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

r (2.8.3 ) He npesu uiaer

AI

(r) , orxyna ssrrexaer;

'lT

O lim DN ;!)t

t -'> oo

A

~(r)

~

1

Ii

YTBe

p)l{,neHlle

reo

pex u

5lB

Jl51e

TC51

CJle,nCTBHeM JleMMbl

2.6.3 npn m = 1. •

Flye

-r

s

N~

2)

= I{x Ev (r): I

~

(x) I> a (x)} I,

Tor

na npn YC,IlOBHII

X

VI

M

N

~

2

)

=

2

MN~1).

Teopena 2.8.2. B YCJl

OB1I51

X reopener 2.7.4 c. B. IN ;2

)-

-M

M

2J

]IA

z

(r), rne

A

~

(r) = 2 .1 JqJ (a (x)) (p(a (y» a (x) a (y) B2(Ix - y I)dxdy,

v et ) Vet)

(2.8.4)

B C. B.

.\ a (x) qJ (a (x)) Hz

(

~( x))d

x

IA

z

(r)

(2.8.5)

V(r)

npn r -)--

00

HMelOT onnnaxoaue npenensnsre pacnpe

ne.neaua

.

• <DyH KUH

51

0(

2)

(u, x) = X(I u 1> a (x)) y,nO

BlI

e TBo pHCT XIII,

npl1'le

M Co(x) = 2(I -

CD

(a (

x»

, C1 (x)

=0,

C

2

(

x)=

2a (x) qJ (a (x».

vb JleMMbI 2.6.6 cnenyer ,

'lT

O

8(\ X- yj)

D

N

~

2

)

= 2 .

\'

j dxdy S [qJ (a (x), a (y), p) -

,'(r) v(n n

-

cp

(a (x), a (y), - p»)dp = 51(r) + 52(r) ,

rne

51

(r) II 5

z

(r)

c

oor

se

rcr

ayror p a3 6H eHHIO nepsoro

unr

erpana

Ha naa no MHO)KeCTBa M

~l

If /), 2 (CM. n OKa

3aT

eJlbCTBO reo

pexu

2.8.1).

fl

OCKOJlbKy

l1Jl51

Jllo6

or

o e> °

inf

{[a (x) a (y) qJ (a (x)) X

X

rp

(a(y

»], 1

~

Ix

l~

r

,

1 ~

I

Y

I~

r

}

~

k

4

(

n

)

r

-en

,

TO

2r

A

~

(r )

~

k~

(n) + k

6

(n)r

n

(1- B ) j

Zn-I

B2

( z) X

o

X l l- z' /(2r). ((n + 1)/2, 1/2) dz.

Ecnn 8 -

11

3 YCJlOBIIH VIl

np

n m = 2, TO,

Bb1

6p

aB e = 8

-~,

nOJl

Y'lHM

2r

51

(r)

IA

~

(r )

~

k

7

(n,

~)

r

n6

r.1zn-

1

B2(z) X

o

X l l- z' /(2r)' ((n + 1)/2, 1/2)

dzr

1.

Ilo

cnennee

aup

aaceua

e CTpeM

ll

TC

51

K Hy JlIO npn r

~

00

B CHJlY

YCJlOBHH

VII .n

pa

m = 2.

gS

l-lcnonssya TOrKl1eCTBO

[

rp

(a,

b,

p)

-qp

(a , b, - p)]/[

<p

(a) '

rp

(b)] =

exp

{ -

(a

2

+b

2

) p2

X

X (

2(l-p

2»

-

J}[

ex:p{abp/(I _ p

2)}_

- exp {- abp/(

1-

p)2

)}

)(I - (

2)

- 1/2, (2.8.6)

aCHM

flT

OTH'l

ecKyr

lO 3KBH BaJleHTHOCTb (2.7. 14) Ii COOTlI OUle

Hlf

e

sup

{a (

x)a

(y)}

sup

{B (

s)}

-+(

O,

xoropoe Bb

ITe

Kae

T H3 (2.7.3),

(>,y)E,'(r )

Xv

(r)

s;;>

r

~

3aK JlIO

'laeM

,

'lTO

ssrpax enne

8(1x

-y

\l

52

(r)

I

A

~

(

r

)

= A;-2(r) Hd

.x

dy 5

rp

(a (x),

a(y

), p)dp

<'>

, 0

npn

60JlbWHX

r He n

pessnuaer

B

OX

-lill

A;-2(r) Sj

cp

(a (x)) qJ (a (y)) dxdy S 2pdp

~

I.

!1.

- (2) 2

T aKH M 06

pa30

M, lim

DN

r 1A

2

(r)

~

1, H yrsepxcnenne reopeasi

. (

-700

crienyer H3 Jl

eMMbI

2.6.3 npn In = 2

.•

Ily cr i,

N

~

3

)

= I{x Ev (r): I

~

(x) I< b (xl) I, rne b (x) = b

(I

x /)-

nOJlOrK HTeJlbH

a51

pannansuas epyHK U

II51

, y110BneTBO

p51lO

111a51

YCJlo -

BHIO

XVI. Torna

r

M

N

~

3

)

= 21

s(l)

Ij Z

'l-

l [

CD

(b (r)) - 1/2) dz.

n .

Teopeaa 2.8.3. B yC JIOBHHX reopea u 2.7.5 o. B. [NP)-

-

M

N

~

3

)

I!

A

3

(

r

)

,

rae

A

~(r)

=2

j 5b (x)

b(y

)

qJ(b

(x») qJ(b (

Y»

X

vCr) v(ri

X B2(Ix - y I)

c.

xdy, H o. B.

- j b (x) qJ (b (x)) H

2

(

~

(

x»

dxlA

s

(r) (2.8.7)

r.(r )

npn r

~

00

HMeeT OnHHaKOBb

le

np

en e

nsn

u e

pac

n

pene

ne

a us .

• ,Ll/

l5l

<P

YHf.UHH 0(3)

(U,

x)= X (Iu 1< b (x) Co (x)= 2

C!)

(b(

x»

-

- 1, C

1

(x) = 0, C

2

(x) = - 2qJ ib (x) b (x). B CHJlY n eMMbI

2.6.3

npn

m = 2

11

JleMMbl 2.6.6

)JO

CTaTO

'l

HO []0I<a3aTb,

'lT

O

B(\X- I

IJ

:

~

A3"! (r) j j dxdy S [

rp

(b (x), b(y), p) -

vCr)

v(r l (I

-

rp

(b(x), b (y), - p») dp

~

I.

7·

99

B YCJlOBH5IX reoperau

inf

{b(x)

b (y)

cp

tb (x)) rp (b (y) >

'x

,lj)Ev(rl

X

v(rl

;> /irf-V.

n03TOMY

sr

A5

(r) > k (n)

r"-v

~

zn-If

\-z'/(2rJ'

((n + 1)/2, 1/2)B2

(1)

dz

9

,

(2.8.8)

nYCTb

~

E (0, 1), 3

MHOIKeCTB3

1'.\ H

1'.2

T3KHe

IKe,

K3K

npn

)I,OK33aTeJIbCTBe

reopenu

2. 8.1.

Vic

nOJJb3Y51

(2.8.8),

KaK

H npn

llOK33aTeJlbCTBe

npeLlbl1:tYlll,eH

reopeau,

H3XOllHM

B{jx-yll

A

a

1

(r)

5)

dxdy

~

[q>

(b (x), b (y),

p)-cp

(b (x), b (y),

-p»)dp

~

6.] 1\

zr

:::::;;;

k (n,

~)

r

n(I+Il)

{k

9

(n)

r"-v

~.

zn-

1

B2 (z) X

l o

o

X f

1

-

z

' /(2r) l ((n + 1)/2. 1/2)

dzfl.

Bb16p3B

~

= a-

yin

(&

- 113

yCJJOBI151

VII npn m = 2), nOJJy-

lJaeM,

'lTO

nOCJIeLlHee

Bblp3

IK eHHe CTpeMI1TC51 K

HyJJIO.

Onnp aact,

ua (2.8.8), (2.7.14). MOIKHO

nOKa33Tb,

'ITO

IJ\lx-yP

!~~

A3"2

(r) 5

~

dxdy

~

lcp

(b (x), b (y), p) -

,~,

n

-

cp

(b (x), h

(y),

- p))

dp»)

~

1.

ElrI1M 3aBepWaeTC51 ,J,OKa3aTeJJbCTBO. •

MM

nYCTb

N~4)

=

N~Z)

+M

3

).

Tor

112

,

4

) =

21

s (1) I)

zn-I

{cD

(b (z» -

cD

(a (z)) +

172}

dz.

o

Teopeaa 2.8.4. B

yCJIOBH51X

reopewu 2.7.7 G. B.

lN~4)

~

MM

41

)IA

4

(r), r ne

A~

(r) = J ) a

3

(x)

b3

(y) rp(a

(x»

cp

(b

iY»

X

<.(r) WI

X

B"

(I

x - y I) dxdy/6,

11

C. R.

(2.8.9)

) all (x) <r(a

(x))

H,

(~(x))

dx/{A.(r)

12}

ul,r

flPlI r

-+

00

I1MeW'r

CllHH8KOBbll::

pClwptAeJleHHh.

100

~

VyHKUill51

0(4)

(U,

X) = X(

II

U I< b

(X)

+X (J UI > a

(x»)

y,nOBJleTBOp5leT XIII,

npH'IeM

Co

(Ar)

= 1-

2(])

(a (x» +

,2C!>

(b(x»'

C4 (x) = 2

[cp

(a! (x» H9 (a (x» -

cp

(,b(x» H:I (b (x»]

,--,

2a

B

(x)q>

(a

(x»)

rrpa

YCJJOBH5IX reopenu 2.7.7. B

cany

JIeMM),1 2.6.6

BlIx-yp

DT

r

j"

(4)

= 2 J I dxdy

J

{L] + L

2

+ L

J

+L

4

} dp,

(2.8.10)

v(r)

v(,r)

r ne L

1

=

cp

(a

Qx),

a (y), p) -

<p

(a (x), a (y), - p), L

2

= - [

cp

(a (x),

b (y), p) -

cp

(a (x), b (y), - p)], L

3

= -

rep

(b(x), a (y), p)

-cp

(b(x),

a (y), -

p»),

=

[cp

(b (x), b (y), p) -

cp

(b (x), b (y), -

p»).

L

II

P3306beM

n

epBblH

HHTerp3J1

B (2.8.10) H2

):\B2,

cOOTBeTcTBy'

iounre

MHO)KeCTBaM

l'1

1

H

l'1

2

(CM.

.lllOKa33TeJlbCTBO

reopenu 2.8.1).

Hnrerpan

no

MHOlKecTBY

/to.

1

oueHHBaeTC5I

T3 K IKe,

K2K

H

npH

no-

K232TeJlbCTBe

reopeau

2.8.2. VlCfJOJlb3Y51 COOTHOWeHI151 [exp {zp/

(l

-

p2)}

-

exp

{-

zp/(I -

p2)})/Z

= 2p + 2

p

3 +

Z2

p3/3 + 0

(pfi

+

.....

-,....

--

+ Z4

p

fi

),

exp

{-

zp2/[2 (I -

p2)J)

= 1- z

p2/2

+ 0

(zP"

+Z2

p

")

npn

zp--+ 0, zp

-+

0, P --+ 0,

33nHweM

.

B(!x-yl)

{4

}

2A

4

2

(r) Udxdy

~

i~

L, dp N 2A;-2(r ) Ua(x)

a(y)cp(

a(x» X

B(lx-YIJ

X

cp

(a

(y»

dxdy

~

(I -

p2)-1/2

{exp

{-

la

2

(x) + a

2

(y»)

p

2

1

o

[2 (l -

p2)J)

[exp

{a (x) a(y) p/(l -

p2)}

-

exp

{-

a (x) a (y) pi

(1 -

p2)}]

{a (x) a

(y)}-l

- {a (x) b

(y)}-I

exp

{-

(a

2

(x) +b

2

(y»)

p2/

[2

(1

-

p2)J)

[exp

{a (x) b (y) p/(l -

(2)}

-

exp

{-

a (x) b (y) p/(l -

-

p2)})

- {b (x) a

(y)}-I

exp

{-

(b

2

(x) + a

2

(y»

p2/(2 (1 -

p2»)}

x

X

exp{b(x)a(y)p/(l-p2)}-e~p{-b(x)a(Y)P/(1-p2)})

+

+ {b (x) b

ly)}-I

exp

{-

(b

Z

(x) + b

2

(y)) p

2

/[2

(1

-

p2)J)

X

[exp

{b (x) b (y) p/(l -

p2)}

-

exp

{-

b(x) b (y) p/(1 -

p2)}]}

dpN

N

2A~

(r)

~ ~

a (x) a

(y)

cp

(a

(x»

qJ (a (y)) dxdy X

to,

B(lx-yl)

X J {(a: (x) - b

2

(x»

(a

2

(y; - b

2

(Y)) p3/3 + R} dp, (2.

8.!

[)

101

r'ne npn

(ionb

w

fix

r

R

~

wz,

(x)

az'

(y) p3 llJ15l

J1I

o6

0 ro e > 0. Cne-

1l0BaTe

nb

HO, (2.8.1

1)

He npeBOCXO ll,HT

c (l X

-Y

Il

2(1 + e) A; 2 (r)

~

5a

3

(x) a

3

(y)

cp

(0

(,~))

<p

(a (y))

~

(p3/3) dp.

~

,

0

B CII J1y npOIl3BO.TIbHOCTIl e> 0

noc.ne

naee

ssrpaacen

ae He npe-

BOC

XO.!l.I1T 1, 3HalJHT,

lim

DT~l)

/A~

(r)

~

I,

II

yr

sepacn

enue Teo-

r -->

00

peMbI Bb

ITeKae

T

11

3 JIeMMbI 2.6. 3 npn In = 4

.•

Sau euauue 2.8.1. B cuny

reopeau

2.5 .2 H3Y lJeHH e npenens-

Hbl X p

acn

penea enn s c . B. (2.8.5), (2.8.7), (2.8.9) CBO

.!l.

HTC5l K H3-

YlJeH

11

10 npene

nsns

ix

pacnpenenean

f c . B. (2.7.7), (2.7.15) B

cny·

3

4a e , xorna epyHKUHI1 a (Ixl) , a (Ixl)

<p

(0 (lxl)), a

(Ixl)

cp

(a

(Ix!)),

b (lxl) (P (b (Ixl)) 5l

BJ151

IOTC5l Me,'l.

neHHO

MeH5lIOw.11MI1C

51

npn x

-+

00.

Il

ocnennee 6y )l.eT Bb

InOJIHeHO

, sanpnvep , eCJIl1 a (lxl) =

=

V

log

lo

g

1xJ,

Ixl >

ee

, a b (I

x!)

~

.

0, b (Ix

/)

- nOJIOJKI1TeJIb-

nas

11

Me

,U

JIe HHO

~

\

e

H

51

IO

I

u

a

51

C

51

epyHKUI15l, npa Ixl

-+

00

. n o -

3TOl\IY

npl1Mep

bI npenensn ux pacnpenenenun c. B. (2.8 .5),(2.8.7),

(2.8.9),

Ta K

JKe

K3K

II np

e.a

ensnux p ac

npeneneuus

c . B . (2.7.7) ,

(2.7.15),

MOJKHO CTp OHTb n a OClIOBaHHH

reope

a 2.10.1, 2.10.4.

2.9. KPAT HblE C

TOXAC

TH4 ECKH E HHT

ErP

A

JIbl

B .3TOM n y

iuc

r e

KpaTKO

H3na

ralOTC51 CBe.!l.e HII

51

0 xparuux CTO'

XaCTlI

lJeCKI1X

urr

rerp

anax

(K .C.I1

.)

no

rayccoacxma

cn

Y4aM HblM

~l

e

p a

M

.

n YCTb F (. ) -

cnexrpansuas

Mepa

on

noponnoro

(B03MOJKHO.

o

606

111e

mIO

ro)

cnYlJa

l1rIOrO

non s , T. e.

o-

KOHelJHa

51

Mepa

a a

(R

n

,

\En)

TaKaH,

lJTO F

(~)

= F (-

~)

,

~

E\En

Ii

)IJI5'! HeKOTO

pOrO

p>O

S

(I+

IA

I)

-

PF

(d

'A

) <

oo

.

R

n

np

eJlTIOJIOJKI1M, 4TO F

(.)

-

Hea

TOMlIlJeCKa51 M

ep

a: F({x}) =

= 0

.!I.

JIil

JI

lO6

0r o X E R", P aCCMOTpllM

KOMn

JIeKc

Hy

lO

rayccoa-

CKy lO

0p

TOr

OI-I

aJIb

HYlO

Mep y ZF

(.)

co

CTpyK

Ty

p HOH epYHKUl1eH

p (. ), T . e. CeMeH CTBO COBMeCTHO

rayccoscxnx

C. B.

TaKYlX,

41'0

.!I

JIil

.

TJ

I0

6bIX

~

j E\En, i = 1, ... , r C

KOHe4HO

H F ( . )-MepOH ssr-

no.menu

cBolkTBa

1)- 4) n 1.2.

3aMeT

I1M,

41'0

M3

srax

CBoikTB

cnenye

r:

a) c. B.

Re

ZF (t.)

11

I

rn

Z»

(

~)

H e33Bl1CI1MbI II

I1MelOT

pacrrp

e)\eJIeH l1e N (0,

F

(~

)

!

2

)

;

6) KOr)l.a

~l

U

(-

~l

)'

."

... ,l1

r

U

(-

~r)

-- HenepeceKalOUJ,HeC5I MHOJKeCTBa,

Torn

3 c. B.

Z

F(11

1

) ,

...

, ZF (

I1

r

)

He3aBH

OIMbl;

B) eCJIH

11

n

(-

t.) = 0 , TO

C. B.

ZF

(11

), ZF(- t. ) He3aBI1CI1MbI

Ii

I1MelOT

pacnpe)l,cJIe

H He

N (0, F (6)!2).

102

E cn l1 F (

IJ

) = 1

L1/

, TO ZF(' ) = : W

(-)

Ha3bIBalOT r ay CCOBCKIl M

R

n

6en bIM

UlY

MO~1

IB

L

2

(R

nm

0603

H3 4

Y1M

uepes

,

F)

.!I.e

iiC

TBl1TeJIbHOe

rllJIb6epTOBo

npccr paac-rao lKnaCCOB 3KBHBaJIemTHOCTI1 KOM

nn

eI(

CH03H

a4HblX

A..

of - f «I)

(m

»)

(j)C.R"

'-1

I -

\flYHKU HH m -

rrn

X , .. . , x

,x

\...

,J- ,...,

tn,

})],oBJIeTBOpRIO

(1)

.(m))

- t

«(I

) (m»).

W.I1X YCJlOBH5IM

f

m

«

- X ,

.,.

, - x -

In

X ,

...

, x ,

m

II

t;

IJ2

=

~

It;

(X(l)

, .. . , X(m») InF (dx(

j))

<

00

.

Rnm i= 1

0 4eB

H.!I.I-10,

1/

i-;

1/

ecrs HOpMa B -L

2

(R"

m,

F).

4epe3

L (R

nm

,

F)

c:

2

c:

I

2

(RMt, F)

06

03HalJI1M npOCTpaHCTBO <pyHKUHH t; EI (R nm,F)

2

(I) (m ) _ (i

j

) ,(1m) u

TaKHX, 41'0 tm

(X

, , X ) -

i-,

(X

,

...

, X )

.!I.JIH

JII0 60H non-

CTaHOBKI1

"m

= (ill , i

m

)

H3 rpynrru nO.!l.CTaHOBOK

II

.

H Op Ma

m

nm

B

L

2

(R

, F)

BbI611pae TCH TaKOH »ce, KaK H B L (R

nm,

F).

2

Bo06111e, eCJIH fm E[2 iR?", F), TO CI1MMeTpI13aUI1

51

sym{fn,}=

~ f ( (i )

(im

))! I L (R

nm

F)

~

m X l •

..

. , x tn .

npHH

a)J.n eJKI1T 2 ' Ii

J]m

Enm

IIsyrn

tm

II

~

II

fm

II.

n YCTb

{

~}r

= {l1

j

,

i = + l, + 2, '''}r' r = 1, 2, ... -

ne

nepe

-

cexaio

ume

c s n

pn

KaJK)J.OM

r 113MepHMble MHO/KeCTBa

(l1

j

= -

I1

-

i

•

i = 1, 2, ...),

06pa3YIO

w,l1e MOHOTOH

H.

ylO no r nOCJIe.!l.OBaTeJIb-

HOCTb C

4e

THbiX

pa36

HeHHI1 tr ;

)\JI5I

KOTOpOH

lim

s

up

d

ia

m{M

=

r--> oo 6El61

r

= 0, a

{11

};

' =

{~

m

=

L1

lI

X .. . X

L1

jm

, s, E

{

~

}

r

'

il'

...

.

im=+

1,

+ 2, ...}, r = 1. 2, ... - HH.!I.YUl1pyeMa5J n OCJIe.!l.OBaTeJIbHOCTb pa3-

611eHliH MHOJKeCTBa

R'":

CV

yH

KUII5I

1m

=

fm

(x

(!),

... , x(m») Ha3bI-

BaeTC51

npOCToH . eCJIH n

pa

HeKOTopOM r> 1 OHa n pl1HI1MaeT

nOCT0

51H

lfb

le 3HalJeHIIH

aa

MHOJKeCTBaX BIi)J.a

~

m

=t.

j

l

X

."

X

~.

1

m

ir= ± 1, ± 2, ... ;

1m

OTJI

li

lJHa OT Hy nH n l1Wb

.!I.n5'!

KOHelJHor O

(I)

'l

l1c n a TaKHX I1HOJKeCTB,

.

t-

m

«

X

(I '

, • •• • X

3

m

») - 0 (

X ,

-,

eCJIH '"

(m»

) N { A m A

A'

•

-

...

, X E m = U = U

jl

X

...

X UJ".> Jr = + J

r,

npl1 HeKOTO-

pux

r

=1=

r'}.

MH

OJKeCTBO npOCTbD. cfJYHKUI1H 0603Ha411M vepes

A A

nm

[2 (R )

c:

L

2

(l~nm

,

F). EC JII1

1m

EI (R

nm,

F), TO K. C. H.

orrpe

-

2

.n:enHeTC5I

cjJOpM

ynoH

m

ImU) = :tnl)- l

5'

fm

(X

(I),

..•

),i

m

))

nZF( dx(i») =

Rnm

i=

1

. m

- I - (11)

(fm)

n

= (m!1

~

f

m(;~

,,,

,X

)

ZF(~

j)

i1,..·,i

m

r=!

103

ECJIH

fm

E[ 2(R

nm

,

F) = L

2

(R

nm

• F) nt., (R

nm

,

F), TO t;

(f)

=

2

= l

IT

(sym

n.

M 11 (f) 1= I

tfm

l

lz

/ml.

M H

O>KeCTB

O npoc

rux

ep

YHI\-

m

"

B I

2

(R

nJTJ

UHH

I

2

(R

nm

• F) nJIOTHO • F).

n03TOM

y n OCTpoeHH oe 1

130-

A

- nrn

MeTpl1lJH Oe

(c T04110CTb

JO

.no

MH

O~

H

TeJlSl)

COOTBeTCTBl1e L(R ,

F)

-+

t,

(P)

(7m

-+

i;

f

f))

M

O>KHO

npO

Ao.rl>KH-:b aa

3aMbiK

aHlie

nc

ex

"

¢ YHKUHH I; E[ 2 (R

rtm

• F), I<OTOpo e cosnaaa

er

C I

2

or': F) . T a-

1\l1M

06p

a3

0M,

.nJlH

np

0l13B

Oll

bHOH epYHKUlJl1

fm

EI

2

iR?", F) K .

C.l!.

m

i; (f) = (ml)- I rf (X(I), '" , x

(rtI»)

nZF (dx

(J))

Rnm ;

=\

ecri, H30M e TpHlJH

oe

(c

TO'lH

OCTbJO

no MHOmlJTem

l)

COO

T6eTCTB

lle

(R

rtm

I

2

, F) B L

2

(P). VIHTer paJi

on

peAeJl5JeTC

SI

KaJ< npene

,n

t; (I)=

=

I. i. m. i;

fr<'\

r-ne

-f~)

E1

2

(R

rtm

,

F) - nOCJIe.nOBa TeJIbH OCTb

npoc

ru

x epYHKUHH, aCCOIL!1UpOB3 HHa H C

CH

CTe

~lO

H

pa 36H

emll1

{t1}

r'

r =

1,2,

... .

'F

MomH

O noxa sa

rs.

4TO T

aKot!

npezien cyinecr

syer

H

lie

3

aBIi-

c

ar

Hii

OT c

no

c

o6a

pa arinea

aa,

Hii

OT Bbl60pa nocn

enos

ar

e

ns-

HOCTH npo

crux

ep

YHKU

IiH

.

AH

aJl

OfH4IJO

MO>KHO onpenenu

rb

K.C.11.

11

.nJlH nex

or

op

or

o no.ni\IH O>KeCTBa A c Rnm .

OTi\l

eTI1M H3BeCTHble

CBOHCTBa K .C.Ii . :

1) Mlm(f) = 0, f

EI

2(R

nm

, F);

2) 1 (I) = 1 (sy m f) - BeUl.eCTBeHHaSi C. B., n

pliH

a.D.Jl

em

am aH

m m

nm

L

2(P),

r .ne l EI

2

(R

,F

);

nm

3) Ml

m

(I)t, (

fP)

= 0. ,

*m;

!. cpEI

2(R

,

F);

4)

Mil

m

(1)

1

2

=

II

t;

IP~

/

ml

,

t;

EL

2

(R

ll

m

, F);

rt

Jrt

5)

Ml/

m(f)1

2

:::;

llfml!2/

m

l,

ImEI

2

(R

,F

).

OJ:>

OpMyJlHp

yeM

HaIl6

0.T)ee BIDi<Hoe CBDHCTBO K. C.

H.,

KOTOpoe

sup

asca

er

ca

ep

OpM

y.

7J

OH

Y1

TO:

eCJUl %' ..

.•

CPr

EL;

ur,

F) - oproro-

HaJlbH3H CIlCTeMa ql

yHK

Ullit,

ail'

...

, jr - uerure nonoxorren

snu

e,

il

+-

...+ ir= N. on

pen

e

rma

.nJlll s cex i = I, .,. • N

ep

y HKUHH

gt =

CP

s

.D.JIH

il ;- ." + i'

_1

< i :::;il + ... + is ' Tor na

m

n H

if

( )

(fr (x) ZF(dx» = re, (X(lJ)

...

gN (X(

N»)

X

r

=!

N N N

X n

ZF

(dx<

r»)

= rsym

In

s,

(Xlf

))]n

2

F

(dx(r»),

r

=1

,.=

1

r=1

r ne Hi (u) - j-il nOJIHH OM 4e6blUlesa -

3pMHTa.

104

Mepa

F na

IR

"

Ha3bIB<le,

C51

JlO

K3

J1

bHO-KOHe'l

Ho

H. e

CJlI1

F

(t1)

<00

.n

Jl51

JlKl

6

oro

O

Ilp

aHH4eH noro

~

E

~

n

.

fO

B

Op51T

, 4TO ce MeHcTBo JIo-

J<

a JlbHO xonexn ux Mep

l

F

~

}

,

).l.

> 0 CXO

.n.

H

TCR

JlOKaJIbHO CJl

a6

0 K

.1J

0K

aJlbH

O

KOHe:1JHOH

Mepe

F

o

,

eCJlII

J~~)

f (u) F

Il

(du) =

~

f (u) r,(du)

Rn Rn

.nJl

51

JlKl

60H nenpepu

aao

j

ep

y

HKU

HH f (.) C or

pa

H

li

4e HHbIM HOC

ll-

ren

ew

.

Ce

Mei

l c TBo Ell JlOK

aJlbH

O KO

lle

4HblX Mep CXO.nHTCH JIOKaJlbHO

cJla6

0 K

Mepe

F

o

,

eCJIH

.ll.

JI5I JIIo6 or o Or paHH'le H

HOrO

t.\

I F

o

(oM -

= 0 lim F

il

(t1

) = F

u

(t1)

.

~

oo

OTM

eTHM Ba )KHOe CBOHCTBO K. C. H

.,

aup ascaexoe CJIe.nYfOUl.HM

yr

sep

xo

eanea .

JIeMMa 2.9.1. n YCTb {F

JJ}

'

f.l

> 0 -

ceMeHc

TBO n

ear

oaaxec-

KHX cnexrpansnux Mep na R", xoropue CXOAHTCH JIOKaJlbH O c.na-

60

K H

eaT

OMHlJeCKOH cnexr pam. uoa

Mepe

F

o

aa

R"

,

KI

t =

= K

Il

(x(l), ...

,x

(

m),

t1

),

f.l

> 0, I

t1

!< 00 - ceMel1CTBO H3MepHMbiX

R

rt m

L

2

(R

rt m

na epYHKlfHH, npananne

acaumx

,

F

IJ.)

Ii

CXO

,lJ.HUl.

IIXC

SJ

K H

enpe

rblBHOH

epyHKUli1i K

o

(x(l), ... ,x(m),

t1)

paBHOM

epHO

B Kam -

.nOM MHOmeCTBe

t1

m

(A) = {x ER?" j x(

J)

E

t1

(A), j = 1, ...

,m},

,I

m

lim 5 I«; (x(l), ... , x(m" t1) 1

2

n

FIJ.

(dx(j) = 0

(2.9.1)

A-'>oo .

Rnm"

t/ r.(A) /= 1

paBHOMepH O no

f.l

E (0, 00) Ii

).l.

= O.

Torna cyiuecrayer K. c. H.

m

r

Ko(

x(l)• ... , x(m),

t1)

n2

F

" (dx(

J)),

(2.9.2)

f am ;=1

T. e . Ko EL2(Rnm,Fo), npHlJeM K. C. H.

m

rK

Il

(X(!),

...

,X(m

),

t1)

nz,(

dx(J)

)

Rnm

npu /!

-+

00

c~60

CXO)l,RTCSI K K. C. H. (2.9 .2).

\05

2.10. Y

CJlOBI

HI

nPHT5I

)I(

EHlHI

I(

nOJI

YY

CTOll'HiBbIM

npOU

ECCAM 4lYH Kl.lHO HAJI O B OT O

jlHOPO

AHblX

IB

O

TPOnHbl

X

rAYCC

OB CKHX

nO

JIEll

Bsnue

H3Y4

eHll

e npenern.nux pacnpenenemoi umpox

or

o

KJl3CC3 lPY

HK

UHOH3JlOB OT ra

yccoscs

nx nOJleH C CH

JlbH

Ol

1

3a

BH'

( li MOCTb lO CBOJUiJlOCb K

accnenoeaamo npenem.iurx pacnpene -

JJ

eH

fIl'i

lPYHKU

HOHaJlOB j g (x) Hm

(S

(x)) dx, r ne g (x) - BCCOBa 5I

6(~1)

¢ YHKUH5I , H m - m-i'i nOJlHHOM

LIe6b

lweB

a - 3 pMHTa,

~

(

/-l)

ce

o

MeH0TBO paCWli p5l101ll,HXC5I 06J1aCTel\.

Onnuiev npe

nensnu

e pacnpenenenus

TaK

HX CPYHKUlWHaJlOB

e KOpp C.15I

UII

OHHbIX Ii cnexrpansau x repvanax.

ll

Jl51

sro

ro B

Be

·

.n.e

Mraxoe onpenenesne.

C Jl

Y4aii

Hb

Il

1

npoue

cc

X:

Q XR

2r

-+

Rl Ha

3blB

aeTC5I nOJlYYCTOI

:'r

<j

li Bbl

t>

1 c nap

aa

erpon x > 0: a) B Y3KOM CMbICJle, eCJlH

.uJl51

Jl106o '

d

ro Q > °X (at) = ar X (t); 6) B WHpOKOM CMbICJle, eCJlH

MX(t)

=

= 0, P (f, s) =

MX

(t) X (s) <

00,

P (at, as) = a

2x

p (t, s).

0 4eBHJlHO, Jl

Jl51

rayc

co

scx

n x npoueccos a) H 6 ) 3K

BHBaJleHT

·

r

fl hl . 113 onpeneneans c

.ne

nyer, 4T O p (f, s)= (sf)" P Cli s

it

, V tis) =

= (sr)" P(s

it)

.

E CJlH

~

(t), t ERl - crauuoaapnsn; B Y3KOM CMblCJle npouecc,

'TO npouecc X (I) = t"

r"

(In t), t ;» 0, X (0) = °n. H. HBJl5IeTC5I

nOJly y cTOH4l1BblM B Y3KOM CMbICJle

aa

R~

c nap

aa

erpov x > 0,

d d d

n OCKOJlbK Y X (at) = (at)x.

r"

(In t + In a) = a"

<t

x

r"

(In t) =

a"

<X (I).

Hao

ri

opor, e CJlH X (I), t E

R~

_

- rroJl y

ycTOH4HBblH

B Y3KOM C"

lb

IC-

ne

CJ1

YlJaHHb

Il

I npouecc C napaver po a

x>

0, TO npouecc

r"

(t) =

= exp { - xi } X (exp {t}) , t ER

1

6y

.ueT

cTaUH

OHapHblM

B Y3KOM

d d d

xt xa

X

(h

a

xt

X (e

l

C:-1bICJle, TaK

KaK

r"

(t +

0)

=

e-

)

= e-

)

=

~

(t

)

.

AH

aJI

Of

H4H bIe COO

HlOWe

HH5I COXP3H5IIOTC5I H

Me)K

JlY norry-

Y

C

T

O

r

I

4

H

D

b

l

~

tH

Ii

cra

unoaap

uua

n npoue

ccaxa

B

WHpOKOM

CMbIC-

ne. T a KHM o

6p

a3

0M,

oroo

pax

ea ue X -+

r"

:

r"

(In

t) = t - xX (I)

3a;13eT COOTBeTCTBHe

Me)KA

Y CeMeHCTBaMIi rrOJlYYCT

oH4HBb

lX npo-

u eCCOB aa

R

~

ft cr aun oxap

nux

npoueccos na Rl 8 Y3KOM

}jJl

H

IIJ

ll

p OKOM CMbICJle . E CJlH

oc060

He o ro so peao , B .uaJlb

Hc

i'!

We

M

p aCCMaTp H

saI<lTC

51

nOJlYY C

Tol

l4

11

BbIe n

po

ueCCbl B Y3KO M CMbICJIe.

Flpocrenum

re

npllMep

OM

nOJlY

YCTOH

4HBor

o

np

oue

cca c n

a-

p a M€Tp OM x = 1/2

HBJl51

eTCH B

liHep

oBcKHH

npouecc w (I) , t E

E

R~

,

T. e . npouecc

6p

oYH

o8

cKor

o

.Q

BH)KeHH

51

C b = 1. ¥ CTOI!'

Ij

ll

Bble npoueccu C napaae

rpo

» y E (0,

2J

HB

Jl51IO

TC51

nony y

cr

o a-

IjJi

BblM

li

C

[l

a

p

a

~

Ie

T

p

O

M

x = I

Ll'.

::nI

l n

p0ll,e

CCbI HMelOT He3

38H

-

10 6

C

HMb

Ie n

pnpauie

n n s Ha uen epecexaio umxca I1HT

epB

a Jia x . O

,nH3-

KO c YlUecTB Y

IO

T

nOJl

y

ycT

OHI:IH

Bble

mpoueccu , npapauieaa s KOTO'

ps

ix

JlI1Wb C'JpeM 5ITC5I K Hy JlIO. Tax , nYCTb X (t), t E

R

~

_

- nOJIY-

YCTOli 4HEbl}f

npouec c C napauerposr x E(1/2,

I),

n

p

l1

4

e

~

1

X (t)

lIMee-r crau nonapnsie npapauienna Ii X (0) = 0,

MX

(I) = 0,

2X

MX

2

(t) = 1. Torna

MX

2

(I) = t

,

.MX (t) X (s)=

!/

t)2x + Is1

_

-

It

-

51

2

><.

]12.

TIp» s

-+

00

M (X (Ii +1) - X (t)) (X (t + s +

1)-

- X (t + s)) N x

(2x

- I) 5

2

;<

-2.

n O,JlYYCTOHYHBhle npoueccsr MO)\(·

HO

sanaaars c rrOM

OLU

blO K . C. H.

B n. 2.

2-

2.7 H3 Y4

eHHe

npenem.nux pacnpenenenan nox

ans

-

HbIX epYHKUJlOHaJlOB OT rayccoscxux nOJleH CBe.ueHO K H3

yyeHHIO

npe.aensnsrx p

acnpe

.ueJIeHI1H c . B. (2.2

.1

) H c. B. Y rn.u .g

(1

) (CM.

n. 2.5).

Flpnsene

s, npaa epsr pacnpeneneaua

YKa3a

HH

blX

c . B•

B C

nexrpan

snux

T

C

p

~

IH

H

a

x

.

Bse

ne» Ta Koe

ycao

s ne.

XXI. n Y CTb S: Q X R

fl

-+ R

1

-

C. K. nenpepusu oe

O,.'.l.

HOP0 .u-

Hoe 1I

30

TpOmi

oe

ra

Y

CC

OBCKoe none

co

cnexr pansnof n JlOTHOCTblO

[(I

AI) = h(l /, D/ IA!fl- a TaKOH, 4TO

f(!

/I)

=

f(p)

=l

s(l)!

pn

-l

g(

p),

- 1 -

pn

-!g

(p) EL

1

(R+), g (p) = h (p)/p

l-

n, a E(0, (n + 1)12), h (p) -

nenpepsisa aa

CPYHKUH5I

B nexoropof oxpecruocrn Hy .'IH, h (0)

*0

H h (p) orpaunuena aa

R~

.

0 60 3HaYHM C

2 1

(n, a) = 2

u

n

/2

r

(0..

/2)/1' ((n - a)/2).

J1eMMa

2.10 .1. Il pn

Bbm

OJlHe

Hl111

XXI (B03M O)KHO

6e3

npen-

n O

ll

O)KeHH 5I 0 rayccosocrn nons) H r -+

00

K.

cp

. n O

Jl51

B (r) =

=

h (0) C

21

(n, a)

r-

U

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

.& vI3BecTHo [5, c. 59] COOT

Ho

we

HHe

00

2

P

1

5J

).l

(at) tP-1dt =

-

r

((~~

+ p)/2)/[a

p

r

(1

+

(/1

- p)/2)],

o

-

Re

/1

< Rep < 312, a> 0.

C Or JlaCHO (1.2.8) }j JleMMe \.5.4 npH r -+

00

B (r) =

00

5Y n (pr) Is (I )Ih (p)

pa-

1dp

=

o

= h (0) C

21

(n, a)

r-

(t(l + 0 (

I))

. •

Flp

a YCJl08HI1 XXI nYTeM, aHaJID

rH4H

bIM

,noKa3a

TeJlbCTBY JleM'

M bl

2.1.3, n

OJ1

YQaeM npa a E(

O,

min {nlm, (n + 1)/2}),

m>

\,

r

-,.

00

a;n (r) = DI SHm(s (x)) dx I= C

22

(n, m, a) h

m

(0) X

"(f )

u

2n

ma

>

-

(1

-+

0 (l )),

r,Ue

C

22

(n,

m, ex) =

c~

(n, a )c

2

(n

, m

,»,

v (I)) .

: 107

O YCTb t E[0, 1). Pa cca o

rpaa

C

Jl

YlJaHHbl e npouec

cu

x; (t) = C

23

(n, m, a)

liT

r[Jn

l2

(i

1.(1

) +

...

+ A(m

J

It i

l"

) X

R"m

m m

X

11..

(

1)

+ .,. + A

(m

),

-"

IZ) [

Fl

IA

U)

j(

cx

_

n)/2

} nW (dA

U

) ,

(2. 10. 1)

1=1 i=1

r ne B

np

aso f

'l

aCTIi (2.1 0 .1) CTOUT K. C. H.

110

rayccoscxony

6e JIOMy Ul y MY W

(.)

B (R

",

\8n), a E (0, min {nlm, (

11

+ I )/2},

"/2 V

C

2J

(n, m, a) = (2n) / C

n

(n, m,

a).

Hc

no

.7Jb

3YH CBOHCTBO nOJIy y cTOHlJHBOCTH rayccoscxoro 6eJIOr o

d

wy

Ma

no

psn xa x = nl2 (rpopa an suo W (d (ax)) = a

n/2

W (dx»,

y6e)l{

.!l.a

eMC51

B TOM,

'l

TO npoueccu X

m

(t) 1l0JIYYCTOH'lHBbI B

y3K

O~

1

CMhICJIe 1l

0p51.!l.

Ka x = 1- am/2n ,

np

li

'leM

x E(1/2, 1).

npl1

BJI

eKa51

v

COOTHOWe

HH51

Jv( Z) = o(l /V z), z

-+oo,

J,,(z

)Nk

1

z ,

2-+

0,

MO)l{H O

np

os

epar

s,

l.J

TO

MX

~

,

(t) =

C

~3

(n, m, a) t .\

J

~

/2

(I

A(l

) +

...

+

').,

}m) Itil" ) X

Rnm

m m

X I

",

(1) +

...

+ ",(m) I-

n

{nI

AU!

la- n} nd",(j) <

00

(2.10.2)

j= 1 i=1

np

n K

a)l{

.!l.

OM t EI

O,

I) . O TMeTl-lM,

'lT

O npu m > 2 npoueccu X m(t)

He raYCCO Bb

l.

Teopesra

2.10.1. O YCTh B

hln

OJ

lHe

HO YC

.1JO

B

lie

XX

I fl a E

E(0, min {nlm, (n + I)/2}, m > I. T

or

na npn r

-+

00

KOHe

ll

HO·

MepH bJe

pacnpeneneuns CJIy

'la

HHbIX

np

ou ecc

os

Xm,

I(t) =

~

nm (£(x» dx/[,

n-

mu/2 X

v(r/

l/n

)

xVC

2 2

(n, m, a) h

m

(0))- 1,

t E[

0,1)

cJIa60

CXOjl.51

T

CH

K KOHelJHOMepHblM pacnpenenenu

sa

n po u ec ca

X m(t), t EI

O,

I) .

• C YlJeTOM (1.2. 16) Ii epOpMyJIhI YlT O (CM. n . 1.9) HaXO

).l,

HM

m m

Hm

(~

(x) = .\ I e

i

(x. l.(I )

+.

..

.j-l.(m» > [II Vf (l A

(/)

\)}n W {dA(i».

Rnm

;-1

;= 1

YIcn OJIb 3Y

Sl

(I .4.5), n o.n

yxaea

'I

, )

V-

ma

(2-n(2

r'J

I~

l

\.

X

m ,1 l

t)

= (.2 8 (n, m, a

t

, J 'l IZ

('"

+

...

Rnm

108

...

+

')..

(

m)

It

l/ll

,.

) 1f.(1 ) +

...

+ A(m)l-

n

(2 Vnh (IA

U)

i)lh (0) X

1= 1

m m

x{nI ).(J) /(

U-

n)/Z} nW(d

"Y\

(2.1 0.3)

i= 1 j= 1

B

bIn

O

J1H5I51

aaaeay nep

ea

eaa

u x ",(})r =

~

(})

,

j =

I,

'"

.m

H

IICn OJlb 3YH

n

o

.~

yy

cTOHlJH:B

O

CTb

rayccoscxor o 6e Jl

or

o uiya a, H3

(2

.10.3) H

aX

OJlHM

d

X»;

(t) = C

23

(n, m , a.) }

/T

S'

J n/2 (1),,

(1

) +

...

+

A(m

) It

Il")

X

Rnm

m m m 1/2

X I

~

AU)/

-n/2

nIA

ll

) I(U- n)/2 {n li (I"'(/) II,) II- I (

O)

} x.

i= 1 1= 1 ;= 1

m

X n W(dA

(i»

).

(2.I OA)

1= 1

11

3 (2.10.1),

(2.10

.4)

Rm,r (t) = M (X

m

.

r

(t) - x; (t))2 =

C

~

3

(n. m, «) t X

m m

ttl

All) 1 t

l/n

n

{n IA

lj

X 5

J

~

/2

(I

~

)

I

~

All) I-

)

I"-'ll X

Rnm 1=1 1=1

;=

1

m

«o

.

(1.

(1

" '" , f,,(m ) ndA

(/

"

(2.10 .5)

i=1

r-ne

Q, (A

'"

....

,

A(

ml)~

[ Vbh (I

A"

) I

,-

')/h (0)) - I r

n YCTb 'l'(r) -

00,

HO

'P

(r)

lr

- 0

np

n r

-+

00

, Pa30

6beM

HH-

r erpaa B (2.1

0.5)

aa

naa, B nepBOIA

11

3 KOTOpblX (

06

03Ha4HM er o

11)

ntrrerp

np

oaan

ne BeJleTCH no MHo)l{

ecTB

Y B1= {A(

Il

E R

n

:1

A

(/

)I

~

'If1

(r), i =

I,

... .

n},

a BO BTOpOM (o603H

a'lHM

e

re

1

2

) -

no

MH

O)l{eCTBY 8

2

= R

nm

"'8

1

•

B CHJlY YCJl

OBHH

XX

I

npn

JII060

M

1(1)

1 (m )

E>

O cy

tu

ecr sy e

'r

v, r

ax

oe,

'l

TO npn r

>,

o

Q

r

(I'.

, ... ,

I'.

<

1::,

(/.,(1

),

...

,/.,

(

m»

EB

l

.

Torna H3 (2.1 0 .2)

11

< ek

2

H, CJIeJlOBaTeJIb-

RO,

nnr

erpax

II

MOlKer

6h

ITb

caena

n KaK y

rOJlH

O MaJIbIM BMec -

'Ie

C e. E C

JlJ1

(A

(I),

... , A(

m»

E8 H3 yCJIOBHH

XX

I ssrrexaer

2

,

Qr (;'

(1

), .. . ,

A<

nl»

< k

s

'

H

cnonss

ya BH

).I,

MIIO)l{CCTBa 8

2

H COOTHO-

109

uienae (2.10 .2), MO)l{H O noka

sars

.

'lTO

lim 1

2

= 0. T a

KIl

M

o6pa-

(-

'H"

30 M, lim Rm

.r

(t) = 0,

orx

y a a MO}f{HO 3aKJIIO

'l

HTb. lJTO .1I}15I JIIO-

r

~

oo

6blX

a j ERi, j = 1,

...

, p

p

Jim M :

~

a j (X

m

.

r

(tj

) - x; (t

j

))2 = 0. •

r~

oo

j =1

M O}f{HO

nOJIy

'lH

Tb npyrue rrpeJI,CTaBJIeHJ15I npe

JI,

eJIbH

Or

O npo-

uecca

X

m

(t), t E[0,

I]

, OCHOBaHHble na IlCnOJIb

30Ba

HHH BMe CTO

epOPMYJIbl

(1.2.16) npeJI,CTaBJIeH

J15I

(1.2.19) (IIJI H

(1

.2.12) npn

n = 2 II

(1

.2.13) npu n = 3). P a3JIOm eHIHI npoueccos X (t),

m

t E[0, I] 6 y

JI,

y T COCT05ITb H3

p5IJI.OB,

conepacaumx K. c.

Ii

. no

ceM

e

ikTBa

M B3aHM HO aexoppennponauusrx rayccoscxax 6 eJlbIX

WyM

OB B

(

R

~,

\5

(R~))

.

- ..

nO

BTOp5l51 JI.OKa3aTeJIbCTBO Te

opeMb

l 2.10.1 C HCn

OJIb30

BaHH eM

(i'

~

2

.

1

2

)

H (6

.561

) [44

1,

npn m = I, n = 2 n oJI

y'l

HM rayccos-

CKII

11

npo

ue

cc

00

Xl

(t) = C

24

(a)

Vi

fpca-

3)/2

) 1 (p

VI)

W(dp), t E[0, I].

o

rne

W(.) -

rayc

coB cK

lli

i 6eJIblH WYM B

(

R~,

\5

(R

~»

,

a c (a)=

24

= l/ (2- a) r ( I - a /2) r (3- a12

)1

[

(r

(a/2) r ((3- a)/2)}1/2

n

I/4

J

•

0 < a < 3/2.

y

!3

YlJHM sonpoc 0 npenensuux pacnpenenennax

~

n. Y

m.

~

.g

(I )

(CM. n. 2.5). 0 YCTb aunoanensr y CJIOBII5I IX, XXI. Tor n a npn

fl

-+

00

B CIfJIY JIeM

Mbl

2. I0. I H reop e

as

r 2.5. I

D [ .\ g

(I

x I)Hm (

1;

(x))

dX

] = C

25

h

m

(0)

fl2ri-

ma

g

2 (fl) (I + 0 (I) .

6(~

)

rne C

25

= C

25

(n, m, a,

g

,~)

= c;\ (n, a)c

20

(n, m, g,

~)

.

XXII. Cymecrsyer epyHKU

If51

g

(I

x I)

TaKa

5I, 'l TO

m m

f nI,,(

f)

la- n I f e

l

(

I..

(I)+ ...+I..(m

l

•

x

)g(x) dx

l2

nd

').,

C

f)

<

00

,

Rnm

j

~

l

6(11/nl ;= 1

n pHlJeM

m

J~

n;,

I nI" Ci)

/a-

n I J / (AO)+

".+

t

.cm

).X) X

Rnn1 i= l

6(

/l/nl

m

X {g- I (fl) g (u ] x DnV h (I

'A

U) I

f;L

-I)

/h (0) - g

(x

»)

dx 1

2X

j

=!

110

m

X n d" C/) = °

i= 1

,nJI5I scex t E[0,

I]

.

Teop

eMa 2.10.2. n YCTb

nb

lIlOJI

HeHbJ yCJIOBlH1 1, IX, XXII n

a E(0, min tnlm , (n + I )/2}. Torna npn

/-1.

-+

00

KOHe1

lHoMep

Hb

ie

pacnpeneneuas CJIy

'laiIHb

lX nponeccon

V

m

.

~. g

(t) = S g (I x I) Hm

(

~

(x)) dxl [

f;L

n-

ma

/2X

6C

~

l

l/n\

x V C

25

h

m

(0) g (fl)],

I.

E[0. IJ (2.10.6)

cJIa60 CX

O,ll,51TC5I

K KOHelJHOMepH blM pacnpenenennaa npouecca

m

Y

m,g

(t) = (C )- 1

1?

r nI

').,(J)

/c

a- n)/2 X

25

(?fUn ;

=1

m

X . r g(x) eIP.(I)+..+ty nl"

"d

x jnW (d

').

Y\

t E[0.

Jl.

(2.10.7)

6

(1

l/fI

,

;=1

LI.oKa33

re

JIbCTBO reopexsr 2.10.2 aaanoruuno

ll

o K33a TeJIbCTBY

reope

uu

2 I0.I

R CIlJlY

YCJlOBJ151

XX

II

MY~,

g(t)

<

00

,QJl

51

scex tE

[O

, II.

ECJIH epyHK UH

H

gttl X D= IX

lB

.

~

0. 3 MHO)KeCTBa

~

(u)

npe,uCTaBJI5IIOT

co60ii

ceMeHCTBO uiapoa V (r), TO yC JIOBH5I IX.

XX

II BhInOJlH5IIOTC5I. npJ1 3TOM g1

(I

X

J)

= IX lB.

B>

0. C =

25

=

~

(n, a) mJ f

~

IX I

BI

!J

I

BI

X - Ij

I-a

mdxdlj, a E(0, min i nlm,

O

(1)

V()1

(n + 1)/2}).

Ilepexcns K cepepHlJe CKHM xoopnaaar ae (1.1.1), Mom HO nOJIy.

lJIJTb COOT

HOme

HlJe

JIx IVel( A.

xl

dm

(x) = pv (2np/l

A.

\)n/2 ) (

n-

2)/2 (p I

A.

D. v> O.

s(p)

Hcnonsaya epupM

yJIy

(dldp) rt(x) dx = rt (x) dm (x), H3

no-

t( P)

$)

cn

ezur

er

o COOTHow eHm l HaXO.ll.IfM

p

SI x IVi (I.,

X)

dx = I, v (2n

rl

l x

J)

n

12

J(n_ 2)/2 (r I

A.

Ddr ,

t{P) 0

III

Teneps

npene

JIb Hh

li

l

np

ouec

c (2.10 .7\ MOIKIIO sanu

cars

B BHJl.e

m

fll

Y

m,g,

(t) = (C

25)

- I/Z J' nI'Ali! t

a

-

/2

X

R

fU'7l

; =1

m

X [ 5 Ix

If!

el(A.

(I)+...+J

JtnJ

,X

)dx 1nW

(d",Ci

I) =

O(/ I/n,

;=1

m

/

I

~

fl

= (2n)"tl (C

)

- 1/2 r nI

').,

lI) l(a-

l/2 {

5

rfl

12

+fl

X

25

" flm i= 1 0

m m m

X J (n - ZI/2 (r I

~

",(j )

I)

dr} I~ ",(J)

l-

n

/2

nW (d",(/\

;

=1

i= 1

,=

1

...

t E[0, I] .

(2. 10.8)

Ilpouecc Ym, lh (t) . t E [0, I]

5lBJ151eT

CH nOJIYYCTOH4HBbIM B ya-

KCJM

CMbICJIe c

rra

pauerpoa x E(1/2 +

~

/

n

,

1+

~

/n).

n pH

~

= 0

npouecc Y m.g, (f), t E[0,

I]

cosnanaer C nponeccox (2. 10. I ).

CAe

8

cf1U3ue

2.10. 1.

rIYCTb

BblIlOJIlleHO yCJIOBHe

XXI

, g l(lxl)=

= Ix f,

~

> 0,

ex

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

r-

aa npu r -)-

00

KOH

e4

HoMepHble

pacnpenenenas CJIY4aHHblx npoueccou

Ym ,F.g, (t) = .\' Ix

If!

Hm

(~

(x)) dx

/[

r"-

ma

/2

X

v(

rt

\f n,

X Jf C

25

h

nl

(0) r ll] ,

IE [0,11

(2. 10.9)

c

JJ

a60

CXO,ll51TC

51

K KOHelJHOM

epH

bI\-1

pacn

pe

nenenaaa

npouecca

.Ym,g, (t), 1E[0, I] (

CM

. (2.

10.S))

H

o6meM

cn y

xae

npcueccu Y m,g (f) 5lB

JI51IO

TC51

nOJI

yy

cTOH4H-

BblMH C

napaa erp ov x E(1/2 +

~

/n.

1 +

~

/n

)

JIKWb

r o

r-

aa , x or-

na

¢ YH1

{UH51

g(x) B y CJIOBl1

11

XXII

5lBlI51eTC51

o n

aoponn

oa:

if(alx!) =

f!-

= a g (j t

I),

~

> O.

B

reopenax

2.10. 1, 2.10.2 yCJI

OBH51

epO

pMyJIHpO

BaJIHCb B

cn

exrpaasnu

x repsomax. E CJllI

H3KJIa)lb

IBaTb yCJIO

Bfl

51

H3 K.

<p

.

B (x)

flOJl51

S (x), x E

RfI

, MOIKHO nOJIY4HTb 60JIee

ornnae

ytsepa«

.llCIIIlH .

n YCTb Bb

In

OJIHeHhl YCJlOBl-1H I Ii I V n

pu

a E (0, n) . P 3CCMOT'

pllM

cnex

rpansuu

e np

enc-r

aaneaus (1.2.1) K (1.2.3), B KOT

Opb

lX

f(

·)-cneKTp

aJIb

Hall

Mepa

n0J111

se

x), x E R

n,

a

ZF(

') -

0P

To

ro

1l3

J1

bHa

51

rayc

coscxa

s Mep a co

cr

p

yxr

ypn

on

rpyHKUKei{

F (.). B

naJlb

HelllJ1eM

npe

ano.n

ar

aea , 4TO F (.) - HeaTOMH4ec·

1J2

KaH

:>

Iep a. B

Be,lJ,eM

ceMeHCTBO

JJOK

<

aJlb

IiO KOHe4HblX

Mep

F, (A) = rOL - I (r )

F~

r

-1t:.),

t:.

E

58

n

.

(2.10. 10)

PacC

MOTpllM ceMeHcTBO

lny

4aii HbIX npoueccos

X

m

.,

(I)

= rma/ Z-

nL

- If' /! (r)

C:

Z

IIZ

5

fI

m

(~

(x)) dx,

vir ! II

fI

,

tE [O, r]. (2.10.11)

rne

C

2

= C

2

(n. m ,a ,v (I») )l,3erC51 epOpMyJIO;1 (2.1.8). Hcnonssya

¢OPM)'JIY VITO

(CM.

n. 2.9), rrOJly 'l

aeM

m m

Hm

(~(

x))

= r

exp

{i

~

(",W,

x>

} nI F

(d",(j»).

(2.10 . 12)

R

nm

j=1 ;

=1

n O)l,Cr3B

Jl51

il (2.) 0.12) B (2.10.I I) H

IIc

nOJlb3Y

51

(I .4.5). Ha-

XO

,L

Ui );!

d m

Xm

,r

(I) = c

25

VI rJ n

/2

(I

~

",(

I) It

ll

fl

)

X

Rnm i= 1

m m

X I

~

A

U)

/

-n

/2 nZF

(d

"Y\

(2. 10.13)

1= 1

i=1

rn

e C

25

= C

25

(n, m,

ex

) = (2rt)"/ Vcd n,m,

ex,

v (I )), 3

lF

(.)

_

r

Opr OrO HaJIbH3 H rayccoecxaa CJlY4aHHaH Mepa co

CTpyKTypH

OH

epYHKUllel1 Fr ( . ) .

B n 3JIbHeHWeM HaM

aaao

conepunrrs n p e

)l

eJIb Hbl H nepexon

no r

-'>-

00

B epopMy JIe (2.10.13).

n YCTb

m

fr (Z(l l,

...

, z(m); t) = .\ ex p

{i

~

(", Iil , zu»}X

Rnm i= 1

m

X K (),}!J, ... .

}.,,(

m); t) nr.(d",

<J

\

i= t

rne K (tYI. ... .

1.,(m

J; t) =

tJ

~

/

2

(I

}."

II) + '" + ,, 1

m

) It l/n)/I ,,(1) +

..

.

... +

'A

(

m)

,n.

JleM

Ma

2.10.2.

npfJ

YCJIOBliHX 1, IV K

ex

E(0, n)

,D,JIH

KIDI<)lO-

fa t E[0, 11

n

l I

n,

t (

Z

I I)

,

•.••

Z

(m ).

,

t)

=

j (

z

(I )

, ... z

(m ).

,

t)

=

f

-4

OQ

J J {n

m

Ix-

y

-z(J

)/-a} dxdy

(2. 10.14)

L'('

l/n

) V(tl/nl j = 1

8-6

·110

11

3

paBHOMepHO B

Ka>K,ll,

OM KOHeLJH

O"M

JJHTepBaJIe, npHLJeM f (z(I), . ••

... , zlm); t)

--

aenpepsraaas epyH

KL1.

1HL

A B h

In

OJIlIYIB saae

ay

nepex

ea

a ux, C y tIeTOM (1. 4.5) nony-

tI

liM

ma

I, (z(l), ... , z (m ); t) = r

-

2n

L

-m

(r)

.1

j d

xd

y X

c(, t l

in)

ti(,t

lin ,

m m

X

~

ex p

{i

~

(

f.}i

l, X - Y +

rz<

11

)} n F

(d'}.,(J))

=

l; r1m

;=

1 1=1

m

=rm

aL- m (r)

~

.\

[l

B (r !x - y +

z(J)

\) dxd

y=

<~

t

l/

nl

* 1/n) 1= 1

m

~

{n IX- y + Z(f)

I-a}

X

<.(lI/

r1

)

{J

(t l/n)

1=1

X {n

m

[L (r IX - y + ZU) \)

IL

(r)l} dxdy.

1= 1

MCn

OJIb 3Y fl reopeu y 2 . 1. 1, us nOCJIe,UHer o

COOTH

OW CHHfI MO

>K

HO

BblBeCTU yTBep>K,UeHHe JIeMMbI 2 . 10 .2

.•

JIeMM8

2

.10.3.

ECJlH Bb

In

OJ

IHe

HbJ YC.l 0BH

51

I, IV npnc E(a , n),

TO ceM

eikTB

O JIOK

8JIbH

O KOHe

LJ

HblX

Mep

(2.

10 .

10

) CXOJl,H

TC51

JIO-

n

KaJI bH O cnaro K JIOKaJIbH O KOHetIHOH xtepe F

o

(~

)

,

~

Em

,

KOTO-

pas yJl,OBJIeTBOpHeT y e.rIOBHlO O,UHOpO,ll,HOCTU F

o

(a~)

=

aa

F

o

(

~

)

,

n

fl

Em

,

a E(a, 00) H

COO

THourenmo

n

~

ex p {i (z, A)}

J

~

!l

(I

"A

J)

I

')..

I- F

o

(dt,) =

Rn

= I

~Ix-y+zl-

adxdy.

(2. 10. 15)

[

(I)

t(1)

A n YCTb K (z) = K (z;

I).

l-lcn

ons

ayn reopen y 0 6 3K BH-

BaJleHTHOCTU CJIaOOM CXO,ll,HMOCTH Mep H CXO,UHMOCTH

HX

npeoo·

pa

30Ba

HHl

1

<t>

ypb

e [32, c. 2

81,

ns JJeM

)I,(

bJ 2

.10

.1 npn m = I, t =

= 1

Bbl

BOA

li

M, tITO ceMeHCTBO

KOHe'lHblX

xep

f1,

(6) =

~

K (z)F,(dz),

n "

t! Em

CJI

a60 CXO,ll

li

TCfI K nexo-ropox K

OHetIHo

H ' Mep e

f.10

(~),

~

E

IB

n

Ta

K

O~

,

sro

ee

npeoepasoea

na

e

<1>

yp

be

~

exp

{i

o;

z)} X

X

f.1

0(dz) = f (z; 1) (OM. (2. 10.

14)

.

~

n

~Y

C

T

b

1'-

mepsna HyJJb

cjJ

y HKIUHH I

n

/2 (u) u

:«

. Torn a

,UJI51

RIO

o

or

o

<.\

c V

()')

r.a x o

ro

, 'lTO

~L

O

(

01".

)=

0,

r

~~

F, (

I".

)

=

r

~

l~

j

[K (x

)J-

'

X

t:.

X

~l

r

(dx) =

~

IA'

(X

)]- I

fJo

(dx) = F;o(

1".),

rne F

o

(.) - MCpa H3

!1

18

(v (

'Y

)).

j]

Jlil

.n

106oro a

>O

, &Em",

I".

C V

(,\,

) TaKOrO ,

'IT

O

F

o

(a

~

)

= 0,

lim

Fr (a

l".

) = a U F0 (

I".

).

(2.

10

. 16)

'--->ex>

Ll.eHc TBliTenb l!O, nyc

n,

v = ri a

..

T

orn

a npH r

=-+

00 F

r

(a~)

=

(r:,,)a (L (v)

IL

(r)) F, (av

Mr

)

-+

aaF

o

(

<.\)

, orxyna ssrrexaer

(2 .

10

. 16 ) . 113

(2

. 10 . 16) cnen

yer

, UlTO

,UJ151

RlOObJX

~

1

'

1".

2

E

~

n

.

1.\

1'

<.\

2C V (,\, ),

~1

=

Q

~2

nJi

Sl

nexoroporo

a>

0 Bb

Ill

OJIH

51

eTCfI

COO

THow

eHHe

r, (

<.\

1) = aUF0 (

1".

2)'

(2

. 10 .1 7)

Onpenenua epyH KUHIO MHOtKeCTSa F

o

(

I".

) ,UJIfI JllOoor o orpanuseu-

n

er

o

~

Em

tl

CJIe,UYfO

lllJI

M

o6pa

30M: eCJIH

~

C v (5

1')

, TO F

o

(

~

)

=

= saF

o

(Ms). npO,UOJDKHS F

o

(. ) na

Q3

n, 1l0JlY

LJ

IlM

JlOJ<

a.

T\bH

O 1<0-

H

ellH

YlO Me

py

F0 ( . ) . Coor nomenne (2 .1

0.16)

Bsm0 .'1II 5leTC

51

6e3

Orpa

HH

LJ

eH

li

51

I".

C v

(1').

M

epbl

F, CXO

JJ.

flTC

51

JIOKa

JJ

bHO CJlaOO

J<

J1

0KaJlbHO KOHetIH011

Mep

e F

o

II BbI

DOJlH51eTC51

(2.1 0

.17

). n 03TOMY

flo

(~)

= j K (x) F0 (d\;). B CHJIY r

eopeau

e,UHIICTse HII OCTII

nm

l

!1

npeoopaaosannf <Pypbe KOHetIHblX

sre

p n

peoopaaosa

nne

C]

l YPbe Me-

PbI floecrs f (z; 1) (CM.

(2.

10 .

14)).

O TclO ,Ua nu rexaer (2 . 10 .1

5)

.•

113 JlCMMbl

2.1

0

.3

CJI

e,lJ.

y eT , xro YUCJIO

o:

E(0 , n) sanaer

npe-

06p

a3

0saHH

e <P

ypb

e Mepbl K (z) F

o

(dz). CJle,UOBaTe.

1Jb

HO. II cax y

Mepy

r;

nY

CTb ZF. ( .) - oproroaam.aas

raY

CCOBC

Ka51

CJlYtl Jli Ha

51

Me'

pa

co CTPYKTy pHOH epyHKuHeiJ F0 ( •

).

P aCCMOTpHM CJlYtJaHHbIe np

o-

uec

cu

m

Xm (t) = C

25

Vt

rJn lZ (/

~

I,Y)

It l/

n

)

X

r<nm 1=1

m m

XI

~

f.

U

)

,-n

/2 nZF. (d'i.(j)), t E[0, II, m ~ 1,

j = 1 j= !

xoropu e 5lBJI

51101

C51

n

OJJ

YYCTOH4HBb1l11l B WHpOKD:\\ CMbl

CJJe

G na-

paa

erpox x =

1-

am

l2n

E(1/2, 1).

Teopeaa

2.10 .3.

ECJJH

Bbm

O

Jll

~

H O

YCJlOBue I, IV npn

etE

E(0. nlm), m

;»

1, TO

npn

r-+-oo

KOH

e4H

OM

epHbi

e pacnpen

ene-

HWI CJlY4aMHblX

npouec

con X

m

.,

(t) C

XO,lJ.

HTC

51

K KOHetIHOMepHblM

pa c

npe

Jl,eJleHHSlM CJIYlJ3MHbiX np

ou

ec

cos

X (t), t E[

0.

I].

m

8*

115

114

A n YCTb f; E[0, 1], b, E

RI,

; = I, "

.,

p.

llo

CTaTolJHO

nOK8

3aT

b,

'ITO

npH

r

-+

00

p _

'!J

p -

Sf =

~

bj./

'<.

m.

r (I

J

)

-+

~

s.x

; (tJ).

(2.10.18)

i=J i= J

C or JlaCHO (2.10.13)

m

Sr = rR (,

.,0"

.., ,

,

yn

~

n

ZF;

(d

'),

Y\

nm ;=1

rne

p m

K

(;>..

(1

1

, .." ,

,e

m)) = C

25

~ b, V

0Jn

/2

(I

~A,

(J)

Itl/

n

)

X

i

=1

i= J

m

X 1

~

", (J) l- n/

2.

;= 1

1-'

13

.'l

e~lM

b

l

2.10.3 cnenyer JlOKaJIb HO cnaoas CXO

jl

HMOCTb Mep F,

K Mepe F

npn

r

-+

00

, a

113

JleM Mbl 2.10.2 BbITeKa

eT

,

'l

TO ceo

o

Mei

ic

TBo I<OHeqHblX Mep

~T

(~

)

= S(K (,

..0

), ...

".

(m

))]2 n

m

r, (d ",(})

t:, i= \

CXO,Tl

IITCH K KO

ll

e4l1o;1 Mepe

m

~

l

O

(~)

=

~

[I<

(",(I), '" ,

,,,

(m

l

]2

nr,

(d",

lJ\

~

E

5S

n

,

6

i=

1

nOCKO.'l bl<Y CXO,llHTCH Il X x apaKTepHCTHQeCKl1e <pyHKUHII . n o-

,HOMY Db

In

O,

lII

eHO YCJlODHe (2.9.1) H

(2.1

0.18)

BblT

CKaeT

113

reo-

PCMbI 2.9.1

.•

r n

AB

A 3

OUE

HHBAH

liE

MATEMAHff 4

ECKOrO

O)l{HJl.A H

H5I

3.1.

ACI1I\\I1TOTH4E

CKHE CBOH CTBA OU EHOK

HA"~'\EHbWHX

KBA)].PATOB

K08<IDcIlMUJ1EHTOB J1HH

H1HOA

PErPE

CCHH

PaCCMOTpl1M

cJly4

aHHoe n

on

e

~

(x) = £8(x) = g (x, 8) + e (x ),

(3.1. 1)

r nc g (x , 8) : R" X e

c

-+R

l

_<p

yHKU II

51

, 3aBI1C

51

111a

51

OT

ne

nsse

cr

nor o

napasier pa 8Ee E

\B

q,

e (x), x ER

n

- cJlyqa(lHoe

non

e C

Il

Y

oi

1eBbIM

R

n

C

pe

.n.

HHM.

Ilp

enn o.noacna, 'ITO B asrnenes a HeKOTOpa

51

CIl CTCMa

11

3-

Mep HMblX orpanaaennux MHOlKeCTB 'im=

{~}

Ii

~

-+

00

. 3 a)\3lJa

3aKJlf04aeTC

51

B TOM, 4T0

6b

I

no

ll

a6JlIO).je

llli

fiM CJlyqaAHOrO 1l0JIH

£(x ) sa MHOlKeCTBax

~

-+

00

o

ue

HIl Tb

n

ap

a

~

I

e

Tp

8.

xeva perpeccnn (3.1.1) Ha3bIBaeTCfI JIIIHe

J"IH

OH, ecrm

<PY

HKUI1

f1

g (x , 8) - JlHHeHHa5I

<p

op

Ma nepeaenuu x 8 = (81' .... 8

q

):

g (x, 8) =

q

=

~

8

j

g

j

(x). B eKTop-<p YlI KUII5I g' (x) =

(g

l (x), ..., gq(x» , KO-

~

I

.

opnnnar u KOTOpOi'I JlHHeHHO He3aBHCHMbI aa

~

ESIn, C

LJ

II TaeTCfI

S

ajl

aHHoA. B

LJ

aCTHOCTH, npH q = I JlIlHel"lHafi cx

ex

ia np

ll

HIIMaCT

BIl )!.

£(x) =

Og

(x) + e (x)

(3.1.2)

11

axmosaer B ce6H

san

asy O

ue

HI1BaHIl H neaaaecr noro cpen ner o fl

nOJIH

s(

x) (g(x) = 1).

B n. 3.2- 3.6

pa

CC

MaTp l1BaeT

CH

IIe H IHe

rllI

aH cxeaa (3

.1.1

)

II

3CHMnTOTHQeCKHe

npn

~

-+

ex:>

c

ao

ncra

a o

uea

o x

lIamle

ll

bl

llllX

KBa.n.paTOB H Hal1MeHbWYlX MO

)J.

y.l eH nap av e

rp

a 8. 5 0Jlb

1U

a

51

4a

CTb

rr

pus enennu x TaM p e3Y Jl

bT3T

OB OCTaeTCfI con ep aca

ren

s-

HOII Il )\Jl H CXel1bI JlI1HCI1HOH

perpeccun. B sa

crosui

ea ny a xr e

np

IlBo.n.HT

C51

CBOHCTBa onen

xn

II aHMe

HbW

IlX

KBa.n.

pa

TOB na

pa-

q

Mer pa 8 )IJl

51

mm ei1HoA <p

yHK

UHH g (

x,

8) =

~

8jgj (x).

j=

1

npe

)\n

OJlOIKHM, 'lTO e (x), x Er - 1l3Mep}IMOC, c. K. a

en

pe-

pusn

oe O

.n.

HOpO,LHoe CJlYQaHHOe n

on

e C K. <p. B (x), a

<PY

HKUI1

5I

g (x, 8) B (3.1.1) nsxepaxa no x

.n.

Jl

51

JII06

or

o 8E

El

c

Yl

u

en

pe-

psr

sn

a no 8

.n.Jl5

Kall<

)\

Or

O x E

tr

. T

aKIl

M oopaaoa , g (

x,

8) HSMe·

11

7

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

Подождите немного. Документ загружается.