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

C

XO

A

51

W.

IIJ

~

1c5

1

B npOCTpalHCTBe L

2

(RI.

<p

(u) au),

npH4e

M n CHJlIY pa-

Be HCTBa Il apcesan s

~

C

~

/

ql

= MG2(

[;

(

0»)

<

00

.

(2 .1. 2)

q=O

II

1.

¢

yH

KUHH G

(u)

ynoaner sopaer

yc.n

OBHH II H cy

uiecr

ayer

UCJIo e m> 1 TaKOC.

'IT

O C

l

=

...

= C

m

-

I

= 0, C

m

* O.

ECJIH !3

bHl

OJIH

eIl

O III. TO m Ha 3hlBa

eT

CH p

aHrO

M ep

YHKlUIH

G

II 0

60

3Ha

'lae

TCH m =

ra

ng

G. Hanp

au

ep. <p

yHKUHH

G(u) =

li

m

(U).

III

> 1 HMce T panr m.

B ,naJIbHCIIWeM HCfTOJIb3YCTCH nJIOTHOCTh

rp

(U, v, p) )l.ByM ,ep Ho -

ro rayccoacxoro !3eKT Op a (

S,1'])

C M I; = M

1')

= 0.

M~

2

=-

M11

2= I,

MS

11

= p. O Ha HMeeT B

lI)!

2

<p

(u, v. p) = ex p

{-

(u

2

+ v - 2uvp)/[2 (1 - p2)]}/[ 2n I

II

- p

2J

.

(2. 1.3)

JleMMa

2.1.1. E CJlH (

S.

11

) - rayccoecxun sex ro p a nJIOTHO'

CTbfO (2.1. 3), TO AJIH scex m> 1,

q>

1

MH

m

(s) H

q

(1'])

=

I)~p

mml

. (

2.1.4)

A. Y'l3Bec THo [110, c . 25 1] cnenyiouiee pasnoacenue epyHK UHH

(2.1.3);

00

q>

(u, v, p) = (

2n)-

1ex p

{-

(u

2

+ v

2

)/2}

~

ovH; (u)

H;

(v)/vl .

v=o

(2.1.5)

C

YlJe TOM (2.1.1), (2.1.5) HaXO,QHM

MH

mCs)H

q(T])

= 5H

m(u)H

q(v)rp(u

.v,p)du

dv =

R'

00

=

~

pv

.\

n; (u)

Il

; (u)

<p(u)du

.I

H

q

(v)

H;

(v)

<p

(v)

dV

=I)

~

I

P

"'

ml

. •

v=o RI

[

~

I

6 y ,neM HCfTOJIb 30 B

aTb

060

3

Ha'l

eHHH n.

1.4

Ii 1.5. n o JIeMMe

2. 1.1 npn m > 1 nonyn aea aH a JIOr HlJH O (1.5 . 1)

a;,

(Ao)

= M

.I

Hm (

1;

(x) dx 5Hq (I;

(y»

dy =

6 (AI 0,

(i

,,1

deAl

=

m

l

l)~

~

B

m(1

x - y

J)

dxdy = ml

8~

I~

2

f"zn

.I

B"'(z) dF

t.tJ.,1

(Z

),1

o,(A)

t,(

)") 0

6YlleM

nu

c a

n,

a

~

(r) BMCCT O a;l

(A),

xorzu

~

(A)

ecr s w

ap

v

(r).

58

n o .'1

eMM

e 1.4.2

Zr

a

~(r)

=

clr

n~z

n-IB

m(

z)I

I

_

2

'

/

/ (

2r

)I

«n

+

1)/2, 1I2

)dz

, (2.1.6)

o

rne

'C

l

= C

I

(n. m) = 4m! m"r -

2

(n /2)

/l

-l.

(2.1. 7)

a Ix (

p.q)

onpenense

rcs

no

(104

.8).

H

an

oMHHM, 'ITO ep

YIlKUI1

51

L : (0, 00)

0-+

(0, 00) Ha3bIl3QeTC

51

MeA-

JIeHHO MeH

51IOw.eH

C

51

ua 6eCKOH

e'lH

OCTH, eCJIH ,nJIH scex

A>

0

lim L (At)/L (t) =

I.

Hanpuxep, epyH'KUHlI ln at, lnln« (I +t)

MeA-

t- .

oo

JIeHHO MeH

51

IOT

C51

Ha 6eCKO

He'l

HOCTlI ,nJIH JIl

oGo

ro

epHKCHp

O!3a

llHOrO

a ERI.

Cdio

pxy

na

pye»

aexoropue asae

cruu

e CBOHCTBa

w.

e,nJIeHHO

r-feH51IOIUH XCH epYH

KUlIH

.

Teopeaa 2.2 .1 [52, 103], n YC

Tb

L - mrr

erp

upyea as na JIIO-

GOM KOHe

'l

HOM HHT

epB

aJIe Me,nJIeHHO

MeH51

IO

U~a51CH

na 6ec KoHc4 -

HOCTH epYHKl

lll

H. Torn a: 1) COOTH

ow

eHHe L (At)IL (t) -->- I npu

t

0-+

00 BbIDOJIHeHO pasnoaepno OT

IIOClITe

JIbHO AE[a, bJ, 0 < a <

< b «; 00; 2) ,nJIH JI

I06

0ro

I) > 0 r

t.

(I)

-,.

00, r

6

L (t)

-

~

0 npn

t

~

00; 3) ,nJIH JI

I0

6bl

X 0< a < b < 00

~

~?:

'"

.

b

{L (ls)

IL

(t)} ds = J

b

}

~

{L (ts)IL (I)) ds = b - a;

a a

4) ,n

JI51

acex 0 < a < b< 00 1im sup IL (s) L- I (I) - I I= 0;

t

-*

oo al< s<bl

5) eC

JIlI

L - n

nre

rpnpyexaa no

P

H

~

a

H Y

HJIl! J le6e r y

(B

n OCJI

eA-

HCM cnysae B onpenenenuu epyH KU H

ll

npennonaraercn ee

11

3Me-

pHM

OC

Tb)

B

Ka)j{

)l.OM KOH

e'lH

OM nnr epeane [a, b

J,

TO L nonycxaer

KaH OHH4 eCI<oe npencraanenue L (t) = C(t)

ex

p

{f

e (s) S- l

dS

}, rne

c (t) , e (I) - aexor

opu

e epy

HKUHIl

, T

aKH

e,

'IT

O npu t -,. 00 c (t) -

--+ c > 0, e (t) .-+

O.

n YC

Tb

B - KJIaCC Me,nJleHIIO Me

IlHl

ow.

HXCH aa 6eCKOHe lJHO'

er

n

¢

Y

II

KllH~

I

,

orp

aanvennsix n Ka)K,n OM KOHe

'l

HOM HHT

ep

EaJIe.

IV.

nJIH

nex c

rop

or

o a > °B (Ix D= Ix

l-a

L (Ix i)

npu

Ix I>

> 0

11

Ix I_ 00 I rne L EB.

JleMMa

2.1.2. ECJlH B

bIn

OJIHeHbI yCJIO

!3I1H

I. IV

np

n a E (0, nl

1m).

n>

I, m> I, TO cyiuecrnyer KOHCTil

HTa

" ... Cln l

C

2

= C

z

(n, tn, a,

~)

= m! . J Ix - y 1- dxdy =

\

0, I>

d( o,)

2

= m! I

~

1

.I

z-wndF 0, (z),

(2.1. 8)

o

59

T21Ka5l, qTO npn

'}.,

--+

00

a

~,

('}.,)

= c

2

'}.,

2,,- maLm

(J..)

( I + 0

(1

).

'(2. 1.9)

. B bl

no

JIH

Ii

M aa xieny nep

eae

anux U

=Z

/'}.,

B (2.1.5)1. He-

nOJIh

3

Yfl

COr JIa CO

B3

HHOCTh p

annoa

epuoro

pacnpenenemaa C

npe

oop asoaaunen ronor er

nn

Ii

cocrnourenne 3) reop erau

2. 1.1,

nO

JI

yYaeM

(2.1.9) . •

E CJI

Ii

MO>l, HO

HaHTli

H

BHoe

asrp

axce

u

ne

j(JI

H epy HKU

Hl

i

pac

-

np enca

euas

F",(z) = Fp

a.B

(z) ( CM.

JIeMMbI

1. 4. 2, 1. 4. 6) ,

Bb

l 'l HC

,l

eH

ll

e

KOHCTaHTbI

MO)K

Il

O

rrpo

n o .rl

)KIHb

. Ha n paa

ep

, ec-

CJlH

6.=0(1

) - urap, TO s

aaeua

neperaeunsrx

u=z

/2r B rpop -

My

JIe (2.1.6) Ii a CII M nTO

Tl

I QeCKHH 3 Ha JlH3 5lTOH ¢ OPMY Jlbl C

HC

nOJIb30B

aHH

eM

reopexru 2.1.1

npIiBO)l,51T

K

rax

oxry yraep:«-

JJ.eHHIO .

JI

eMM

a 2.1.3. Il

ycr

s

BhIlI

OJIHe

HhI

I, IV H r -+-oo. Torn

.a

npn

a E (0, n/m)

a;, (r)

=c

2r2n-maLm(r)(1 +

o(J)),

(2.1 .10)

rne c

2

= c

2

(11, IU , a, v (I)) = m! 2,,-m

a+l

n

n-l/

2r ( n -

ma+

I )/2) X

X

[(n -

mal

I' (n/2) r « 2n - mix

+-

2)/2)]-1 .

flpli

a = nltn

O

~

,

(r) =

crLm

(r) In (2r) (1 + 0 (1

»).

(2. I . J I)

Ilpu ct> nlm

00

C

a

(n, m) = I

zn

-I

IB (z) r'dz <

00

(2. 1. !2)

o

00

Ii , eC

JIIi

C

4

= c

4

(n, m) = rzn

-I

e: (z)dz

=1=

0, TO

J

o

a~

(r) = c

4

c

1r

n

(I + 0 (1» .

(2.1.13)

JI

eMMa

2.1.4. Flyers B

bIn

OJIHeH bI YClI OBH5I I, II I, IV n

pn

a E

E(0, n/m), n

~

I, m > I. Tor aa npn I,

-

~

00

D l \G (; (x))

dX]

= C;,c

2

(n, m, a ,

~

)

(ml)-2

X

M I.)

X

~

2

n

-

.m

aL

m(/,)(

1

+0(

1))

.

• V!

cn

OJ

lb3y

51

JIew-My 2. 1.1 Ii yCJIO B ne III, npen cr a a a n .n.IiC-

nepcmo

13

13J.1.llE'

D l

~

G(6(x»

dx

J=

Sd

A.

J+

.)2

(A),

ac).)

eo

r n e B CH

JIy

JIeMMlb

I 2.1.2 npn

"--""

'00

51

(A.)

=C

~a

;'

(A.)

/(ml)Z=

= C;,c

2,,

2n-

rn

a

L

m

(1"-)

/(m!)2(I

+ 0 (I)), a

Bblpa)K

eH He

oc r

52

(I,)

=

~

C

~a

~

(h)/(q!)2

:::;;;

a:;'+1(A.) [(m + 1)!]-1 X

Q=

m+l

00

X

~

C

~ /

,q

l

Q=m+l

nO

CJ1

e

.n.

eJleml51 na

'}.,2n-ma

CTpeMHT

CH K H

yJIIO

npn A. --""

00

.

!l

cx-

oo

C

TBIl

Te.TlbllO, B CIlJnY (2.1. 2) k

1

=

~

C

~

/q

l

<

00

.

fl

o YCJIOBHlO

Q=m+l

'1 ,ll..'1

51

.

111

0 6

0r

o e> 0 cyiuecrn

yer

Ai

o

> 0 T3Koe,

qT

O IB

(\

x -

-y

DI

< e npn Ix - y l> A

o

·

nY

CTb

6.

1

= {(x,y)E6.(A.)X6.(A.) 1

Ix

-y

l

~A

o}

,

L\.

2

=

{

(

x

,

Y

)

E

~

(

A.

)

X

6.

(

A.

)

'

lx

-

yl

>

Ao

}

'

Torna

m

1

a

~n

+

l

(A.)

/(m + 1)1 = U+ f}B

+

(I x - y I) dxdy =

ti l s,

= 53(I,) + 54

('}.,)

.

m

+

1

Hcn

oma

ys

na

MlI

om eCTBe 6.

1

ou e ux y IB (.) I :::;; I, a n a MII O-

)K CCTBe

~2

OUeHKy

1 B (.) I

m

+

1

< e I B (.) r". nOJIy 4aeM

I5 s(A) I = I

~

B

m

+

1

(IX - yJ) dxdy I

~

k

2A."

;

0"

154 (A.) I = \

~

B

m

+

1

(IX - y Ddxdy

I

~

ek3

A.2n

-rna,

0"

r n e k

2

= k

2

(n, m,

~

,

A

o

) > 0, k

s

= k

s

(n, m, a ,

6.)

> O. B C

Il

JIY

rrp O

Il

3B0

J1

b HOCT

Ii

e>

0 aupaacen ne 52 (A.)

~

k

1

(k

2

Iv

n +

eA.

2n

-

rn

aks)

A.

2

n-

ma

rrOCJIe

.n.

e

J1

eHH5I

aa

CT

pe

M

Il

TC5I K HyJIIO

npn

a E(0, n/m) II

A.

-+-

00

. •

JI

eMMa

2.1.5. n

YCTb

BbII10J1HeHb

I y CJIOBH5I I, III II n JI5I

HeKO

-

r oporo a E(0, nlni: npn r --+

00

cnpaeennaao (2.1.10),

rne

L EB.

T or ,U3

npn

r

--+oo

D [

~

G(s(x» dX] = C

;n

(m!)-

zc

2

in, m, a,v (1)) X

vC

r)

x r

2n

-

ma

Lm (r) (I + 0 (I

».

(2.1.1 4)

• C

oor

aotneuae

<X

D

[~

G (s (x)) dx J=

~

C

~a

~

(r)/(ql

)2

(2. 1.15)

U(

r)

Q

=rn

61

npn r

-+

00

aHaJll1311pyeTCl1

aHa

Jl

OrJPIHO

)l.OKa

3aT

eJlb

CTBY Jle M-.

MbJ

2.1.4 C HCnOJlb30BaHHeM (2.1.6) H (2.1.10) . •

_ n YCTb r

1

< r

2

.

Hcnonsaya Jle

MMb

J 1.4.3, 2.1.1, npu m

~

1,

q > I nos

y-r

ae»

o

~n

(r

1

, r

2

) = M I H

m

(

~(

x)

)

dx

~

H

q

(s(y)) dy =

tJ(rd vCr,)

(1+'

2

=

B

~/

nl

2

n

/2rt r-

J

(n/2) (rh t

/2

SB

m

(z)zn

/2

X

o

X

[I

00

p-

n

/

2

J

(n

-2i

/d

zp) Jn/d

r

IP) Jn

/2

(r

2P

) dp] dz. (2.

I.

16)

o

00

V.

CY

Il

~eCTBye

T

uenoe m

~

I raxoe,

'lTO

C

3

(n, m) = S

zn

- I X

o

X IB (z)

Imdz

<

00.

JIeMMa

2.1.6. n YCTb BsmOJIHeHbJ

YU

JO

BHH 1, III, V. Torna

.n.Jll'I

scex q

~

m (m

113

YCJlOBI1H

Ill

, V) CYW.eCTByIOT upenensr

Co (n, q) = 1im

{r

-

Jla

~

(rn

11

r --'>oo

00

Cs

(n, m) =

~

C

~

C

5

(n, q)/(q!)

2.

<

00.

(2.1.17)

Q

=m

E CJII1 C

s

(n, m)

=1=

0, TO npn r

-+

00

D [ SG(s

(X

)) d

X]

= cd n, m)

(n

(I +0 (1)).

(2.1.18)

vCr )

npil

r, <

(2'

r, + r

2

-+

00

D [ f G (s (

X))

dX] =

C5

(n, m)

(r

~

-

r~)

(I + 0 (1))'

vCr ,)"

t{r

,)

(2.1.19)

..

n

pll

q

~

m H3 1, V

11

(2.1.6)

2r

lim

r-na

~

(r)

= lim c

1

(n, q) r zn-1B

Q

(z)

(1

+ 0 (I)) dz = C

6

(n, q).

r-.

oo

r

~

oo

d

n OCK OJIb

KY

Co (n, m)

~

c

5

(n, q), TO 1I3 (2.1.15)

11

(2.1.2)

c,

1

eAYe

T

(2.1. 18). COO

TIIOWe

HlJe (2.1 .19) )l.OKa3bIEaeTCH C 11CnOJlb

30

EaH HeM

(1.2. 16) a HaJIO m q HO JIeMM e 1.5.1. •

VI.

CY

ll~

eCTBy

e

T

uenoe m

~

I raxoe,

'lTO

<P

YH

FU

IIH L[ (r) =

=

r

-

"

o

~

(r) E

~,

npH'leM

)l.JHI scex q

~

m

cy

uiecr

s

yior

npene

nu

lim [a

2

(r)/o; (r)]

= c, (n, q).

r--+-

00

Q

YCJl

OBl1e VI

IBbJO

OJlHeHO, nanpuwep, xorna

Bblll

OJIHeH O IV npn

a = nlm (CM. (2

..

1.11)).

JIeMMa

2.1.7. Ilpn YCJlOBIHlIX I, III, VI CS (n, Ill ) =

00

= ~

C

~

C7

(n, q)l'(q!

)2

<

00

. E CJlIl

Cs

(n, m)

=1=

0, TO npn t -+ =

Q

=m

D[

f G(

£(

x)) dxJ

=c

8(n,m)r

nL

1(r

)(1 + 0

(1

)).

v(r )

n OKa

3aT

eJIbCTBO JIeMMbJ 2.1. 7 aH<lJlOrI1

'l

HO noxaaarensc

rsy

JIeMMbI 2. 1.6.

B ce

cjJ

aKTb

l,

nepenncnennue ssnne

)l.JI51

cpennnx n o mapav

su-

r Jl

51

JJ.

51

T

rio

nee npocro

.n.Jl

H

C¢e

plJ 4eC I<HX cpeaa nx.

cepOp

MyJlll

py

-

eM JlII Wb

aa anorn JleMM 2.1. 3 II 2. 1.4.

nY

CTb dm (x) on peaen s-

eT C

51

no

(1.1.2). T o rna, lJCnOJlb3Y5I JIeMMY 1.4.4, npn n

~

2 npu-

n

eM

K

~,

;

(r) = M I H

m

(£ (x)) dm(x) .\ n , (£ (y)) dm (y) =

sCr ) « rl

2r

= 6

'!n

c

9

(n, m) r

n-

I

f

zn

- 2(I -

z2

/(2r

)2.

)("-

3)

/2

B" (z)

dz,

(2. 1.20)

o

rne

CD

in , m) = m! 2"rr

n

-

I

/(n - 2)

1.

JIeMMa

2.1.8. n YCTb n

~

2, In

~

I, r

-+

00

11

ssmonnenu

YCJlOBI1

51

I, IV.

TO

fJJ.a npn a E (0, (n - I) /m)

cr;,.

(r) =

= c

10

in, m) r2n- 2-m uLm(r) (I +0 (I )), r

-a

e c

10

(n, m) = c

g

(n, Ill) X

2

n 1

mu

B ((n -X

-

I - ma)/2, (n - 1)/2). Ilpu (J. = (n - 1)/m

~

(r) = [CD(n, m)/4)

r:':

" (r) In (2r) (I + 0 (I)). Ilp» O',>(n- I )/m,

00

eCJlI1 C

ll

(n, m) = I

z"-2

Bm (z) dz

=1=

0, TO

;~

(r) = Co (n, m) X

o

X C

ll

(n, m)r

rJ

- I (I + 0 (I)).

...

n OCT

aTO'l

HO cnen

ars

3aMe

Hy

nepeaennu x U = z/(2r) B

(2.1.20)

11

npOBe

CTI1

aCI1M[]T

OTH'l

eCK

'lif

anan

ns

3TO

fO

nn

rerpana

C

H

Cn

OJl

b

3 0BaHlle~1

reope

nu

2.1.1.•

JIeMMa

2.1.9. n YCTb asmonrreasi YCJlOBHH 1, III, IV n

pn

0: E(0, (n - I)/m), n

~

2,

m~

I. Teena npa r-s-s» D [f c(s (x)) X

5(r )

X dm (x))

=C~

Cll

(n, m) (ml)-2r2

n-

z- muL

T7I

(r) (I + 0 (I)).

n OKa3aTeJIbCTBO JleMMbI

aHaJIO

~11'lHO

JJ.

0Ka3aTeJ1bCT

BY JICM-

MbI2.1.5.

63

62

2.2. YCJlOBH5I

PE,ll

YKU.HH

,llJ151

CJlYIJAAHblX

nO

JlEH

C CH

Jlb

H

OA

3AB" CHMOCTbIO

Il paa enea yCJlOBH

lI

,

npn

KOTOpbIX npenensnsre pacnpene.nen HH

II

Hre

r pa JIOD

.

~

G

(£

(X

)) dx n Jl

51

nocraroxn o Wl1pOK

Or

O

KJIa

CCa

v

Cr

)

(PYHHUIIH G(.) TaKHe me, KaK npene

nsnue

pacnpenenenns HH-

r er panos

~

H m (£(

X))

dx, r n e Hm ( • ) - m-ii nOJllIHO M lJ:e6

blweBa-

vCr )

3 pMHTa, a £ (x) - O.n.HO

pOn

HOe

113

0TponHo

e raycconcxoe none

raxoe,

'I

TO

JIB

(x) I dx =

00

.

R"

P

a

c

C

~IO

Tp

HiI1

CJlYlJal1HOe none £' Q X R"

-+

RI, K.

<p

. KOTOpO-

r o Y

,lI,

OBJleTBOpHeT caenyioiuexy YCJlOBHIO.

VIr.

Cy

mec

TDYIOT /) EtO, 1) H uenoe lJHCJIO m> I Ta

KHe,

lJTO

2r

Ii

In

r - "

tJ

5zn

-I

B" (z) I I- z'/(2r)' ( n +

1)

/2, 1/2)dz =

00

,

'

~

oo

0

np ll 4eM B (z)t0 npn Z

-+

00

.

B YCJlOBII H VII

cjJ

Y

ll

KU I1H I x (p, q)

onpenenaer

cs no (1.4 .8).

Teopea a 2.2. 1. n YCTb

BbInOJIHeHbI

YCJIOBI1H I,

III

H V

II,

B

KOTOpOM

m> J B3HTO 1i3 y CJIOBHH III. Torna npn r

--+

00

ripe-

n e.1b Hble pacnpeneneuns c . B.

x,(I ) =

{

~

G (£ (x)) dx -

Crf

n Iu (I) I

}/[J

c; Iam(r)/m! ]

~

rI

JI C. B.

X

m

•

r

(I)

=

sgn

{C

m}

~

H

m

(£

(x)) dx/a

m

(r)

(2.2 .1)

vC

r )

cosnanaior * (am(r) onpenenaercs uo (2. 1.6)).

.A

3 anH w eM

S (r) = 5 G

(

~

(x)) dx - CornIu (I) I = c; (ml)- l X

u(r l

00

X

~

H

m

(

~

(

x

)

d

x

+

~

CQ(ql)-1 SH

q

(£(x)) dx

=S

I (r)

+S

2(r).

u(r ) Q

=m

+ 1 U(

r)

B CII .ll Y Jle:'I Mbl 2. 1.1 OS (r) = DS

1

(r) + OS 2(r). n YCTb /) E(0, 1)

11

3 yCJIOBHlI VII. Y!

cn

)JIb3

Y51

(2.1.2), OU eH

Ky

B (z)

~

I npn 0

~

• B ljJopMyJlHpOBK

ax

r eopen raxo

ro

rana

co

nnaa

euue

npenem.u

ux pac-

npe)leJIeHHil 03HB"lBeT

C

Jle~

Y

IO

Ul.

e e

:

e

CJl

H CY

l.J..le

C

TB

yeT npeaeasaoe pacnpenene-

HHe

cnn

oro ceMeAcTBa C. B., TO cyuiecr ayer

npcn

e

nsno

e

pacnpea

e.nea re npy-

ro

fI OIlH COBlJa.llBIOT.

64

~

Z

~

(

2r)~

,

a raxace coor uomemee

(2r )0

- II

~

5

Z

n

l-

z' /(2r )' ((n +

1)

/'2, 1/2) dz

~

h, (n) r n

o

(nnr er pan DbIlJHlCJIHeTCH

nyr

e»

3aMe

HbI nepe

ra

eim

srx U = z/ (2r)),

n OJIylJ aeM

00

DS

z

(r)

~

(C

q

/q!)2

cr

~

(r)

~

m ax

{S

JB

q

(i

x-

y I)d

xdy}

X

q= m + 1

q

~

m+1

v(r) vCr )

00

(2r)

0 2r

X

~

C

~

/

q

l

~

k

2

(n)

m

a ~

{Tn [ J + 5]Zn- 1B

q

(Z

) X

Q= m+1

Q

~

m

-r

l

0 (2r)tJ

X

11_

;' /(2r)' ((n + 1)/2, 1/2) dz} ~ k

3

(n) {,

no

+

il

) +

2r

1

+ r" j

zn-

B

m

+

1

(z) I I- z'/(2

,,

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

z}.

(2r )tJ

Pa

3neJIHM npaayio lJaCTb nOJlYlJ

eHHOrO

HepaBeH CTBa sa a;' (r) . B

CHJIy yCJIOBIl

51

VII

lim

r

n

(1

+O

)

/

a

~

.

(r) = 0,

'

-'

00

2r

n

1

{r Jzn-I B

m

+ (z) 1

1

-

z

'/(2r )'

((n + 1)/2, 1/2) dZ}

/cr

;, (r)

~

1

2r

)

l

~

k

4

(n) sup {B (z)}

--+

0, r

--+

00

.

z;. (2r)0

Vl 3

npuneneunux oueno« cneny

er,

lJTO Iim

MS

~

(r) = O.

3H

a lJ1lT.

r

-.

oo

npenensnoe pa

cnpeneneu

ae X; (I : raxoe

xc

e, KaK X

m

•

r

(1) -

=

S1

(r)/[i c, Ia;' (r) (

ml)-I

j

.•

VIII.

Hal1

,U)'I'c5I /) E(0, 1) H u e .roe m> I r

axn

e, lJTO

2r

lim r- (n- llil S

zn-

2

B

m

(z) (1 - z2/(2r)

2)

(n

-

3)

12d

z =

00,

r

-.oo

0

r-n

e B (z) +0 npt z--+

00.

5 - 6- 110

65

Teopeaa 2.2 .2 . D y CTb npa n> 2 BhIDOJIHeHhI I, III. V III .

T

or

Aa np a r

-+

00 npenensnsie pacnpenene

aa

n c. B.

X,

(I)

=

[.\'

G

(

~(

x))

dm (x) - c

.r:'

Is (I)

iJ

IIc;

10m

(r)/ml ]-1

« f )

H C. B.

X

m

.

f

(I) = sgn {C

m}

I n;

(~(

x))

dm (x)/G

m

(r)

S(f )

COs nanalOT, rne 0m(r) naerca

epO

p

My

JIOH

(2.1.2

0).

Teopexa

2.2

.2

nOKa3hIBaeTCH aHaJIOr WlHO reop

eae

2.2.1.

Baueu

anue 2.2.1.

OpH

m = 1

11

r

-+

00

p

acnp

enenea a s De-

JIHtI llH X

m

.

r

(1) H X m,r

(I)

CXO

n51TC

H K CTaH

napT

HOMy HOpMa

JIb

HO-

My

pacnpenea ea mo.

DP

H m > 2 npene

asnue

pacnpezienea

aa

He

HBJlHIOTCH ra yccosc

xaaa.

B n.

2.1

0 n aHbI npn v

epu

n

pen

ens nsix

pacnp

enenea

aa

c . B. (2 .2.1) .

Sanesanue 2. 2. 2.

HCn

OJIb3 y H JleM

My

1.4.6,

ananoru reopea

sr

oro pasnena

aerpyn

no nOJIytIHTb B cnyxae, x

or

na n = 2, a

nnr

erpan

u pacca a

rp

aaaiorca

no

xs a

nparaa.

2.3. U

EHTPAJIbHMI

nPE,LI.EJIbHMI

TEOPEMA

,LI.JI5I

HEJIHHEAHblX

nPE06PA30BAHHA

rAYCCOBCKorO

nOJI5I

YIcCJIen y eM YCJlOBHH aCHM

nToT

H

4ec

KOH HOpMaJlhHOCTH cepe-

PH4ecKHX

Cpen

HHX JlOKaJlbHbIX ¢ YHKUHOHaJlOB OT

ray

cc

os

cxax

nonea

co

CJl

a60

H 3aBHCHMOCTblO.

Teopeaa

2.3

.1. O YCTh su non ne

nu

YCJlOBHH I,

III

, V,

npa-

qeM C

6

(n, m) > 0 . T

or

na npa r

-+

00

KOHetIHOM

epHbl

e pa

cnpe

ne-

JleHHH CJIYlJaHHbIX npoueccos

Y

r

(t) = ,- n/2{

~

G(~(

x))dx

-C

o

r

n t

lv(l)I

},

tEIO,I)

t>(

rl

11n)

CXO,lUITCH K KOll etIHOMepH blM pacnperie.rennaa

np

ou

ecca

o

po

y -

HOBCKO

ro

n BH)Ke HHH Wb (t), t E

10,

II,

rue

b = Co (n, m)

on

p en e-

JIHeTCH

no

(2 .1. 17) .

• Bbln eJlHM HeCKOJlbKO 31anOB

,ll,OK

a3aTeJIbCTBa.

I. .D.

Jl51

Jl

I060ro

q > I

Bhl6e

pe M TOtIKH

O=t

o

< t

l

<

...

<

t

q~

I.

q q

O OJlO)KHM

S,

=

~

A

jY

r

(tj) =

~

aj

(Y

r

(t

j

) -

Y ,

(t

j- l

))'

r ne

a)=

j

=1

j

=1

=

A)

+

...

+ A

q

•

AJ

ER', j = 1,

...

, q. Paccaor

pns

c. B. S=

q q

=

~

A

jWb

(tJ)

=

~

a ) (w

b

(t)) - W b

(tl-li).

COrJIaCHO MeTOAY MO'

1

~1

j-

I

MeHTOB A . A.

MapK

oBa

,ll,O

CTaTOQHO noxa

sars,

'l TO llJlH JI

I060

r o

uen

or

o p >2

• p I(P - 1

)11

[±

ba

;(tj -

l

)

tn, p = 2",p t

j

_

lim

MS:

=

Ms

= i= 1

r

~

oo

0, p =

2v

+ I,

(2

.3

. 1)

r

,ll,

ev

=I

, 2,

.....

2. 6e 3 Or pa Hl14eHIJ

51

ooraao

cra

C4l1TaeM Co =

0.

OY

CTb

A

j

=

v(

r

t

yn

)""v

(

r tJ~I)

'

11

;·(r ) = 5

H

l

(

~

(

x

))

d

x,

YJ(r) = DnJ(r ).

tJl

3aMeTliM,

'lTO E CH

.7

IY (2.1 . 18)

lim V - ny

t

(r)} = C

s

in, l) (Ii -

tf-J

)'

(2.3 .2)

r-+oo J

.D.JlH N>m sarnnuea

Sr

=

~

;

+

s;

, r ne

q N

a

~;

=

~

QI ~ C

1r

-

n

/2

nf(r )I/I,

~

;

=

~

A)

~

Cjf- n/2(l!)

-I

X-

j

=1

1= 1

j= l

I>

N

x

~

HI

(~

(x)) dx.

(2.3.3)

o(f l

~

/n

)

J

Co

r JIaCHo (

2.1.4)

q

m;

~ ~

Ai

~ c

70

i (rt]i

n)

r- nll!

~

j

=1

J> N

2,

~

k]

(~

ci //1)B

N

-

m

(0) fzn-I IB(z)

Im

dz.

(2. 3 .4)

I>N

a

0 pI1HI1MaH BO BHI1MaHlIe

(2.1.2),

a TaK)Ke V, \l3 (

2.3

.4) YCT8H8B-

JIHBaeM, 'lTO n Jl!l Jil06

0r

o

e>

°cyuiecr ayror N

a

,

r

o

rax

ne, 'lTO

npn

/ > No, r > r

o

D

s

;

< e. B

)l

aJIbHei'lw eM c

xurae»,

4TO B

(2.3.3) N = No. Tor na nOCTaTOlJHO

Il

OKa3aTb, 4TO (2 .

3.1)

HMeeT

N

MecTo, eCJlH 33MeHHTb S, aa s

;,

a b s a b' =

~

C

~

.

c

s

(n,

'\I)

/('\I!

)2

.

v= m

3.

Ha

3

0BeM

,ll,HarpaMMoH nopsnxa

(u

l

' " ' '

up)

rparp

I' 0

u] +

...

+ up BEpwHHaMH, eCJlH:

p

a) MHO)KeCTEi) BepWI1H V r

parpa

r HMeeT Bl1n V = U WI, r

ne

1=1

5-

67

66

\

V'

J = {(j, I

);

1 ~ I

~

UI) -

i-

V.

yposens r-p

ae

a r, 1

~

i

~

P (.ll.J

IH

u J

:=

0 ClIl1

TaCM

\V

j

= 0 );

6)

f<a)j{

,ll

a51

Be

pW

I1H

:a r

JM

eeT crerieus I;

B) e

c.mt

Wi, 1

1

),

(j2:' 1

2

))

Er, TO il

=1=

i2

' T. e. pe6pa rparpa r

~

lO

r

Y

T

npoxonn

rs

TO

JJI

bl<O

Mell<.uy

pa3

JIl1

tIHbIM

l1

y

pOBH51MH

.

n YCTb

T = T (Ui, .. ., Up) - M

HOlI<ec

TBOnl1arpaMM

I'

nop sn-

" a

(Ui

, .. ., up). 0 603H

aIJ

llM R (V) M

HOIKeC

TBO pe6ep r pa

rp

a r E

E T . )1.

iJ

5I

pe6pa w =

((j

l, 1

1

) ,

(j2

' 1

2

) )

E R (V), h < i2 n

OJI

O

)f{I1M

d; (w) = ii, d (w) =

i2

' H a30B

CM

nHa, paMMy

I'

peryJI

51pH

OH,

2

eCJHI ee ypOBH

I1

MoryT 6blTb pa

36l1Tb

I aa napu TaK, liTO HE On

HO

pe6po He npoxonnr Me

)K

ny y

pO

B

H5IM

JI, npHHanJIell<a

ll

Ul

MI1

pas-

JIHtIHblM nap

aM

. MHOlI<e

CT

SO pery

.a

spasi x n

l1

arp

aMM

T* s;: T (U

1

,

.

..

, Up

).

ECJIli P HelleTH

O,

TO

T* = 0 . 5 yn

eM

I1C

nOJIb30B

aTb CJIe-

nyiouiee

YTBe

p

lI<

,llemle [187, 1261.

n Y

CTb

(Sl " '" sp),

p>

2 - rayccoac

xn

a aexrop G M

si

s

j

=

= r (i, j), i, i = 1,

...

, p, a HI, (u

1

) ,

•••

, Hlp(up)

nO

JIli

HOMb

l Ye-

p

ObIUJeSa

- 3 pMHTa. T or na M {n H

lj

(Sj

)}

=

~

If, r-ne T =

i=\ f

ET

=

T(l

l' ... , l p), a I

r

= n r (d

l

(w), d

2

(w)).

(L'

EN(Vl

Ilpnaenennas

11l1arp

aMMH

a

51

¢o

pMyJIa npa p = 2 cosnana

er

a

(2.1 .4)

4. n YCTb D

p

= {(J, L) i J =

(il

' ... , j

p),

1 ~ ii~ q, i =

-= 1, ... , pi L = (l

l'

...

,l

p),

l

~l

v

~

N,

v=I,

... , p},

K(J

,L)

=

p p

= n {b

JiC

I

/l

i!i ,

~

I

P

)

•••

= n J..

..

To

r-

na

COrJIaC

HO

ll,Ha

rpa

MMHoii

1=

1 /= 1 Ai

¢ opMY

Jle

npu p> 2

11

3 (2.3 .3) HaXO

ll,H

M

o

np

lP

M

(Op

=

~

K (J, L) r- !2 f P) {n M H I;

(1;

(XU)))} dx(\)

...

dx

)

=

D

p

j=1

~

K (J

-i-

nor: r (P){

~

n }d (I) d (Pl

= .kJ

,L)

r J

~

B (X d, IU'1- xd,,(W)) x ••. x =

Do

rE

T w ER( V)

(2.3.5)

=

~

K (J, L)

~

F

r(J,

L, r),

Co f ET

r-

ne

F

r (

J

,

I

_, r) = r

- ro/2

J

r (Pl n

B (X dt(Wl - x d,(w) dx

(\

)

.. . dx

(?)

.

w ER( V)

I

'l

ycrs T* - MIIO}KeC1BO pe

ry

Jl5lp

HbIX nmrpaMM. Pa306beM

CyM

MY

~

.., =

~

... +

~

.

..

. To

rn

a cy vaa (2.3.5) pa306ber

C51

na

r

l'

r f ET" f Eh , TO

nse lIa

CTH

. Ile

jp

ayio H3 H

I1

X, o'naexa

ro

uryio

~

... , 0603

Ha

llH'1

f ET"

~

:<

r ); a sropyro, o

rae

uaio

uiyo

~

... , 0603HallHM

~

0 (r ). Flpu

f ET' T'

r

-+

00

npoaaannsnpyea ItX

nOB

e

.neHHe

OT.neJlbHO.

5.

ECJl

II p= 2v + 1, "

=1,

2, .. , T

OT'

= 0 Ii lim

~

:<r

)

=

O

.

r-> oo

)J.JlH

lIe

TH

oro

fJI

= 2v H peryJI

HpH

O(1

ll,

Ha

r

pCl

M~

lbJ

r ET" pac

cx

or -

pHM napu (i(1)I, i(2)), ... ,

(i(p

- I),

i(

p)),

r-

ne U (1), ... , i (p))-

TaK

a51

nepec

raa

oaxa

MH

O

:tK

eCTBa ( I, ... , p), liTO pe6pa rp

aep

a C

O-

e.n

I1H

H

IO

T TO

JJ

bKO

COCe)I.Hl1

e yposnn i (2m - I) H i (2m), m =

= I, 2, ... , v. n pe

)I.

nOJlO

:tK

HM,

xro

yposnn i (2m - I) 11 i (2m)

r IO)

liMeK'T MO

W.

HO

CTb r (m), m = 1, .•. , v, T

or-

na nnr erpan J

'"

pacnanaercs B nponaacnemre nap fIH rerpaJlOB . AHaJlOfHlIllO (2.3.2)

n

lim

r-

J J B r(m)

(I

U - v I) dudu =

r

~

oo

A · A-

/i(2m- l l /i (2m1

= 8

;'

(2m ) c

5

(n, r (m))

(t)i

(2

I1l

l - t1;12

11

1)

- 1). (2.3.6)

i(2m-l)

OTCIO

)I.

a

v

n

rtm)

(I

Ur-(J,

L,

r) = n {

r-

J J B - v I)dudv},

m= 1

'\

(2m_ l) A

i;

(2ml

H

11

3 (2.3.6) SbITeKaeT

. I

~

I

c6 (n, r (m)) (

ti;

12m l - t

j i

(2m

l-

1

),

ecnn

lJ

mF

:r

(J,L,r

)=

i.

.

r

~

oo

IJi (2m

-l

) = J

i(2m

)' m =

I,

... , v,

l O B

OC

Ta

JJbH

blX C

JlY

lI

aH

X.

CJlenOBa

TeJlbHO

,

I(f

) = lim

~

K(J,

L) Fr (J, L , r) =

r

~oo

/

Q v

= (2 a; (tj - t

;-,

))') nC;(m )C

5

(n, r (m))/lr (m)]!

i=J

m=

l

Ii

lim

~

:

(r)

=

~

I

f'

r ne CyMMI

;pO

BaH

He

np

OB

O

)I.liTC51

no

SCe

M

r-v

co f ET'

p

CrYJJ

51pHhIM

ll,mrpaM

MaM

C fJ YP0

3H

5IMH

.

69

68

K OJIH4eCTBO per yJI5

IP

ll blX

nnar

pav

x,

conep

xc

aunrx 2mj ypO BHeH

s

MOW.HOCTH rJ' j = 1,

""

s,

~

m, = v (Bee m, paanauasr), pa BHO

i=1

s

(2v)1[(2m

\)1

.._(2m

s)I

J-

1

[n (2m) - 1)(2m) - 3) ...

3·1

j X

i=

1

x n

(rll)m

; = (2v -

1)11

vl

[mIl

...

mslr

1

n

(rtIt

l.

1= 1

i=1

.am

I TaKIlX ,nH

arpaMM

(c <pIlKCIl P0BaHll bIMH S Il r

i

,

i =

I,

... , s)

Q S

I (f ) =

(

~

a

~

(t j -

ti

_l)

rn

{C

~

(i)

C5

(n, r (i))/r (i)

I}m

l

.

i

=1

1=1

BCI10M

Hlla

H,

'lT

O p = 2v, Ha XOJlHM

li

m

~:

(r) =

~ ~ ~

(2v -

1)11

v! X

,~

oo

I ';;s

";"

m,+ ...

+m

s= v l

';;'

j

";N

\";j';;

s

q s

X [m11...

m.lj

-l

(

~a~

(tj

-ti

_

J)

n {C

;(Oc

5

(n,

r(i))

/r(i)I}m

l

=

1=1 (= 1

Q N

= (p -

1)11

(

~

Q

~

(tj - (

i_I)

t

2

(

~

C;c

6

(n, i)/il ( 2. (2.3.7)

i =1 I= rn

6. n OK

a/K

eM,

'l

TO

lim

~

P (r) =

O.

(2.3.8)

r~

oo

YTB

ep/KJl

eHHe r eopeasi 6y,neT cne

no

sa

r s

113

(2.3.7), nOCKOJIhKY

8 CHJlY (2.3.5) H3 (2.3 .8) ssrre xaer COOTHoweHHe lim

(

~

)P

=

,-,00

= lim

~

:

(r).

r

~oo

BB

eJlH n

ex

or

op

yro

nepecr

aa

osx

y n -MHO/KeCTBa (1,

..

. . p).

MO/KHO H3 JlHar paMM bI

I'

f:T (l l' ..

.•

l p) nOJI

y'lHTb

,nH

arp

aM My

nf

co

C

Jl

e

Jl

Y

lO

u

~

II

M

H

CB

OI

ICToa MH: n (j)-ii

yposem,

,nHa

rpa

MMh!

nr

HMeeT MOW.HOCTh l)

(T

. e. M

OlllH

OCTb

yp

OBHH j ,n

Har

pa MM hl

f).

j = 1, ... , P H W = (

iI

' " 1)'

(j

2' '\'2))

E:

R tv) r or na Ii TOJI

bK

O

To

r -

na,

x or

aa

n (w) = «n (j I

)'

VI) '

(n

(i2

)' '\

'2))

EnR (V).

060

3 Ha

'l

HM

qr

(j) KOJIH'leCTBO

pe6ep

wE

R(V) TaKHX. n o d

l

(w) = j. 3 aM

eTliM,

4T

O ,nJIH

I'

ET

(ll'''

, l p), J =

(iI'

. i

p

), L

=

(lp

..

lp)

F['(J, L, r) = F (

J1

(J ), 11(L). r), (2.3.9)

nr

70

rne n (J) = (n

Ul)'

,.. , n U

p)).

It «L) = (rt (ll)' ... , n (l

p)).

.amI

J1106

0H M a

rp

aM

~I1b1

I' cyuiecrayer nepecra aosxa n TaKa R,

'lT

O

Jllla

rp

aMMa f ' = ;

nf

06JIaJlae

T CBOHicTBOM

r

'ET(l

;

....

,

l~),

l;

-c

i

;«;

...

«i

.. (2.3.10)

B CIlJIY (2.3.9) .,nOCTaTO

'l

IlO JlOKa3aJTb (2.3.8) JlJIH Jlfia r paMM co

CB

OHCTBOM

(2.3.

nO)

.

3al1HW

eM

np

Fr (J, L, r)

~

r-

/

2

j'

'"

r n

niB

(I X li) -

A

i,

AJp 1=1 wER(V)

d,(w)=

1

- X( d. (W») j)Id X(

I)

...

d x (P). (2.3.11)

Ba yrpeaaee npoasseneane B (2.3.11) COJlep/KHT qr (I)

'l

JIeHOB,

I1pH'l eM

r1 I B (IX Ci) - X (d,(W)) I)

1~[

q

I'

(i)j-

l

~

IB (i

Xl

i) -

wER(V) wER(V }

d

,(w

}=

i

d,(w

}=1

_ X (d. (W» I) IQr (i).

¢IlK

cllp

y eM X(21,

...

, x (P) H

aar

erpapy

ea

(2.3.11) n o dx'»,

3Ty

npoueay py npon

ozox

aea ,nJI

51

d X(2),

....

dx(p

),

BbI6p

SB

k~

TaK,

'lT

O

v (k

2

r)

::J

A

iv

, v = 1, ... . p.

Il

oCKOJIbK y Y - z Ev (2k

2

r), eCJIH

yEA

jv

,

v = 1,

...

, p, zEo(k

2

r). TO

P

np

IFr(J,

L.

r)

l~

r-

/

2

n \

IB(jYDIQ

r<

i)dy~

1

=1

tl(2k.' )

P

2k"

np

~

k

3

r - !2 n rz

n-

l IB (zWr(i)dz.

(2.3.12)

1=1 0

E CJIIi l

i;;;:':

m, TO

24"

rz

n-

I IB (

Z)IQr<

I) dz

~

k,r(l-I/

(i)]n.

(2.3.13)

o

npn

qr (i) = 0

linH

qr

(i) = It, g

(i)

=

qr

(i)/lt· C JJ.pyr

ol!

CTOpOHbl,

2k,.

J'

zn-1

1B (zI

IQ

r

1i)

dz = 0 (r

ll

- I/(O)1) , 0 < qr (i) <

[I'

(2.3.14)

o

nOCKOJIbKY

as

YCJlOB

li

H V c

ne

n

ye-

,

'l

TO

cy

uiec

r sy

er

r

e

TBKoe,

71

00

QTO

jzn-1IB(

z)j'ldz<e,

e>O,

orxyna

B OJ.1lY

Hepa

BeHCTBa

'e

Fe.nsnepa

2k"

~.

z

n-l

\ B

(zt

r(l) dz < k (e) +

eg(l)

(4k r)(l- g(i)" .

s

z

o

T aK 1IM 06pa

30M

,

060

3H

a411B

f.L=n[p

I2-

~

g(i)

],

11

3(

2.3.1

3)

l

<i:

i

~

p

l1

(2

.

3.12)

nonyxaea

I

Fr

(J

,

L

,

r

)l~k6

r

!1

.

y13

(2

.3

. 14)

11

(2.3

. 12)

cae

zr

y

er

,

4T

O

IFr

(J,L

,r)l=

o

(

r

~

t),

eCJII1

O<

qr

(i)<

l

i

,UJlH Ka·

l<

OrO-HB6Y

,Ub i. E CJIB r ET '<;

T*

, TO JII1

60

O

<qr

(i) < l,

,UJI

H HeKOTO-

poro

i,

J11160

,Ul1ar paMMa conepsorr

pe6po

MelK,Uy ypOBH

51M

I1 pa3HOH

MOIUHO

CTI1.

Bb16p

aB pe 6po w E R (V), onpe,UE'JIlI M 4

11

CJla PI (w) H

pz(w) KaK

MOmH

OCTb d

1

(w

)-

ro

11

d

z

(

w)-r

o ypoaneii coora

ercr

seuno.

B

curry

(2.

3.1

0)

Pl(

W)

~PZ

(

w)

,UJl5l

Jl10

6b

IX wE R (V).

Y4

11TbIBaH

onpeneneune g (i),

HaX

O)J.I1M

p

2 ~ qr (i)/ I; = 2 ~ [PI (W

)]-l

;>

;= J wER(V )

;>

~

{[ PI (

w)r-

1

+ [P2

(W)j-l}

= p, (2.3.15)

WER(V )

TaR KaK astpa

xc

enue

l/

I

j

n 05lBJl5leTCH p OBHO I

j

pas npa CYl\lMI1PO-

BaHl111

IIPl

(w)

11

I

1p

2 (w) . ECJII1 nnarpaana c

onepxorr

p e6

pa

ME'IK-

,Uy y pOBHHMI1 p a3HOH MOmHOCTI1, TO B

(2.3.15)

BbInOJIHHeTCH CTpO-

r

oe

Hep aBeHCTBO. 113 (2

.3

. 15) CJIe,Uy eT, '-ITO npn r

ET

"",

T*

,U

.IJH

Jl

1060r

o J lim F

r

(J, L, r) =

0,

o

rxyna

11

BWTeKaeT

(2

.3

.8

) . •

'->

00

Teopeva 2.3.2.

nY

CTb BbInOJIHeHb[ yCJIOBl1H 1, III, VI, npn-

qeM

C

s

(n, m)

=1=

0. Tor na npn r

-+

00

C. B.

17

. (1)

=r-

n/2

Li

ll2(r)

{j

G

(~(x)

)

d

x-

C

or

nl

v(l)

I}

v(r)

11

MelOT aCI1MnTOTI14eCKI1

HOpMaJIbH

Oe p

acnpenenenn

e N (0,

C

s

(n, m)).

1

...

Cxesia ,UOKa

3aT

eJIb CTBa

TaKaH

IKe,

KaK

y

reop

e

-iu

2 .

3.1.

H esaa-nrre

nsn

u e

pa

3nJi4

HH CBH3

aHbI

C TeM, 4TO, ao-n

epsux,

paccxarp

11

Ba lOTCH

JII1111b

o

"U

HoMep Hble

pacnpenenenns

11

t = 1.

3TH OTJII1

'-1

I151

npI

1BO,U

51

T JIHllJb K

ynp

ouieua

su

B

"U

OK

a3a

TeJIbCTBe.

B O-BT

OPbIX,

IICn

OJIb3YlOTCH YCJIOBl1e VI H r eop e

aa

2 .1 .1 . Hanpu-

Mep , B

(2.

3 .1

3)

npa

sas

4aCTb

,UO

MHOlKaeT

C51

aa lL

l

(r)j g(i) Ilpn

,UOKa3aTeJlbCTBe

(2.3.14)

I1HTerp

a.

1J

He06xo,U

I1MO pa3611Tb H3

,UBe

72

lJaCTI1

;

2hz'

0,

2k"

S

Zn

- I IB (z

tr.

(i) dz »

[I

+ S]zn- II B (

zwr

(i)dz = 51 + 52'

o 0

~

.IlaJI

ee, K

060llM

CJIar aeMbIM

aano

np

l1MeHI1Tb HepaBeHCTBO Fens-

nepa

11

npu ouesrxe 52 BOCnOJIb30B

aTb

CH TeM, 4TO npu r

-

~

00

s»,

S

zn

- I IB (zll" d'z = 0 (L

l

(r) )

.•

e,

n YCTb BbInOJ]HeHO YCJIOBl1e IV

11

a E(n/3, nI2). EC

JJ

I1

rp

yru«

U HOHaJIbl 11; (x) = G;

(1;

(x)), i = 1, 2, 3 r axne, 4TO r

an

g G; = i,

TO

co

r JIaCHO reo p

evaa

2.2.

1

11

2.3. 1

,UJJH

(PYHK

Ul1o

HaJloB G,

(1;

(x»

11

G'

J(S(x))

cnpa

Be

,UJlI1BO

U. n.

T.,

a ,UJIH G

z

(S

(x)) OHa He BbIn OJI-

HHeTC5l (cp C

pe3Y

JlbTaTOM n.

1.7).

2.4.

AnnpOKCHMAUIUl

PACnPEll.EJ1EHH}1

rEOMETPH4

ECKUX

l!>YHKUHOHAJ10B OT rAY CCOBCKHX

nOJ1E"

V13y 411M «CJIYlJaHHble

o6be

Mbl»,

orpanuuennue rayccoscxoa

nos ep xx oc

rsro,

a raxxce Mep bl

np

eBbIWeHI15l nOCTOHHHo

ro

yp

OBHH .

P aCCMOTpl1M cjJYHKUl10HaJlW

Vd

r

)

= I max

{O,

S(x)} dx,

Vz(r)

= I 11;(x)ldx,

V(

fl

V(r!

I-1Me

lOml1 e npn n = 2 HCHbl H

reOMeTpl1

4eCKl1H CMbI CJl ,UJlH aenpe-

pusnsr

x

1;

(x). Hanpnvep, npn n = 2

VI

(r) - «CJIY4aHHblH 06b -

eM», O

rpa

Hl14eHHbIH peanuaaunef n OJlH

1;

(x), x E R

2

,

n OJIOCKO-

CTblO

xpy ra v (r) c R

2

11

UIIJII1H,UpI14eCKOIl

nOBep

XHOCTblO, npo-

XO,UHmeH

-rep

ea

rpaHI1U

Y

sroro

xp

yra

nep

ne

H

,U11

KyJI

Hp HO

ero

nJIOCKOCTI1.

Teopena 2.4.1. ECJlI1 BbJnOJlHeHbI YCJlOBI1H I, VII (npu m =

1), TO npn r

-+

00

C. B.

g)

{VI

(r) -

(2n)-1

/2 r

n

Iv

(l)1}

/{

01

(r)/2}

-+

N (0, 1),

rne

0

1

(r) on p

ene

nse

r c s no

(2.1.6)

. B

4a

CTH

OC

TI1, eCJlI1 B

blJ1

0JJHe-

HO IV n

pn

a E

(0,

n) , TO 0

1

(r ) o n p e,UeJIHeTCH n o (

2.1.10

)

npu

m=l.

... BbI411CJIl1M n

epsue

,UBa

KOsepcjJl1

Ul1eHTa pa

3J1

0lKeHI1H epyHK-

UI1I1 G (u) = m ax {

O,

u} B p

sn

no

II

O.IJHH

OMaM

4e

6blweBa

- 9 pMH-

ra.

OHI1

paens:

COO

TBeTCTBeHHO Co =

l

/

V

~n

,

C

I

= 1/2.

T

a K

II

~

1

06pa

3

0M,

reop

ea a 2

.4

. 1 - STO

cn

en cr

sae

r e

ope

asi

2.

2.1

rtpu

m =

1.-

73

Teopeaa 2.4.2.

ECJlH

sunonn

essr YCJlOBHH I. V (npa m =

I),

7 0 npn r s-»

00

1)

r"]

0

(l)

I}

/r

fl12

{VI

(r) -

(2n)-

lf

l

~

N (0, (n)), C

12

cue

00

""

cu (n) = n

1

0 (I )12{4-

1

~

Zfl- IB

(Z

)d

z+

~

{[CD

U

-

1

l(

0)j2/j!} X

., 1=2

00

i

X SZ

,,-IBI

(z) dZ}, CD(z) = I Ip (u) duo

-

00

.Ii.

C

or

JlaCHO

(2.1.3)

00

M

IV

1 (r )12 = I I dxdy

~ ~

uS

Ip

(u, s, B

(I

x - y I)) duds.

l!(

f)

U(F) n 0

Hcnonssy» H3BeCTHOe T

O>K

olleCTBO

[

76,

pasnea 1

0.8]

00

Q

rI(0 - u)q(w - u)q

<p

(0, w, p) dodw =

~

pi

Ul)-

t X

U u 1=0

co

X lql [(q -

i)

I]- l I(0 - u)Q-i Ip (0) do

]2

+

II

00

+

(ql)2

~

pi

(jl)-l

[<D

(/-Q)

(U))

2,

(2.4.1)

I= Q+ I

npn q = I, U = °nOJlY4aeM aHaJlOr l14HO JleMMe 1

.5.1

2,

DV

I

(r ) = n 10 (I)j2 r

fl

rz

fl-

1

l

1

_

l

'

/

(2r )' «n + 1)/2 , 1/2) X

o

cc

X {B (z)/4 +

~

B

I

(z)

UI)-I

[<1>

0-

1

)

(O)F}

dz =

i=

2

= c

12(

n)

r"

(l + 0 (1)).

(2.4.2)

Y lJHTblB8H 3Ha

qe

HHH K

03¢ep

IHl,HeHTOB

Co

, C

1

,

3aKJlI04aeM,

4TO

reo

pev a

2.4.2

HBJlH eTCH CJleollCTBHeM

TeopeMbl

3.2.]

npa

m=

I,

t = ] •

Teopesia

2.4.3.

ECIIl1 BhlO OJlHeHbJ yCJIOBHH I, VI

(npa

m= I),

o DpH r _

00

C. B.

D

{VI

(r)

-

(2n)-l

f2r

10

(I)I}

r-

flf2

L, lr

L

(r ) _

N (0, C

(n).

lS

3H

a 4

eHHe

C

l S

(n) onpenena

er

ca n OCTOHHHOH C

s

(n, I) (

eM.

JleMMy

2 .1

.7

), e

CJ1

H nOJlIO>KHTb C, = r

00

Ulp(u)

Hq(u)du,

q = 1, 2, .

...

o

T eopev a

2.4.:3

HBJlHeTCH CJle

oll

CTBHeM

reo

pex u 2

.3

.2

npn m =

=

I H

JleMMb12.n.7.

E CJlH BbInOJlHleHO YCJlOBHe IV, TO reop

exu

2.4.1

-2.4

.3

orin-

csraaior

xapaxrep

HOpMaJlbH

OH annpoxcnaauaa <PYHKUHOH

aJla

VI

(r).

Ilpa

3TOM sun

HOpMHp

OBKH H KOHCTaHTbl

HOpMHpOB

KH

OTJIHlIalOTCH B c.nyxaax a E

(0,

n), a = n,

a>

n (CM. J1eMMy

2.1.

3) .

Teopeaa

2.4

.4.

ECJlH ssmonaensr

YCJIO

BHH 1,

VII

(npa m= 2),

TO npn r

--+

00

npenensnsie pscnpenenenHH C. B.

{Vi

(r) -

- (2/n)I

/2

r

fll

o

(1 )!}/{a

2

(r)

/}

;"2n} TaK He >Ke, K8K H C.B.

r Hi

(

~

(x)) dxlo, (r), (2.4.3)

<J{r

)

rne a

2

(r) onpenensercs no

(2.1.6).

B 4a CTHOCTH, eCJlH

BbIn

OJlHe·

1i0 IV npa a E

(0,

n/2), TO

a~

(r) onpenens ercs

no

(2

.1.1

0)

npn

m = 2.

.....

BbI'lHCJHlM

nepsue

TpH K0

3<P<P

HUH

eH'r

a pa3

J1

0>KeHHH B

L

2

(RI,

rp

(u) du) <pyHKUHH G(u) = Iu I.

OHH

p

asn

u COOTBeT CTBeH-

HO

CO

= (2

/n)1

/2,

C

I

=

0,

C

2

= (2

/n)1

/

2.

np

HMeHHB re

opexy

2.2.1

npu m = 2, sasepmaev ollOKa3aTeJJbCTBO. •

Teopeaa 2.4 .5 .

ECJlH

aun

cna

enu

YCJlOBHH I, V (npn m =

2)

,

TO npn r

-+

00

C. B.

1)

{V (r) - (2/n) 112 r" I0

(I)J)

,

-n

12 _ N (0. C (n)),

2

li

r

oll

e

Cg (n) = C

I5

(n) r

00

Zn-l

{V 1- B2 (Z) - I + B (z) a r cs i n {B (z)}} dz,

o

1

(n) = 8n

n

-

1

r-

2

(n/2) n-

•

(2.4.4)

C

I 5

.....

3a

nHw eM

M

[V2

(r)j2 = r J M

I;

(x);

(Y)I dxdy, p = B

(I

x - y

J)

.

t

oCr

)

vC

r )

.ll

il Jlee,

M I

;(x)

~

(Y)I

= rrIuz !

rp

(u, z,"p) dudz =

R'

o

00

= [rJ- r

rl

uz

exp

{ - (u

2

+

Z2

- 2puz)/l2 (I - p

2)]}

dudz X

o 0 _

00

0

75

74

X [rt 1

/

1- p2J-1 = n-

I

(I - p2)3/2 (d/dp) X

X ([ S

J'

-

o

J

00

J]

exp

{-

(U

Z

+ Z2 - 2pzu)/2} dudz) =

o 0 -

00

0

=

n-

J

(I - p2)3

f2

(d/dp) [2a rc

sin

{p (I _ p2)- 1/2}] =

=

2

(V

I -

p2

+ p a

rcsi

n p)/n,

orxy.aa

)],JlB nncnepcnn DVz(r ) nOJlY4aeM

BblpaMeHlle

z,

n

DV

z

(r)

= G1d n) r j'z

,,-I

ll

-z

'/(zr,,( (n + 1

)/2,

1

/2

) X

o

X 0/

'"

1

-B

2(Z)- 1 + B (z) ar csi n {B (z)})dz =

= G

1'o

(n) r

n

(I + 0

(1

))

npu r

--+

00

. Y 411TbIBa H 3

11

a

ll

eHII H K03epep

HlUle

HTOB

pa

3JlOiKCII

IlH

epy HK lJ,1I

11

G (u) = lui B L

2

(Rt,

rp

(u) du), Hall,Il,eHHble npu JW Ka ·

3a TeJl bCTBe

reop

eau

2.4

.4

, no

.n

yuae

re

nauiy reopevy

11

3 reo pe-

MhI 2

.3

. 1 npn t =

I,

In

= 2

.•

E CJIIl BhlDOJl H

eJ-IO

l V, TO npn

o:.E

(0, n/2) pacnpenerienue

¢Y

IlK

-

unonana V

2

(r) HeJlb3H

all

npOKCIL\1HpOBaTb HOp

MaJlbHblM

3aKO

-

HOM

(

reopeva

2.4

.4

),

a np» 0:. > n/2 MO)l{HO

(reopeua

2.4

.5

) .

Ilp

n 0:. = n/2 Ta J

()I

{ e no ny

crnva

HO

pMaJl

b

HMl

annp

o KCIIMalJ,1H1.

Xapa

xre

p HOp

MI1p

OBJ(1I

11

nOCTOH

HHb

le

MO)l{HO Ha l1T1l H3

reope

-

MbI 2

.3

.2

.

n YCTb

a;>

O. P a CO

lOT

pl1M

reOMeT

pIJ4eCKI1e epYHK uHOHaJlbI

W

1

(r) = I X(s (x) > a)dx, W

z

(r) = j X(j s(x)1> a) dx.

v (r )

cU)

K 03ep

epH

lJ,

lIeH Tbl pasnoxcenns

¢YHJ(

lJ,

11

G

1

(u) =

1.

(u > a) B

L

2

(

R1

,

cp

(u) du) I1Mel{)T BH)],

C,,(a) = { l - (D(a), q = 0,

(

2.4

.5 )

.

cp

(a) H Q_ I (a),

q;>

I,

a

K0

3epep

ll

lJ,l1eHTbl

pa3

1l0M

eHHH epYHI<lJ,lIlJ G

z

(u)

= X(I u I> a) MOM-

HO

npe

ncr aa a

n,

TaK:

J2

(1

- (1) (a)), q = 0,

C

q

(a) = l2

<P

(a) HQ

:'-l

(a), q = 2r,

(2.4.6)

0, q = 2r +

I,

r = I, 2, ...

n YCTb sd r) = {Wd !) - (1 -

<:1)

(a)) r" Iv (1)

1}

,

1;2

(r ) = {W

z

(r)-

- 2r

n

\ v

(1)1

(I

- (1) (a))}.

76

Teope

na

2.4..6. n YCTb

BbJn

O

.~HeHO

I H r

--+

00

. Torna, eCJl II

!)

cnpaaea

nn

ao

Vl

JI

(np

u

m=

I), TO)

~

1

(

r

)

/

[

r

p

(

a

)

O'

1

(

r

)

J

-+

N

(O,

1),

rne

0'1 (r)

on

pene

.n

a

er

cs

no

(2.

1.6) npa m =

I.

Il pa YCJlOBHH V

!)

(c m = I)

~1(

r)

lf-n

/2-+N(

O

,

G

I6

(n, a)), r ne

00

C

1

6in, a) = qp2 (a)

~

H

~

-I

(a)(qn

)-

2G

1

in, q) r

zn-

I B

Q

(z) dz,

Q= I 0

Teope

va

2.4

..

7. E CJlH Bb

IllO

.m

eHb

l 1,

VII

(npn

In

= 2) , TO

npa r

-+

00

npeziensnoe pacnpeaenenue G. B.

sz

(r)/ [arp(a) 0'2 (r)]

r axoe

iKe

, KaK

If

C. B. (2.4.3 .).

!)

ECJ1H BbInOIlHeHO V (npn m = 2), TO C. B. S2(r) r-

n

/

2

-+

N (0,

C

17

(n, a)), rae

00

C (n, a) = 4r

pz

(a)

~

H

~

Q_I

(a) [(2q)lj - Zc

1

(n, 2q) X

n

Q= I

X

00

j

zn-

IB

ZQ

(z)

dz.

o

Teope

vu

2.4.

6

,2.4

.7 BblBo

nH

TCH H3 reopen 2

.2.

1,2.3.

1 H

coor

nourenua

(1.4.4

) , (

1.4

.5)

.

OTMeT

HM

JlHWb

, 4TO npu YC.1]OBHII

IV n

eps

oe H3 yTBe

piKne

HHH

KaiK.noH

r eope

xu

OXBaTb

lBae

T CJlY-

4 a l1

0:. E (0, n) (cooraercrneuao 0:. E (0, n/2)), a aro poe-i- C,1]y

4aH

Ct..

> n (II

J1

H 0:. > n/2). Onna xo, HC

nOJlb

3yH

reopexy

3.2.2

, ne-

rpyn

no BbJnI1C

aTb

HopMaJlbHYIO

annpO

KCIIMaUl11O epYII KUHO

llaJla

W

1

(r) npa

0:..

=

ri

H W

z

(r) npn

Ct..

= n/2.

C

ytIe

TOM JleMMbI 1

.4.

6 npu n = 2 MO)l\HO Ha HTI1 ananoru pe-

3YJlbTaTOB s

roro

nymcra Jl,JlH epYHKILH OHaJlOB j' G (s(x)) dx, rne

6.

(1..)

~

(I)

- xsanpar.

B

reopev

a x

2.4

.6 , 2

.4

.7 MO:-KHO O

TKa

3aTbCH OT r ayccoso

cm

nOJlH . Torna MO)l{HO nOJl b

30BaTb

CSl

u , n . T. n . 1.

7.

Hanpunep,

pac

c

aor

pnv

ep

~HK

lJ,HOHaJl

"A

(6.) = j' X (s (x) > a) dx. Torna

6.

M"A

(6.) = j P {s (x)

;>

a} ex, 0'2

(~)

= D

'A

(~)

=

6.

= Sj P

{S

(x) ;> a, S(y )

~

a} dxdy - [M

'A

(

~

)j2

.

6.

li

VH.

n YCTb n pn

~

-~

00

0

2

(~)

;::::: I

~

, a

K03

epepH

lJ,H

eHT n

epe

.re

urusa

-

77

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

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