Directory UMM :Journals:Journal_of_mathematics:VMJ:

ˆî«ì{ᥭâï¡àì, 2000, ’®¬ 2, ‚ë¯ãáª3

“„Š 511.3

Ž–…Šˆ ‚‡€ŠŽ€• Ž‹œ˜ˆ• —ˆ‘…‹

„‹Ÿ …ƒ“‹Ÿ›• Œ…’Ž„Ž‚‘“ŒŒˆŽ‚€ˆŸ

”. •. „®¥¢

‚ ¡®«ì設á⢥ à ¡®â, ¯®á¢ï饭­ë嬥⮤ ¬ á㬬¨à®¢ ­¨ï à áᬠâਢ «¨áì ç áâ­ë¥

¬¥â®¤ë. â¨¬ ¨áá«¥¤®¢ ­¨ï¬ ¯à¨¤ ¥âáï ­¥ª®â®àë© á¨á⥬ â¨§¨à®¢ ­­ë© å à ªâ¥à.

 áᬮâ७ ª« áá ॣã«ïà­ëå ¬¥â®¤®¢ á㬬¨à®¢ ­¨ï, ᮤ¥à¦ é¨© â ª¨¥ ¬¥â®¤ë ª ª

€¡¥«ï,—¥§ à®,®à¥«ï,©«¥à ,᪮«ì§ïé¨åá㬬¨¤à. „«ï¢§¢¥è¥­­ëåá㬬ᢥᠬ¨

¨§í⮣®ª« áá ¯®«ã祭ë®æ¥­ª¨¢§ ª®­ å¡®«ìè¨åç¨á¥«¢¢¨¤¥á室¨¬®á⨨­â¥£à «®¢

®â ¢¥à®ïâ­®á⥩ ¡®«ìè¨å 㪫®­¥­¨©. “áâ ­®¢«¥­   á¨¬¯â®â¨ª ¯® ¬ «®¬ã ¯ à ¬¥âàã

íâ¨å¨­â¥£à «®¢.

‚ ¡®«ì設á⢥ à ¡®â, ¯®á¢ï饭­ëå ¬¥â®¤ ¬ á㬬¨à®¢ ­¨ï (=¬. á.)

à á-ᬠâਢ «¨áì ç áâ­ë¥ ¬¥â®¤ë. ‚ ¤ ­­®© à ¡®â¥ ¯®¯ëâ ¥¬áï ¯à¨¤ âì í⨬

¨áá«¥¤®¢ ­¨ï¬ ­¥ª®â®àë© á¨á⥬ â¨§¨à®¢ ­­ë© å à ªâ¥à. ¨¦¥ à áᬮâ७

ª« ááॣã«ïà­ë嬥⮤®¢á㬬¨à®¢ ­¨ï,ᮤ¥à¦ é¨© â ª¨¥¬¥â®¤ë,ª ª

€¡¥-«ï, —¥§ à®, ®à¥«ï, ©«¥à , ᪮«ì§ïé¨å á㬬 ¨ ¤à. „«ï ¢§¢¥è¥­­ëå á㬬 á

¢¥á ¬¨ ¨§ í⮣® ª« áá  ¯®«ãç¥­ë ®æ¥­ª¨ ¢ § ª®­ å ¡®«ìè¨å ç¨á¥« ¢ ¢¨¤¥

áå®-¤¨¬®á⨠¨­â¥£à «®¢ ®â ¢¥à®ïâ­®á⥩ ¡®«ìè¨å 㪫®­¥­¨©. “áâ ­®¢«¥­ 

 á¨¬¯-â®â¨ª  ¯® ¬ «®¬ã ¯ à ¬¥âàã íâ¨å ¨­â¥£à «®¢.

ãáâì 0 < 1. Ž¯à¥¤¥«¨¬ ª« áá ä㭪権 (¨«¨ ¢ á«ãç ¥ ¤¨áªà¥â­®£®

¯ à ¬¥âà  | ª« áá ¬ âà¨æc

k

(n)),§ ¤ î騩 ॣã«ïà­ë¥ ¬. á.:

D

=f0c

k

() 1; k=1;2;:::; >0;

sup

k c

k

()b

1

,

¯à¨ !1;

1

X

k=1 c

k

()!1 ¯à¨ !1;

B 2

()= 1

X

k=1 c

2

k

()b 2

2

,

¯à¨ !1g:

‹¥£ª® ¯à®¢¥à¨âì, çâ® í«¥¬¥­â ¬¨ D

1

ïîâáï ¬. á. —¥§ à® ¯®à浪  r

1 (C;r), €¡¥«ï (A). Œ­®¦¥áâ¢ã D

1=2

¯à¨­ ¤«¥¦ â ¬¥â®¤ë ©«¥à  ¯®à浪 

ãáâìX

1 ,X

2

;::: |¯®á«¥¤®¢ â¥«ì­®áâì­¥§ ¢¨á¨¬ë室¨­ ª®¢®

à á¯à¥¤¥-«¥­­ëåá«ãç ©­ë墥«¨ç¨­ (­. ®. à. á. ¢.). Ž¡®¡é ïª« áá¨ç¥áªãáâ ­®¢ªã

§ ¤ ç¨ ® § ª®­¥ ¡®«ìè¨å ç¨á¥«, à áᬮâਬ ¢§¢¥è¥­­ë¥ á।­¨¥

S()= 1

X

k=1 c

k ()X

k

(S(n)= 1

X

k=1 c

k (n)X

k )

¨ ¢ëïá­¨¬ ãá«®¢¨ïá室¨¬®á⨠¨­â¥£à « 

(";q;t)= 1

Z

1

qt,,1

P(jS()j" (q,1)

)d;

  ¢ á«ãç ¥ ¤¨áªà¥â­®£® ¯ à ¬¥âà | à鸞 P

1

n=1 n

qt,,1

P(jS(n)j"n (q,1)

).

‘室¨¬®áâì í⮣® ¨­â¥£à «  âà ªâã¥âáï ª ª ¨­ä®à¬ æ¨ï ® ᪮à®áâ¨

áå®-¤¨¬®á⨠¢ § ª®­¥ ¡®«ìè¨å ç¨á¥« ¤«ï ¬¥â®¤  á㬬¨à®¢ ­¨ï fc

k ()g.

„«ï c

k

()2D

¢¢¥¤¥¬¢ à áᬮâ७¨¥ á«¥¤ãî騩 ­ ¡®à¨­¤¥ªá®¢ ¯®

áâ¥-¯¥­¨ ã¡ë¢ ­¨ï c

k

() ¯® :

I = fk:c

k

()=O( ,

) ¯à¨ !1g:

—¥à¥§ c, ¨­®£¤  á ¨­¤¥ªá ¬¨, ¡ã¤¥¬ ®¡®§­ ç âì ¯®«®¦¨â¥«ì­ë¥

¯®áâ®ï­-­ë¥.

’¥®à¥¬  1. ãáâì X

1 ;X

2

;::: ¯®á«¥¤®¢ â¥«ì­®áâì ­. ®. à. á. ¢., qt > 1;

q > 1

2 , c

k

()2D

. Šà®¬¥ ⮣®, ¯ãáâì¯à¨ !1

X

k c

t

k

()=O

(1,t)

(0<t<1): (1)

„«ï á室¨¬®á⨠(";q;t) ¯à¨ «î¡®¬ " > 0 ¤®áâ â®ç­®, ç⮡ë EjX

1 j

t

< 1 ¨

EX

1

=0 ¢ á«ãç ¥ t1:

â¨ ãá«®¢¨ï ­¥®¡å®¤¨¬ë, ¥á«¨ ¯à¨ !1

card(I)=O(

): (2)

C ‡ ä¨ªá¨à㥬 § ¢¨á¨¬®áâì (";q;t) ®â ¢ ¢¨¤¥ ­¨¦­¥£® ¨­¤¥ªá 

(";q;t): ®¤áâ ­®¢ª®© = y

¢ëà ¦¥­¨¥

1

(";q;t) ¯¥à¥¢®¤¨âáï ¢

(";q;t):

‘®®â¢¥âáâ¢ãî騩 ¢¨¤ ¯à¨®¡à¥â îâ ¨ ãá«®¢¨ï (1) ¨ (2). ‘«¥¤®¢ â¥«ì­®, ¤®ª

„®áâ â®ç­®áâì. ãáâì EjX

1 j

t

< 1; 0 < t < 1: ‚®á¯®«ì§ã¥¬áï  ­ «®£ ¬¨

­¥à ¢¥­á⢠ £ ¥¢  |”㪠 [2]. ’®£¤  ¤«ï «î¡®£® >0

1

(";q;t)= 1

Z

1

qt,2

P

S()

" q,1

d

1

Z

1

qt,2 X

k P

c

k ()

X

k

" q,1

d (3)

+ ,

e" ,t

1,t

EjX

1 j

t

1= 1

Z

1

qt,2,(q,1)t= "

X

k c

t

k ()

#

1=

d=A

1 +A

2 :

’ ª ª ª ­ á ¨­â¥à¥áã¥â ⮫쪮 á室¨¬®áâì ¨­â¥£à «®¢, â® ¯à¨ ¨å ®æ¥­ª¥

¡ã¤¥¬¯®«ì§®¢ âìáï á¨¬¯â®â¨ç¥áª¨¬¨á¢®©á⢠¬¨c

k

()¯à¨!1.

®«ãç î-騥áï¯à¨ í⮬¨­â¥£à «ë, á室ïâáï¨ à á室ïâáﮤ­®¢à¥¬¥­­® á ¨á室­ë¬¨.

à¥®¡à §ã¥¬A

1 :

A

1 =

1

Z

1

qt,2 1

X

k=1 1

X

i=k P

" q,1

c

i ()

jX

k j<

" q,1

c

i+1 ()

d

= 1

Z

1

qt,2 1

X

i=1 i

X

k=1 P

" q,1

c

i ()

jX

k j<

" q,1

c

i+1 ()

d

1

Z

1

qt,2 1

X

i=1 i

Z

L

dP(jX

1

jy)d;

(4)

£¤¥ L=

" q,1

ci()

y< "

q,1

ci+1()

:

Žç¥¢¨¤­®, L ­¥ ¯ãáâ®, ¥á«¨ c

i

() > c

i+1

(). ãáâì fc 0

k

()g fc

k

()g

ã¡ë-¢ îé ï ¯®á«¥¤®¢ â¥«ì­®áâì ¯à¨ ä¨ªá¨à®¢ ­­ëå . ®áª®«ìªã 2n

P

k=n c

0

k

() ! 0

¯à¨ n!1 ¨ ¯à¨ í⮬ 2n

P

c 0

k

() >nc 0

2n

();â® c 0

i

() =o( 1

i

‘«¥¤®-¢ â¥«ì­®, ¨§ (4) ¨¬¥¥¬

A

1

1

" 1

Z

1

q(t,1),1 1

X

i=1 Z

L

ydP(jX

1

jy)d

1

" 1

Z

1

q(t,1),1 Z

y" q

=b

1

ydP(jX

1

jy)d

b

1

(") 2

1

Z

1

q(t,2),1 Z

y" q

=b

1 y

2

dP(jX

1

jy)d

=c 1

Z

"=b

1 y

2

(yb1=(")) 1=q

Z

1

q(t,2),1

ddP(jX

1

jy)cEjX

1 j

t

:

(5)

¥à¥©¤¥¬ ª ®æ¥­ª¥ A

2

. ® ãá«®¢¨î (1), A

2

á室¨âáï ®¤­®¢à¥¬¥­­® á

¨­-⥣ࠫ®¬

1

Z

1

qt,2,(qt,1)=

d:

‹¥£ª® § ¬¥â¨âì, çâ® ¯à¨ <1

A

2

<1: (6)

‚ ᨫ㠯ந§¢®«ì­®á⨠<1 ¨§ (3),(5) ¨ (6), ¯®«ãç ¥¬ ¤®ª § â¥«ìá⢮ ¤®áâ

 -â®ç­®á⨠¤«ï0<t <1.

à¨ ¤®ª § â¥«ìá⢥ ¤®áâ â®ç­®á⨠¤«ï ®áâ «ì­ëå §­ ç¥­¨© ¯ à ¬¥âà  t

á«¥¤ã¥â¢®á¯®«ì§®¢ âìáïᮮ⢥âáâ¢ãî騬¨¢ à¨ ­â ¬¨­¥à ¢¥­á⢍ £ ¥¢ |

”㪠.

¥®¡å®¤¨¬®áâì.  ¬ ¯®­ ¤®¡¨âáï

‹¥¬¬  [7]. …᫨ fX

n

g ¯®á«¥¤®¢ â¥«ì­®áâì ᨬ¬¥âà¨ç­ëå ­¥§ ¢¨á¨¬ëå

á. ¢., â® ¯à¨ 0ja

k jd

k

; k =1;2;:::;n;¤«ï «î¡®£® ">0

P

n

X

a

k X

k

"

!

2P

n

X

d

k X

k

"

!

Ž¡®§­ ç¨¬ ç¥à¥§ e

X

n

| ᨬ¬¥âਧ®¢ ­­ë¥ á. ¢. e

S

n =

n

P

k=1 e

X

k ;

e

S() =

P

k c

k ()

e

X

k

. ® ­¥à ¢¥­á⢠¬ ᨬ¬¥âਧ æ¨¨

e

(";q;t)= 1

Z

1

qt,2

P

j e

S()j" q,1

d<1:

à¨¬¥­¨¢ «¥¬¬ã á

d

k =c

k

() ¨ a

k =

(

c

k

(); k 2I;

0; k 2I;

¯®«ã稬

e

(";q;t) 1

2 1

Z

1

qt,2

P

X

k2I c

k ()

e

X

k

"

q,1 !

d:

‘«¥¤®¢ â¥«ì­®, á室¨âáï ¨­â¥£à «

A= 1

Z

1

qt,2

P

X

k2I e

X

k

c"

q !

d:

‘ ãç¥â®¬ ãá«®¢¨ï(2) ¡ã¤¥¬ ¨¬¥âì

A= 1

X

n=1 n+1

Z

n

qt,2

P

e

S

[]

c" q

d

1

X

n=1 n

qt,2

P

e

S

n

n q

c"

1+ 1

n

q

!

1

X

n=1 n

qt,2

P

e

S

n

"

1 n

q

;

£¤¥ "

1 =2

q

c":

Žâá ¯® ¨§¢¥áâ­®© ⥮६¥  ã¬  | Š æ  [5] á«¥¤ã¥â E

e

X

1

t

<1.

‘®£« á­® á«¥¤áâ¢¨î ¨§ ­¥à ¢¥­á⢠ᨬ¬¥âਧ æ¨¨ ¯®«ãç ¥¬ E

X

1

t

3{18

”. •.„®¥¢

…᫨ ¢¬¥áâ®

fc k

(

)

g

¢§ïâì ¬¥â®¤ á।­¨å  à¨ä¬¥â¨ç¥áª¨å (

C;

1), â® ¨§

⥮६ë 1 ¯®«ãç ¥¬ ⥮६㠁 ã¬  | Š æ  ¨§ [5]. ’¥®à¥¬  1 ¤«ï ¬. á. (

A

)

¡ë«  ¤®ª § ­  ¢ [4] ¤«ï

q

= 1,

t

= 2.

’¥¯¥àì à áᬮâਬ  á¨¬¯â®â¨ªã

(

";q;t

) ¯à¨

" !

0. Žç¥¢¨¤­®, ¤«ï ¬. á.

¨§

D

¢ë¯®«­¥­  ­ «®£ ãá«®¢¨ï ‹¨­¤¥¡¥à£ :

1

B 2

(

)

1

X

k=1 c

2

k

(

)

Z

jyj" B()

c

k ()

y 2

dP

(

X k

y

)

!

0 ¯à¨

!1:

’ ª¨¬ ®¡à §®¬, á¯à ¢¥¤«¨¢  業âà «ì­ ï ¯à¥¤¥«ì­ ï ⥮६  (æ.¯.â.) ¤«ï

S

(

). ‹¥£ª® ãáâ ­ ¢«¨¢ ¥âáï ®æ¥­ª ,  ­ «®£¨ç­ ï ¨§¢¥áâ­®© ®æ¥­ª¥ €.

¨ªï-«¨á  ¨§ [1].

ɇǬ

EX

1

= 0,

EX

2

1

= 1, â®

jP

(

S

(

)

xB

(

))

,

(

x

)

jc

(

;x

)sup

k

c

k

(

)

(1 +

jxj

)

3

B

(

)

;

(7)

£¤¥

(

;x

)

Z

juj

(1+jxj)B()

sup

k c

k ()

juj 3

dP

(

X 1

u

)+(1+

jxj

)

B

(

)

Z

juj

(1+jxj)B()

sup

k c

k ()

u 2

dP

(

X 1

u

)

:

Ž¡®§­ ç¨¬ ,

l

=

,(l,1=2)

(l,1) p

,

qt,

2q,

=

s

, £¤¥ ,(

z

) | £ ¬¬ -äã­ªæ¨ï.

’¥®à¥¬  2

.

ãáâì EX

1

= 0

, EX

2

1

= 1

. ’®£¤  á¯à ¢¥¤«¨¢ë ᮮ⭮襭¨ï:

 ) lim

"#0

(

";

1

;

1)

ln

1

"

=

2

;

¡) lim

"#0 "

2s

(

";q;t

) =

(

p

2b

2 )

2s

(2q,1)

,

s+1

¯à¨ EjX

1 j

t

<1.

C

‚¢¨¤ã á宦¥á⨠à áá㦤¥­¨©, ®£à ­¨ç¨¬áï ¤®ª ¦§ â¥«ìá⢮¬ ¯ã­ªâ  ).

à¥¤áâ ¢¨¬

(

";

1

;

1) ¢ ¢¨¤¥ áã¬¬ë ¤¢ãå ¨­â¥£à «®¢

(

";

1

;

1) =

1

Z

1

1

P

(

jS

(

)

j"

)

,

2

,

1=

b

2 "

d

+

1

Z

1

,

1=

b

2 "

d

=

1

+

2 :

Žæ¥­ª¨ ¢ § ª®­ å ¡®«ìè¨å ç¨á¥«¤«ï ¬¥â®¤®¢ á㬬¨à®¢ ­¨ï

3{19

®ª ¦¥¬, çâ®

lim

"#0

1

ln

1 "

= 0

:

(9)

‚롥६

n 0

(

"

)

>

0 â ª, ç⮡ë

n 0

(

"

)

!1

,

n0(")

ln 1

"

!

0 ¯à¨

"!

0. ’®£¤ 

1

=

Z

1<expn

0 (")

+

Z

expn

0 (")

=

0

1

+

00

2

:

(10)

Žç¥¢¨¤­®, çâ®

0

1

2

Z

1<expn0(")

1

d

= 2

n 0

(

"

)

:

‘«¥¤®¢ â¥«ì­®, ¯à¨

" !

0 ¢ë¯®«­ï¥âáï

0

1

ln

1 "

!

0

:

(11)

 áᬮâਬ

00

1

¨ à §®¡ê¥¬ ¥£® ­  ¤¢  ¨­â¥£à «  ¯® ®¡« áâï¬ (exp

n

0

(

"

)

; "

,2=

)

¨ (

" ,2=

;1

):

00

1

=

Z

expn

0

(")" ,2=

+

Z

" ,2=

=

11

+

12

:

(12)

Ž¡®§­ ç¨¬ (

) = sup

x

jP

(

S

(

)

xB

(

))

,

(

x

)

j:

® æ. ¯. â. ¤«ï

S

(

), (

)

!

0 ¯à¨

!1

. ®í⮬ã

lim

"!0

sup

expn

0

(")" ,2=

(

) = 0

:

‘ ãç¥â®¬ í⮣®, «¥£ª® ¯®«ãç ¥¬ á«¥¤ãîéãî ®æ¥­ªã:

11

sup

expn

0

(")" ,2=

(

)

"

,2=

Z

expn

0 (")

d

=

sup

expn

0

(")" ,2=

(

)

2

ln1

" ,n

0

(

"

)

:

‘«¥¤®¢ â¥«ì­®, ¯à¨

" #

0

11

ln

1 "

3{20

”. •.„®¥¢

„«ï ®æ¥­ª¨

12

¢®á¯®«ì§ã¥¬áï ­¥à ¢¥­á⢮¬  £ ¥¢  | ”㪠 á® ¢â®àë¬

¬®-¬¥­â®¬. à¨ í⮬, ¤«ï «î¡®£®

>

0 ¯®«ã稬

12

Z

" ,2=

1

X

k P

(

c

k

(

)

jX

1

j"

)

d

+

c" ,1=

Z

" ,2=

1

"

X

k c

2

k

(

)

# 1

2

d

+ 2

Z "

,2=

1

,"

=2

b

2

d

=

1

+

2

+

3

:

(14)

®«ì§ãïáì ⥬¨ ¦¥ ¯à¨¥¬ ¬¨, çâ® ¨ ¯à¨ ¤®ª § â¥«ìá⢥ ¤®áâ â®ç­®á⨠⥮६ë

1, ¢ë¢®¤¨¬

1

1

" Z

" ,2=

,1

Z

b

1 y"

ydP

(

jX 1

jy

)

d

= 1

" 1

Z

" ,1

b

1 y

(b

1 y=("))

1=

Z

" ,2=

,1

ddP

(

jX 1

jy

)

c" 1

Z

" ,1

b

1 y

(b

1 y=("))

1=

Z

" ,2=

,1

ddP

(

jX 1

jy

)

=

c 1

Z

" ,1

b

1 y

2

dP

(

jX 1

jy

) +

c

1

" 1

Z

" ,1

b

1

ydP

(

jX 1

jy

)

c 1

Z

" ,1

b

1 y

2

dP

(

jX 1

jy

)

:

Žâá á«¥¤ã¥â, çâ®

lim

"#0

1

= 0

:

(15)

ˆá¯®«ì§ãï ᢮©á⢠

c k

(

), ¡ã¤¥¬ ¨¬¥âì

2 c"

, 1

Z

,2=

,1,

2

d

=

c 1

Z

1 y

,1, 1

2

Žæ¥­ª¨ ¢ § ª®­ å ¡®«ìè¨å ç¨á¥«¤«ï ¬¥â®¤®¢ á㬬¨à®¢ ­¨ï

3{21

¯®áª®«ìªã

>

0 ¯à®¨§¢®«ì­®. ‘«¥¤®¢ â¥«ì­®,

lim

"#0

2

ln

1 "

= 0

:

(16)

Žç¥¢¨¤­® ¨ ¤«ï

3

¢ë¯®«­¥­® ᮮ⭮襭¨¥

lim

"#0

3

ln

1 "

= 0

:

(17)

ˆ§ (10){(17) á«¥¤ã¥â (9).

 áᬮâਬ ¨­â¥£à «

2

, ª®â®àë© ¯®¤áâ ­®¢ª®©

1

b2

=2

"

=

p

x

¯à¨¢®¤¨âáï

ª ¢¨¤ã

2

= 2

1

Z

1

1

,

=2

b

2 "

d

= 2

1

Z

" 2

=b 2

2

1

x

(

, p

x

)

dx

= 2

p

2

,"=b

2

Z

,1 e

, t

2

2 t

2

Z

" 2

=b 2

2

1

x dxdt

= 2

p

2

,"=b

2

Z

,1 e

, t

2

2

ln

t

2

dt

+ 4

p

2

ln1

" ,"=b

2

Z

,1 e

, t

2

2

dt

+ 4ln

b 2

p

2

,"=b

2

Z

,1 e

, t

2

2

dtc

+ 2

ln1

"

(18)

¯à¨

"!

0.

ˆ§ (8), (9) ¨ (18) ¯®«ãç ¥¬ ã⢥ত¥­¨¥ ¯ã­ªâ  a). ’¥®à¥¬  2 ¤®ª § ­ .

B

à¨

t

= 2 ¨

q

= 1 ¤«ï ¬. á. (

C;

1) ¨§ ¯ã­ªâ  ¡) â¥®à¥¬ë ¯®«ãç ¥¬ १ã«ìâ â

•¥©¤¨ [6]. à¨

t

2 ¨

q

= 1 ¤«ï ¬. á. (

C;

1) ⥮६  2 ¤®ª § ­  ¢ [4].

‘¯à ¢¥¤«¨¢ à ¢­®¬¥à­ë© (¢ á¬ëá«¥ ¨á室­®£® à á¯à¥¤¥«¥­¨ï) ¢ à¨ ­â

â¥-®à¥¬ë 2.

ãáâì

F

t

| ª« áá ä㭪権 à á¯à¥¤¥«¥­¨ï

F

(

x

) =

P

(

X x

) ®¡« ¤ îé¨å

᢮©á⢠¬¨:

1

Z

,1

xdF

(

x

) = 0

; 1

Z

,1 x

2

dF

(

x

) = 1

;

lim

a!1

sup

F2F Z

x 2

dF

(

x

) = 0

; 1

Z

,1 jxj

t

3{22

”. •.„®¥¢

Ž¡®§­ ç¨¬

(F)

(

";q;t

) =

1

Z

1

qt,,1

P

F

(

jS

() j"

(q,1)

)

d;

£¤¥

P

F

| ¢¥à®ïâ­®áâ­ ï ¬¥à , ᮮ⢥âáâ¢ãîé ï ä㭪樨 à á¯à¥¤«¥­¨ï

F

(

x

).

’¥®à¥¬  3

.

ãáâì c

k

(

)

2D

. ’®£¤  ¢¥à­ë ᮮ⭮襭¨ï

a) lim

"#0

sup

F2F2

(F)

(";1;1)

ln 1

" ,

2

= 0

;

¡) lim

"#0

sup

F2F

t

" 2s

(F)

(

";q;t

)

, (

p

2b

2 )

2s

(2q,1) A

s+1

= 0

; t

2

.

‚ ®â«¨ç¨¥ ®â ⥮६ë 1, à áᬮâਬ ªà¨â¥à¨© á室¨¬®á⨠¨­â¥£à «®¢ ¢

â¥à¬¨­ å ¢¥á®¢®© ä㭪樨 ¨ £à ­¨æë.

ãáâì ­  [1

;1

) § ¤ ­ë áâண® ¯®«®¦¨â¥«ì­ë¥ ¨ ­¥ã¡ë¢ î騥 ä㭪樨

f

(

x

) ¨

'

(

x

), 㤮¢«¥â¢®àïî騥 ãá«®¢¨ï¬

f

(

x

)

'

2

(

x

)

";

f

(

x

)

'

3

(

x

)

#:

(19)

Ž¡®§­ ç¨¬

H

(

) =

=2

'

(

)

;

(

f;H

) =

1

Z

1 f

(

)

P

(

j

S

(

)

jb 2

H

(

))

d;

£¤¥

b

2

¨§ ®¯à¥¤¥«¥­¨ï ª« áá 

D

,

H

,1

(

x

) | äã­ªæ¨ï ®¡à â­ ï ª

H

(

x

).

’¥®à¥¬  4

.

ãáâì X

1 ;X

2

;::: | ¯®á«¥¤®¢ â¥«ì­®áâì ­. ®. à. á. ¢.

à¥¤-¯®«®¦¨¬, çâ® ¢ë¯®«­¥­ë ãá«®¢¨ï (19), EX

1

= 0

; EX

2

1

= 1

; c

k

(

)

2D

, ªà®¬¥

⮣®,

E

[

H ,1

(

jX

1 j

)]

f

(

H ,1

(

jX

1 j

))

lnH

,1

(

jX

1

j

)

<1:

(20)

’®£¤  à ¢­®á¨«ì­ë ãá«®¢¨ï

 )

(

f;H

)

<1

;

¡)

1 R

1

f()

1,=2 H() e

, H

2

()

2

d<1:

C

‡ ¯¨è¥¬

(

f;H

) ¢ ¢¨¤¥ áã¬¬ë ¤¢ãå ¨­â¥£à «®¢:

(

f;H

) =

1

Z

f

(

)

P ,

j

S

(

)

jb 2

H

(

)

,

2

,

,'

(

)

+2 1 Z 1 f() , ,'()

d=I

1 +I

2

: (21)

‚®á¯®«ì§®¢ ¢è¨áì ­¥à ¢¥­á⢮¬ (7), ¢ë¢®¤¨¬

I 1 c 1 Z 1 f() ,=2 ' 3 () H() Z 0 u 3 dP , jX 1 ju

d + 1 Z 1 f() 1 ' 2 () 1 Z H() u 2 dP , jX 1 ju

d=I 0 1 +I 00 1 : (22) Œ¥­ïï ¯®à冷ª ¨­â¥£à¨à®¢ ­¨ï, ¯®«ã稬 I 0 1 =c 1 Z H(1) u 3 1 Z H ,1 (u) ,=2,1 f() ' 3 () ddP , jX 1 ju

c 1 Z H(1) u 3 f , H ,1 (u) ' 3 , H ,1 (u) H ,1 (u) ,=2 dP , jX 1 ju

=c 1 Z H(1) f , H ,1 (u) H ,1 (u) dP , jX 1 ju

cEf H ,1 , jX 1 j h H ,1 , jX 1 j i <1: (23) €­ «®£¨ç­® ãáâ ­ ¢«¨¢ îâáï®æ¥­ª¨ I 00 1 =c 1 Z H(1) u 2 H ,1 (u) Z 1 f() ' 2 () ddP , jX 1 ju

c 1 Z H(1) u 2 f , H ,1 (u) ' 2 , H ,1 (u) lnH ,1 (u)dP , jX 1 ju

=c 1 Z H ,1 (u) f , H ,1 (u) lnH ,1 (u)dP , jX 1 ju

cE

H ,1

,

jX

1 j

f

H ,1

,

jX

1 j

lnH ,1

,

jX

1 j

<1: (24)

‘«¥¤®¢ â¥«ì­®, ¯à¨ ãá«®¢¨ïå ⥮६먧 (22){(24) ¨¬¥¥¬

I

1

<1: (25)

’ ªª ª ,

,'()

1

p

2'() e

,' 2

()

2

¯à¨ !1, â® ®¤­®¢à¥¬¥­­ ï

á室¨-¬®áâì ¨ à á室¨¬®áâìI

2

¨ ¨­â¥£à «  ¨§ ¯ã­ªâ  ¡) ®ç¥¢¨¤­ .

Žâá, ãç¨â뢠ï (21)¨ (25), ¯®«ãç ¥¬ ã⢥ত¥­¨¥â¥®à¥¬ë. B

‚ ç áâ­®áâ¨, ¤«ï ¬. á. á।­¨å  à¨ä¬¥â¨ç¥áª¨å, ¨§ ⥮६ë 4 ¯®«ãç ¥¬

ᮮ⢥âáâ¢ãîéãî ⥮६㠨§ [4].

 áᬮâਬç áâ­ë©á«ãç ©,ª®£¤ ' 2

(x)=(2+")ln lnx,">0,f(x) =' 2

(x).

‹¥£ª® ¯à®¢¥à¨âì, çâ® ¯à¨ x!1

H ,1

(x)

x 2

(2+")lnlnx

1

:

’®£¤  ãá«®¢¨¥ (20) ⥮६ë 4 ¯à¨­¨¬ ¥â ¢¨¤

EX 2

1 lnjX

1

j<1: (26)

‚¢¥¤¥¬ ¢à áᬮâ७¨¥ á. ¢.

" =

1

Z

e

lnln

I

n

jS()jb

2 p

(2+") ,

lnln o

d:

ˆ§ ¯à¥¤ë¤ã饩 ⥮६ë á«¥¤ã¥â, çâ® ¯à¨ ¢ë¯®«­¥­¨¨ (26) E

"

< 1 ¯à¨

ª ¦¤®¬ " > 0, ­® ¢ â® ¦¥ ¢à¥¬ï

"

à áâ¥â ¯à¨ " ! 0. ®í⮬㠯।áâ ¢«ï¥â

¨­â¥à¥á  á¨¬¯â®â¨ª 

"

¯à¨" !0.

’¥®à¥¬ 5. ãáâìX

1 ;X

2

;::: |¯®á«¥¤®¢ â¥«ì­®áâì­.®.à.á.¢.,EX

1 =0,

EX 2

1

=1,¢ë¯®«­¥­® (26). ’®£¤  ¯à¨ " !0

E

" =

p

2

" p

"

(1+o(1)):

C à¥¤áâ ¢¨¬ E

"

¢ ¢¨¤¥ áã¬¬ë ¤¢ãå ¨­â¥£à «®¢

E

" =

1

Z

lnln

h

P

jS()jb

2 p

(2+") ,

,2(, p

(2+")ln ln) i

d

+2 1

Z

e

lnln

(, p

2+"ln ln)d=A(")+2D("): (27)

®ª ¦¥¬, çâ® " p

"A(")!0 ¯à¨ " !0: „«ï í⮣® à §®¡ê¥¬ A(") ­  ¤¢ 

¨­â¥£-à « 

A(")= exp ("

,3=4

)

Z

e

lnln

h

P

jS()jb

2 p

(2+") ,

lnln

,2

, p

(2+")lnln i

d

+ 1

Z

exp(" ,3=4

) lnln

h

P

jS()jb

2 p

(2+") ,

lnln

,2

, p

(2+")lnln i

d=A

1

(")+A

2

("): (28)

Žç¥¢¨¤­®

A

1

(")2 exp("

,3=4

)

Z

e

lnln

d2" ,3=4

ln" ,3=4

:

Žâá á«¥¤ã¥â, ç⮯ਠ" !0

" 3=2

A

1

(") !0: (29)

„«ï ®æ¥­ª¨A

2

(") ¢®á¯®«ì§ã¥¬áï ­¥à ¢¥­á⢮¬ (7):

A

2

(")c 1

Z

exp (" ,3=4

) lnln

,=2

(lnln) 3=2

H()

Z

0 u

3

dP(jX

1

j u)d

+c 1

Z

exp (" ,3=4

) lnln

1

lnln 1

Z

H() u

2

dP(jX

1

ju)d=A 0

2 +A

00

2

: (30)

Œ¥­ïï ¯®à冷ª ¨­â¥£à¨à®¢ ­¨ï, ¡ã¤¥¬ ¨¬¥âì

A 0

2 =c

1

Z

,3=4 u

3 1

Z

H ,1

(u)

d

1+=2

p

lnln

dP(jX

1

’ ªª ª >0, â®

A 0

2 c

1

Z

H(exp" ,3=4

)

u 3

p

lnlnH ,1

(u)

H ,1

(u)

,=2

dP(jX

1

ju):

ˆá¯®«ì§ãï ®¯à¥¤¥«¥­¨¥ H(),«¥£ª® ¯®«ãç ¥¬, çâ®

A 0

2 c

1

Z

H(exp" ,3=4

) u

2

dP(jX

1

ju)cEjX

1 j

2

: (31)

€­ «®£¨ç­® ¤«ï A 00

2 ;

A 00

2 =c

1

Z

H(exp" ,3=4

) u

2 H

,1

(u)

Z

exp" ,3=4

,1

ddP(jX

1 ju)

c

1

Z

H(exp" ,3=4

) u

2

lnH ,1

(u)dP(jX

1

ju):

®áª®«ìªã H(exp" ,3=4

) ! 1 ¯à¨ " ! 0, â® ãç¨âë¢ ï  á¨¬¯â®â¨ªã H ,1

(),

¯®«ãç ¥¬

A 00

2 c

1

Z

H(exp" ,3=4

) u

2

lnudP(jX

1

ju) cEX 2

1 lnjX

1

j: (32)

ˆâ ª, ¨§ (30){(32)á«¥¤ã¥â, çâ® ¯à¨ "!0

" 3=2

A

2

(") !0: (33)

‘«¥¤®¢ â¥«ì­®, ¨§ (28),(29), (33) ¨¬¥¥¬

" 3=2

A(")!0 (34)

¯à¨ "!0:

‘ ¯®¬®éìî í«¥¬¥­â à­ëå ¯à¥®¡à §®¢ ­¨© ¯®«ãç ¥¬ ¯à¨ "!0

D(") = 1

p

+o(" ,3=2

Žæ¥­ª¨ ¢ § ª®­ å ¡®«ìè¨å ç¨á¥« ¤«ï ¬¥â®¤®¢ á㬬¨à®¢ ­¨ï

3{27

Žâá, á ãç¥â®¬ (27) ¨ (34), ¢ë⥪ ¥â ã⢥ত¥­¨¥ ⥮६ë.

B

‹¨â¥à âãà 

1. ¨ªï«¨á €. ’. €á¨¬¯â®â¨ç¥áª¨¥ à §«®¦¥­¨ï ¤«ï á㬬 ­¥§ ¢¨á¨¬ëå

m

-à¥è¥âç âëå á«ãç ©­ëå ¢¥ªâ®à®¢ // ‹¨â. ¬ â. á¡.|1972.|’. 12.|‘. 118{

189.

2. ƒ äã஢ Œ. “. à¨¬¥­¥­¨¥  ­ «®£  ­¥à ¢¥­á⢠ £ ¥¢  ‘. ‚. ¨ ”㪠 „. •.

¤«ï ¢§¢¥è¥­­ëå á㬬 ­¥§ ¢¨á¨¬ëå á«ãç ©­ëå ¢¥«¨ç¨­ ¯® § ª®­ã ¡®«ìè¨å

ç¨á¥« // Banach center publication, Warszawa.|1979.|V. 5.|P. 260{271.

3. ƒà ¤è⥩­ ˆ. ‘., ë¦¨ª ˆ. Œ. ’ ¡«¨æë ¨­â¥£à «®¢, á㬬, à冷¢ ¨

¯à®¨§¢¥¤¥­¨©.|Œ.: ”¨§¬ â£¨§, 1963.|1514 á.

4. ‘¨à ¦¤¨­®¢ ‘. •., ƒ äã஢ Œ. “. Œ¥â®¤ à冷¢ ¢ £à ­¨ç­ëå § ¤ ç å ¤«ï

á«ãç ©­ëå ¡«ã¦¤ ­¨©.|’ èª¥­â: ”€, 1987.|140 á.

5. Baum L. E, Katz M. Convergence rates in the law of large numbers // Trans.

Amer. Math. Soc.|1965.|V. 120, No. 1.|P. 108{123.

6. Heyde C. C. A supplement to the strong law of large numbers // J. Appl.

Probab.|1975.|V. 12, No. 1.|P. 173{175.

7. Sztencel R. On Boundednes and convergence of some Banach space valued

random series // Probab. Math. Statist.|1981.|V. 2, No. 1.|P. 83{88.