Directory UMM :Journals:Journal_of_mathematics:VMJ:

‚« ¤¨ª ¢ª §áª¨© ¬ â¥¬ â¨ç¥áª¨© ¦ãà­ «
ˆî«ì{ᥭâï¡àì, 2000, ’®¬ 2, ‚ë¯ã᪠3

“„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­ëå ¬¥â®¤ ¬ 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¨å á㬬 ¨ ¤à. „«ï ¢§¢¥è¥­­ëå á㬬 á ¢¥á ¬¨
¨§ í⮣® ª« áá  ¯®«ãç¥­ë ®æ¥­ª¨ ¢ § ª®­ å ¡®«ìè¨å ç¨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 áᬠâਢ «¨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« áá ॣã«ïà­ëå ¬¥â®¤®¢ 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¥« ¢ ¢¨¤¥ á室¨¬®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®£®

ck (n)), § ¤ î騩 ॣã«ïà­ë¥ ¬. á.:
D = f0  ck ()  1; k = 1; 2; : : : ;  > 0;
, ¯à¨  ! 1;
sup ck ()  b1 

¯ à ¬¥âà  | ª« áá ¬ âà¨æ

X1 c  !
1
X

c   b ,
k

k=1

B2(

)=

k(

)

2

k=1

k(

1

)

¯à¨

2
2

‹¥£ª® ¯à®¢¥à¨âì, çâ® í«¥¬¥­â ¬¨

C; r), €¡¥«ï (A). Œ­®¦¥áâ¢ã D1=2
q > 0 (E; q), ®à¥«ï (B ) ¨ ¤à.

1 (

c 2000 „®¥¢ ”. •



 ! 1;
¯à¨

 ! 1g:

D1 ïîâáï ¬. á.

—¥§ à® ¯®à浪 

r

¯à¨­ ¤«¥¦ â ¬¥â®¤ë ©«¥à  ¯®à浪 

3{14

”. •. „®¥¢

ãáâì

X1 , X2 ; : : : | ¯®á«¥¤®¢ â¥«ì­®áâì ­¥§ ¢¨á¨¬ëå ®¤¨­ ª®¢® à á¯à¥¤¥-

«¥­­ëå á«ãç ©­ëå ¢¥«¨ç¨­ (­. ®. à. á. ¢.). Ž¡®¡é ï ª« áá¨ç¥áªãî ¯®áâ ­®¢ªã
§ ¤ ç¨ ® § ª®­¥ ¡®«ìè¨å ç¨á¥«, à áᬮâਬ ¢§¢¥è¥­­ë¥ á।­¨¥

S () =

1
X
k=1

ck ()Xk

Sn

( ( )=

1
X
k=1

ck (n)Xk )

¨ ¢ëïá­¨¬ ãá«®¢¨ï á室¨¬®á⨠¨­â¥£à « 

 ("; q; t) =

Z1

qt,,1 P (jS ()j  "(q,1) ) d;

1

  ¢ á«ãç ¥ ¤¨áªà¥â­®£® ¯ à ¬¥âà  | à鸞

P1

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

n=1 n

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

fck ()g.
ck () 2 D ¢¢¥¤¥¬ ¢ à áᬮâ७¨¥ á«¥¤ãî騩 ­ ¡®à ¨­¤¥ªá®¢ ¯® á⥯¥­¨ ã¡ë¢ ­¨ï ck () ¯® :
¤¨¬®á⨠¢ § ª®­¥ ¡®«ìè¨å ç¨á¥« ¤«ï ¬¥â®¤  á㬬¨à®¢ ­¨ï
„«ï

I = fk : ck () = O(, ) ¯à¨  ! 1g:
c,

—¥à¥§

¨­®£¤  á ¨­¤¥ªá ¬¨, ¡ã¤¥¬ ®¡®§­ ç âì ¯®«®¦¨â¥«ì­ë¥ ¯®áâ®ï­-

­ë¥.

q>

X ;X ;:::

’¥®à¥¬  1. ãáâì
¯®á«¥¤®¢ â¥«ì­®áâì ­. ®. à. á. ¢.,
1
2
1
,
( )
. Šà®¬¥ ⮣®, ¯ãáâì ¯à¨
2

ck 

2 D

!1



X t
ck () = O (1,t)
k

 ("; q; t)
EX1 = 0 ¢ á«ãç ¥ t  1:
„«ï á室¨¬®áâ¨

¯à¨ «î¡®¬

">

â¨ ãá«®¢¨ï ­¥®¡å®¤¨¬ë, ¥á«¨ ¯à¨

I

C

‡ ä¨ªá¨à㥬 § ¢¨á¨¬®áâì
®¤áâ ­®¢ª®©



=

y

< t < 1):

¤®áâ â®ç­®, ç⮡ë

(1)

E jX1jt <

1

¨

!1

card ( ) =

 ("; q; t):

0

(0

qt > 1;

O (  ):

 ("; q; t)

¢ëà ¦¥­¨¥

®â  ¢
1 ("; q; t)

(2)

¢¨¤¥ ­¨¦­¥£® ¨­¤¥ªá 
¯¥à¥¢®¤¨âáï ¢

 ("; q; t):

‘®®â¢¥âáâ¢ãî騩 ¢¨¤ ¯à¨®¡à¥â îâ ¨ ãá«®¢¨ï (1) ¨ (2). ‘«¥¤®¢ â¥«ì­®, ¤®ª § â¥«ìá⢮ ⥮६ë 1 ¤®áâ â®ç­® ¯à®¢¥á⨠¤«ï á«ãç ï

ck () 2 D1 :

Žæ¥­ª¨ ¢ § ª®­ å ¡®«ìè¨å ç¨á¥« ¤«ï ¬¥â®¤®¢ á㬬¨à®¢ ­¨ï
3{15
„®áâ â®ç­®áâì. ãáâì E jX1 jt < 1; 0 < t < 1: ‚®á¯®«ì§ã¥¬áï  ­ «®£ ¬¨
­¥à ¢¥­á⢠ £ ¥¢  | ”㪠 [2]. ’®£¤  ¤«ï «î¡®£®
> 0
Z1

qt,2

1("; q; t) = 
Z1

 
+

e",t
1,t E jX1j

P S   "

q ,1



d

1

qt,2

1

,



 ( )

t
1=

X

Z1

k



 
k ( ) k 

q ,1

P c  X  "


qt,2,(q,1)t=

1

"
X
k

ctk ()



d

#1=

(3)

d = A1 + A2:

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

c 

!1

¡ã¤¥¬ ¯®«ì§®¢ âìáï  á¨¬¯â®â¨ç¥áª¨¬¨ ᢮©á⢠¬¨ k ( ) ¯à¨
. ®«ãç î騥áï ¯à¨ í⮬ ¨­â¥£à «ë, á室ïâáï ¨ à á室ïâáï ®¤­®¢à¥¬¥­­® á ¨á室­ë¬¨.
à¥®¡à §ã¥¬

A1 :
Z1

qt,2

A1 = 
1

Z1



qt,2

 

qt,2

=
1

Z1

1
1 X
X



1X
i
X



q ,1 
q ,1
"

"

P c ()  jXk j < c () d
i
i+1
k=1 i=k
q ,1
q ,1 
"

"

P c ()  jXk j < c () d
i
i+1
i=1 k=1

1 Z
X



i dP (jX1j  y) d;

i=1 L

1

£¤¥



 y < c"
+1 (,1) :
Žç¥¢¨¤­®, L ­¥ ¯ãáâ®, ¥á«¨ ci () > ci+1 ().

L=

"
q,1
ci ()

(4)

q

i

fc0k ()g  fck ()g ã¡ë2n
P
¢ îé ï ¯®á«¥¤®¢ â¥«ì­®áâì ¯à¨ ä¨ªá¨à®¢ ­­ëå . ®áª®«ìªã
c0k () ! 0
¯à¨

n ! 1 ¨ ¯à¨ í⮬

2n
P

k=n

ãáâì

k =n

c0k () > nc02n (); â® c0i () = o( 1i ) ¯à¨ i ! 1.

‘«¥¤®-

3{16

”. •. „®¥¢

¢ â¥«ì­®, ¨§ (4) ¨¬¥¥¬

1

Z



q (t,1),1

 "


q (t,1),1

A1  "
1

1

Z

1

Z

i=1 L

1

1

1

X

Z

y "
q =b1

1

Z
b
1
 ("
)2 q(t,2),1

=

c

1

y2

ydP (jX1j  y) d

Z

y "
q =b1

1

Z

ydP (jX1j  y) d

(yb1 =Z
("
))1=q

"
=b1

y2dP (jX1j  y) d

(5)

q(t,2),1ddP (jX1j  y)  cE jX1jt :

1

¥à¥©¤¥¬ ª ®æ¥­ª¥
⥣ࠫ®¬

A2.

® ãá«®¢¨î (1),

1

Z

A2 á室¨âáï ®¤­®¢à¥¬¥­­® á ¨­-

qt,2,(qt,1)=
d:

1

‹¥£ª® § ¬¥â¨âì, çâ® ¯à¨

‚ ᨫ㠯ந§¢®«ì­®áâ¨
â®ç­®á⨠¤«ï 0

< t < 1.

0

!

dk X  " :

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

Sen

Ž¡®§­ ç¨¬ ç¥à¥§ en | ᨬ¬¥âਧ®¢ ­­ë¥ á. ¢.
P

k

=

ck ()Xek . ® ­¥à ¢¥­á⢠¬ ᨬ¬¥âਧ æ¨¨
1

Z



3{17

n
P
k=1

Xek ; Se() =



e("; q; t) = qt,2P jSe()j  "q,1 d < 1:
1

à¨¬¥­¨¢ «¥¬¬ã á
(

dk = ck ()
¯®«ã稬

e("; q; t)  2
1

1

Z

1

I;
ak = ck0(; ); kk 22 I;

¨

qt,2P


X



k 2I



k ( ) e k 

c X 

"q,1

!

d:

‘«¥¤®¢ â¥«ì­®, á室¨âáï ¨­â¥£à «

1

Z

A = qt,2 P
1

!



X


e 
k


k 2I

X  c"q d:

‘ ãç¥â®¬ ãá«®¢¨ï (2) ¡ã¤¥¬ ¨¬¥âì

1

X

A=



n=1 n

1

X

n=1

"1 = 2q c":





qt,2 P Se[]  c"q d






nqt,2P Sn  nq c"


£¤¥

nZ+1



e 

1

X

n=1



1+

1

n

!
q 



nqt,2P Sen   "1 nq ;


E Xe1 t < 1.
t
‘®£« á­® á«¥¤áâ¢¨î ¨§ ­¥à ¢¥­á⢠ᨬ¬¥âਧ æ¨¨ ¯®«ãç ¥¬ E X1  < 1.
’¥®à¥¬  1 ¤®ª § ­ . B
Žâá ¯® ¨§¢¥áâ­®© ⥮६¥  ã¬  | Š æ  [5] á«¥¤ã¥â





3{18

”. •. „®¥¢

…᫨ ¢¬¥áâ® fck ()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£ :
Z
1
1 X
2 dP (X  y ) ! 0 ¯à¨  ! 1:
2
c
(

)
y
k
k
2
B () k=1
jyj" cBk(())

’ 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].
…᫨ EX1 = 0, EX12 = 1, â®

(; x) sup ck ()
jP (S ()  xB ()) , (x)j  c (1 + jxjk)3B () ;

£¤¥

(; x) 

Z

juj3dP (X

1

Z

 u) + (1 + jxj)B ()

(7)

u2 dP (X1  u):

x )B()
juj (1+
sup ck ()

x )B()
juj (1+
sup ck ()

j j

j j

k

k

l,1p
=2) qt,
Ž¡®§­ ç¨¬ ,l = ,(
(l,1)  , 2q , = s, £¤¥ ,(z ) | £ ¬¬ -äã­ªæ¨ï.

’¥®à¥¬  2.

ãáâì

EX1 = 0 EX12 = 1

 ("; 1; 1) = 2 ;
 ) lim
1

"#0
ln

"

,

p

2s

. ’®£¤  á¯à ¢¥¤«¨¢ë ᮮ⭮襭¨ï:

¡) lim
"2s ("; q; t) = ((22qb,)1) ,s+1
E jX1jt < 1
"#0
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= 
Z1 1 
 ("; 1; 1) =  P (jS ()j  ") , 2 , b " d
2
1
(8)
Z1 1  1= 
+   , b " d = 1 + 2:
1

2

¯à¨

2

.

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

1
lim
= 0:
"#0 ln 1"

3{19
(9)

‚롥६ n0 (") > 0 â ª, ç⮡ë n0(") ! 1, nln0 ("1") ! 0 ¯à¨ " ! 0. ’®£¤ 
Z
Z
1 =
+
= 10 + 200 :
 0 ¯à®¨§¢®«ì­®. ‘«¥¤®¢ â¥«ì­®,

2 = 0:
(16)
lim
"#0 ln 1"
Žç¥¢¨¤­® ¨ ¤«ï
3 ¢ë¯®«­¥­® ᮮ⭮襭¨¥

3 = 0:
lim
(17)
"#0 ln 1"
ˆ§ (10){(17) á«¥¤ã¥â (9).
p
 áᬮâਬ ¨­â¥£à « 2 , ª®â®àë© ¯®¤áâ ­®¢ª®© b12 =2" = x ¯à¨¢®¤¨âáï
ª ¢¨¤ã
Z1 1 p
Z1 1  =2 
2
 , b " d = 
(, x)dx
2 = 2

x
2

1

"2 =b22

,Z"=b2
Zt2 1
2
2
= p
dx dt
e, 2
x
 2
2 2
,1
t

" =b2

,Z"=b2
,Z"=b2
2
2
1
2
4
e, 2 lnt2 dt + p ln
e, 2 dt
= p
 2
 2 "
,1
,1
t

t

,Z"=b2
2
2 1
4ln
b
2
+ p
e, 2 dt  c + ln
 "
 2
,1
t

(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 ⥮६ë 2.
ãáâì F t | ª« áá ä㭪権 à á¯à¥¤¥«¥­¨ï F (x) = P (X  x) ®¡« ¤ îé¨å
᢮©á⢠¬¨:
Z1
Z1
xdF (x) = 0;
x2 dF (x) = 1;
,1

lim sup

a!1 F 2F

Z

jxj>a

,1

x2 dF (x) = 0;

Z1
,1

jxjtdF (x) < 1:

3{22

”. •. „®¥¢

Ž¡®§­ ç¨¬



(

F ) ("; q; t) =

Z1

qt,,1PF (jS() j  "(q,1) )d;

1

£¤¥ PF | ¢¥à®ïâ­®áâ­ ï ¬¥à , ᮮ⢥âáâ¢ãîé ï ä㭪樨 à á¯à¥¤«¥­¨ï F (x).
’¥®à¥¬  3. ãáâì
 Fck () 2 D . ’®£¤  ¢¥à­ë ᮮ⭮襭¨ï
a) lim"#0 supF 2F   ln(";" 1;1) , 2  = 0;


p s
¡) lim"#0 supF 2Ft "2s  (F )("; q; t) , ((22qb,)1) As+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¬
( )

1

2

2

2

f (x) "; f (x) # :
'2 (x)
'3 (x)

(19)

Ž¡®§­ ç¨¬

H () = =2'();

(f; H ) =

Z1
1

f () P (j S ()j  b H ()) d;
2


£¤¥ b2 ¨§ ®¯à¥¤¥«¥­¨ï ª« áá  D , H ,1 (x) | äã­ªæ¨ï ®¡à â­ ï ª H (x).
’¥®à¥¬  4. ãáâì X1 ; X2 ; : : : | ¯®á«¥¤®¢ â¥«ì­®áâì ­. ®. à. á. ¢. à¥¤¯®«®¦¨¬, çâ® ¢ë¯®«­¥­ë ãá«®¢¨ï (19), EX1 = 0; EX12 = 1; ck () 2 D , ªà®¬¥
⮣®,

E [H ,1(jX1j)]f (H ,1(jX1j))lnH ,1 (jX1j) < 1:

’®£¤  à ¢­®á¨«ì­ë ãá«®¢¨ï

 ) (f; H ) < 1;
R1
H 
¡)  ,f=(H)  e,  d < 1:
1
C ‡ ¯¨è¥¬ (f; H ) ¢ ¢¨¤¥ áã¬¬ë ¤¢ãå ¨­â¥£à «®¢:
2( )

1

2

2

( )

(f; H ) =

Z1
1

f () P ,jS ()j  b H ()
, 2, , '()
d
2


(20)

Žæ¥­ª¨ ¢ § ª®­ å ¡®«ìè¨å ç¨á¥« ¤«ï ¬¥â®¤®¢ á㬬¨à®¢ ­¨ï
Z1

+2

1

3{23



f () ,

,
'() d = I1 + I2 :


(21)

‚®á¯®«ì§®¢ ¢è¨áì ­¥à ¢¥­á⢮¬ (7), ¢ë¢®¤¨¬
Z1

I1  c

1

+

Z1
1

HZ()

f () ,=2
 '3 ()

f () 1
 '2 ()

Z1

,




u3 dP jX1 j  u d

0

,




u2 dP jX1 j  u d = I10 + I100 :

(22)

H ()

Œ¥­ïï ¯®à冷ª ¨­â¥£à¨à®¢ ­¨ï, ¯®«ã稬
Z1

I10 = c

u3

H ,1 (u)

H (1)
Z1

c

Z1

,=2,1

,

,


f ()
ddP
j
X
j

u
1
'3 ()




,=2
,


f H ,1 (u) 
u 3 , ,1
H ,1 (u)
dP jX1 j  u
' H (u)
3

H (1)
Z1

=c

H (1)

 cEf

,






,

f H ,1 (u) H ,1 (u) dP jX1j  u



,

H ,1 jX1 j


h

,

H ,1 jX1 j


i




< 1:

€­ «®£¨ç­® ãáâ ­ ¢«¨¢ îâáï ®æ¥­ª¨
I100 = c

Z1

u2

H (1)

c

Z1

=c

H (1)

1

,

,


f ()
ddP
j
X
j

u
1
'2 ()




,


f H ,1 (u)
u 2 , ,1
lnH ,1 (u)dP jX1j  u
' H (u)
2

H (1)

Z1

HZ,1 (u)



 ,




H ,1 (u) f H ,1 (u)

,

lnH ,1 (u)dP jX1j  u




(23)

3{24

”. •. „®¥¢



,




,

f H ,1 jX1 j




,




lnH ,1 jX1j < 1:
‘«¥¤®¢ â¥«ì­®, ¯à¨ ãá«®¢¨ïå â¥®à¥¬ë ¨§ (22){(24) ¨¬¥¥¬

 cE

H ,1 jX1 j



I1 < 1:

(24)
(25)

’ ª ª ª  , '()  p2'1 () e ,'  ¯à¨  ! 1, â® ®¤­®¢à¥¬¥­­ ï á室¨¬®áâì ¨ à á室¨¬®áâì I2 ¨ ¨­â¥£à «  ¨§ ¯ã­ªâ  ¡) ®ç¥¢¨¤­ .
Žâá, ãç¨â뢠ï (21) ¨ (25), ¯®«ãç ¥¬ ã⢥ত¥­¨¥ ⥮६ë. B
‚ ç áâ­®áâ¨, ¤«ï ¬. á. á।­¨å  à¨ä¬¥â¨ç¥áª¨å, ¨§ ⥮६ë 4 ¯®«ãç ¥¬
ᮮ⢥âáâ¢ãîéãî ⥮६㠨§ [4].
 áᬮâਬ ç áâ­ë© á«ãç ©, ª®£¤  '2 (x) = (2+")lnlnx, " > 0, f (x) = '2 (x).
‹¥£ª® ¯à®¢¥à¨âì, çâ® ¯à¨ x ! 1
,




2( )
2

H ,1 (x) 



x2
(2 + ")lnlnx

 1

:

’®£¤  ãá«®¢¨¥ (20) ⥮६ë 4 ¯à¨­¨¬ ¥â ¢¨¤
EX12lnjX1 j < 1:

(26)

‚¢¥¤¥¬ ¢ à áᬮâ७¨¥ á. ¢.
" =

Z1

e

lnln I njS ()j  b p(2 + "), lnlno d:
2


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.
X1 ; X2; : : :
EX1 = 0
2
EX1 = 1
(26)
"!0
ãáâì

, ¢ë¯®«­¥­®

| ¯®á«¥¤®¢ â¥«ì­®áâì ­. ®. à. á. ¢.,

. ’®£¤  ¯à¨

p
2
E" = p (1 + o(1)):
" "

C à¥¤áâ ¢¨¬ E" ¢ ¢¨¤¥ áã¬¬ë ¤¢ãå ¨­â¥£à «®¢
E" =

Z1

e

lnln hP jS ()j  b p(2 + "), lnln
2


,

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

p

+2

3{25

,2(, (2 + ")lnln) d

Z1

lnln (,p2 + "lnln) d = A(") + 2D("):


(27)

e
p
®ª ¦¥¬, çâ® " "A(") ! 0 ¯à¨ " ! 0: „«ï í⮣® à §®¡ê¥¬ A(") ­  ¤¢  ¨­â¥£-

à « 

Z"

exp(

A(") =

,3=4 )

e p

lnln hP jS ()j  b p(2 + "), lnln
2

i

, 2 , (2 + ")lnln d
Z1

+

",3=4 )

exp(



lnln hP jS ()j  b p(2 + "), lnln
2

i

p

, 2 , (2 + ")lnln d = A1(") + A2("):
Žç¥¢¨¤­®

A1(")  2

Z",3=4 )

exp(

e

(28)

lnln d  2",3=4 ln",3=4:


Žâá á«¥¤ã¥â, çâ® ¯à¨ " ! 0

"3=2 A1(") ! 0:
„«ï ®æ¥­ª¨ A2(") ¢®á¯®«ì§ã¥¬áï ­¥à ¢¥­á⢮¬ (7):
Z1

A2(")  c
exp(

+c
exp(

Z1
",3=4 )

",3=4 )

(29)

H ()
lnln ,=2 Z u3 dP (jX j  u) d
1
 (lnln)3=2
0

1
lnln 1 Z u2 dP (jX j  u) d = A0 + A00:
1
2
2
 lnln
H ()

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

A02 = c

Z1

H (exp ",3=4 )

u3

Z1

H ,1 (u)



dp

=2

1+

lnln

dP (jX1j  u):

(30)

3{26
’ ª ª ª

”. •. „®¥¢

 > 0, â®
A0

Z1

2  c

 ,1
,=2
u3
H
(u)
dP (jX1j  u):
lnlnH ,1 (u)

p

H (exp ",3=4 )

H (), «¥£ª® ¯®«ãç ¥¬, çâ®

ˆá¯®«ì§ãï ®¯à¥¤¥«¥­¨¥

A0

2

€­ «®£¨ç­® ¤«ï

Z1

c

H (exp ",3=4 )

Z1

u2

H (exp ",3=4 )

HZ,1 (u)

exp

Z1

c

H (exp ",3=4 )

¯®«ãç ¥¬

(31)

A002 ;

A002 = c

®áª®«ìªã

u2dP (jX1j  u)  cE jX1j2:

",3=4

,1 ddP (jX1j  u)

u2 lnH ,1 (u) dP (jX1j  u):

H (exp ",3=4 ) ! 1 ¯à¨ " ! 0, â® ãç¨âë¢ ï  á¨¬¯â®â¨ªã H ,1 (),
A00  c
2

Z1

H (exp ",3=4 )

u2 lnu dP (jX1j  u)  cEX12lnjX1j:
"!0
"3=2 A2(") ! 0:

(32)

ˆâ ª, ¨§ (30){(32) á«¥¤ã¥â, çâ® ¯à¨

(33)

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

"3=2 A(") ! 0
¯à¨

" ! 0:

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

D(") = p1 + o(",3=2 ):
" 2"

(34)

"!0

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 mà¥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.
£. ‚« ¤¨ª ¢ª §

‘â âìï ¯®áâ㯨«  22 ¨î«ï 2000 £.