~

00 . Tor na npenensa oe pacn

pe

nenenne c. B.

[T

~4

)

-

M

T

~4)

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~Az

(f ), rne A

z

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rp

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2

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b2

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TaKOe

IKe, KaK H C. B.

rH 4 (l; (x ))dx to 4 (f ).

(2.7.15)

u(r)

A

3a

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rr

ro B y CJIOBHHX reopevsr C

4

(0

(f

)) N 2 (a

2

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- b

2

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a

2(jlz

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or

na npa r

-+

00

(2 , )tl

zr

D

T

~4

)

/[

C

i

(0

(f

))

O'

~

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1

=cd

n,

2),

" rr+ S ]

Z"-

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- " /C2II

'

X

o r2r,

fl

B(z ,

X «

n+

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2

+ L

s

)

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o

np

ns

e», HCrrOJlbSY

51

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lle

HHH, saeneanu e B JleBbIX

sacrsx

(2.7.13) H (2.7.14), rrOJlY

lla

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2,

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2)2

c

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(n,

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j' z"- II

1

_

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72,

1

72)

dz X

l

(

2,

)8

BCz)

X S

(1

- y2)- 1

12

{a-

2'\jJ

~

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(a) - 2 (aiJ)-l

exp

{-

(a

2

+ b

2

) y2/[2

(1-

n

-

y2

]}

'\jJ

~)

(

a

,

b) +

b-

~

'P

~

)

(

br

dy/[(a

2

-

b

2

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0'4

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~

2J

:(;

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1

(

1'

·, 2) f " f z,,

-l

f

l

_

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.:(2

Pj

, «fl + 1)/2. 1

12)dz

X

(2r)

~

!l5

H\2.

X

~

[y3

{(

a~

- 6a

2

+ 6) - 2 (a

2

b

2

+ 6) + 6 (a

2

+ b

2

)

+

o

+

(

b~-

6

b

2

-1-

6

)}1

3j dy [(a

2.-

b

2

) G

4

(r)j-

?<

I.

K3

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suui

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2 - (4)

110

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ll

fl H3 A

2

(r ) CTpeM

HTC51

K uyruo, C

JJ

e

.ll.

OB

aT

e

JJ

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'

->

00

i

A

~

(r ) < 1,

11

yreep

ac

neuue reopewu c

ne

nyer

11

3

JJ

e m

lb

l 2.6. 1

npn m = 4. III

2.8. YCJIOBIUI PE,UYKUIHI XAPAKTEP HCTHK n PE BbIWEHH

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PA,UHAJI b HO A nOBEPXHOCTH

HC

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M

e

p

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np

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i

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1l

5l

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H

O.ll.HOP0.ll.HbIX ns

orp

orm

u x

raycccscxnx

cny

qaH-

t1blX nOJlCI1 C CH

nbH

Ofl 33 BHCIIMOCTblO. PaCCMOTpHM epy

HKUIi

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Ha.']

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= I {x Ev (r):s (x) > a (xl) I=

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X(s (x) > a (x)) dx,

CIT)

rne

a (x) = a (I x

I)

-

Tl

OnO)J\HTeJJbH3

51

pa

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bH3

51

q

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YHK

U

I151

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ynOBn eTBOp 5lI-oW.3

51

yc

no

s «10 X

VI.

Oves Ii.ll.HO,

r

MN

~I

J

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~

Z,,

-1

[1 -

CD

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))

1dz.

o

r

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2.8 .1. B y eJlOBH5Ix

re

opesu» 2.7. 2 o. B.

[

N

~

\)

-

!V

li

V

~

I

)

J/

A

l

(r), rne

A7(r ) = S I

<p

(a(x

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B(

lx

-y

l) dxdy,

qr

)Vrr)

np

n r

-+

00

Il

MelOT 3CHMnTOTHQeCKH HOpMaJIbHoe pac

npe

zie.neuae

N (0, 1).

•

<P

y HKUH

51

GO)(u, x) = X

(u>

a(x))

Y.ll.OBneTBOp51

eT ycnoaaro

XIII,

npHq

eM

Co

(x) = 1-

cD

(a (x)), C

1

(x) =

cp

(a (x)) Bb1

6p

3B

~

113

(2.7 .3), p3

306beM

MHOlKeCTBO V = v (r) xv (r) H3 )1,81:

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=

= {(x, y) EV. Ix - Y I<

(2r)

~},

,1z = {(x, y) EVi Ix - y I>

(2r)

~}.

Hcn

oJIb3Y51 JIeMM y 2.6. 6, nOJIyQaeM

{;( !X

-!l

li

DN;l) = J ) d

XQ

Y S

<p

(a (x), a (y), p) dp =

SdrJ

+ S2(r),

U(r) e,f l IJ

(:16

nO

CKOJJbKY

npn

II

B

1<

1

B IBI I

5

rp

(

u,

w, p) rdp < 5qJ (u, w, p) tip

~

S

~

(u , w, p) dp <

I,

n 0 0

TO

B

(lx-

yl l

Sd

r

) =

'\sd

xd

y 5 qJ (a (x),

a(y

),

p)dp<5Sd

xdy<

Il, 0 Il,

< k

1

(

n

,

~

)

rn

(l

+tll

. (2.8.1)

H o B Y

CJlO

BHHX

re

opeau

A

~

(r) >

(2n)-'

e- a' (r) 55B (Ix - Y I)dxdy

~

vC

r) v(n

2r

-"({,-fl)+,,

r

,,-1

I ) 1

~k2(n)r

J Z

l-z

'((2r).((n + 1 12, 1

2)B(z)dz,

'I

(2.8.2)

ra

e

~,

~-1i3

(2.7.3).

n03TOMY

1i3 (2.8.1) Ii (2.8.2) cn e

nye

r,

4TO

51

(r)IA~

(r) < k, (n,

~)

r',(I

+fl

) (

li

z(n) r,,(

I+fl-lI)

X

2r

X 5

z,,-

II I_Z'( 2r)' {(n + 1)/2, 1/2) B (Z) -r'

o

nO

CJle

JJ.Hee

astpaaceuae

CTpeMHTCH K HyJJf{)

npa

r __

00

B CHJIy

yCJIOBIi51

VII

npa

m = I.

V1

cn

oJlb

3

~51

T

O>KJJ.eeTBO

qJ (a, b, p)l

n<p (a)

<p

(b))

=exp{

(a

2

+ b

2)p

/[ 2(1 + p)j}

exp{-(a-b)2p

/[

2(l-

-

p2.)j}

/. V I - p2,

JJ.JlH

JIlO6or

o

e>

0

npa

6oJJbw

OM r HaXOJI.IiM

8(\

x-

yl l

.s,

(r) =

Sf

dxdy

j'

~

(a (x), a (Y), p) dp <

Il, n

B OX

-ill

)

< SS

rp

(a (XI)

<p

(a (y)) dxdu 5

(J

+e) exp {[(a

2

(x) +

~

0

+b

2

(y)]

p [2

(I

+

p)]-I}

dp < exp [k

3

(

~

)

a

2

(r)

sup {B (z)}} X

2>rfl

X

55

rp

(a (x))

~

(b (y)) B

II

x - Y

I)

dxdy. (2.8.3)

Il ,

n OCKO

JlbKY

lim

e:p{k3(~)a2(r)

sup

{g(

2)}} = I, TO

npa

60JIbWYlX

r-

H>o z>rfl

7 - 6-110

97

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

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