B-值随机元及经验过程的Kolmogor

n≥1
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>
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*
,
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lim
f (Xi )
i=1 F
2 2s2 n L 2 sn ≤ M
M ≥1
(b.)
2
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n i=1
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p
>
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F
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7
af
F
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jX x
f ( s)
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(1.8)
n
n→∞ s2 ∗ n f ∈ B1
lim
1
sup
E
k=1

f 2 (Xk )I
Xk ≥ε
s2 k L2 sk

=0
(1.9)
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F
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N\ EX (PX ), EX (PX {εn }n≥1 , {gn }n≥1 1Hy(_ (Q ). = {af }f ∈F , p
af
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§3.
{Xn }n≥1
, "
Sn /
{s2 n }n≥1
(1.1)
Xk ≤ ηk
s2 k L 2 s2 k
a.s.
∀k ≥ 1,
(1.5)
9)>d
lim
2 2s2 n L2 sn −→ 0,
p
(1.6)
Sn
2 2s2 n L 2 sn
]3 W o< 5 Kolmogorov A H E ` 2 B E na t > 2 2 {Xn }n≥1 ∈ W M0 k sn → +∞ q ' (1.6). ) \ d EF * g/a t A Z 2 Sn / 2sn L2 s2 4 L 5 {Xn }n≥1 o N ' nlim n 5 →∞ 1.2 $ jdE P;5 E `
, f1 , · · · , fk ∈ F , @ A 6 i = j },
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5
,
k
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B-
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Kolmogorov
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∀ε > 0 N (ε, F , d) ≤ D(ε, F , d) ≤ N s2 n → +∞,
X x (1.5),
Kolmogorov ( Bn→∞
Sn ≤M 2 2s2 n L 2 sn ([10], [7],
a.s.
(1.4) 292 1) s2 n =n i=19h Nhomakorabea3.1),
Y 1
L8
E Xi
2
,
lim
Sn ≤M 2 2s2 n L 2 sn
a.s.
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&,
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E {Xn }n≥1 5> O g Hx -
* , # FW Dl
1.2 F ⊂ B ,
F= ∗
L ? r < v j §1 ~X5 " 2 §2 ~X5 " jf y "
, ## ,
1.2,
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{Xn }n≥1 , {Xn }n≥1 , {εn }n≥1 , {gn }n≥1 ), Eε (Pε ), Eg (Pg ) {Xn }n≥1 , {Xn }n≥1 ,
1991
3
29
z
Sn = X1 + · · · + Xn (n = 1, 2, · · · ).
5
)
Q
EV -
1.1 s2 n
→ +∞ (n → +∞),
k = {Xn }n≥1 E
B-
B
(Ph l ~Z%6 ,, > Og x "H
s2 k L 2 sk a.s.
2 2s2 n L 2 sn n≥1
2
"H
ε , F, d 2 {αn }n≥1 , αn → 0 (n → +∞)
(3.1)
@
(3.2)
Xk ≤ αn
s2 n L 2 s2 n
1 ≤ k ≤ n, n ≥ 1
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6_ EV
s2 n
, +
I
z Xx
z
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#
36
<
1.1
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1.2, B
1.3 → +∞
= {Xn }n≥1 E 9 ∀ε > 0,
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P
n
3
{ Xn }n≥1
g 5.
f 43 }5 _ u
(1.1)
6
Og x
Xn ≥ ε s2 n L 2 s2 n
2 {Xn }n≥1 ∈ W M0 , {s2 n }n≥1
F ≥ε
s2 n L2 s2 n

1 n→∞ s2 n lim
n
E
2 s2 n L 2 sn
< +∞
f 2 (Xk )I
f (Xk )
F ≥ε
k=1
s2 k L2 s2 k

= 0,
F
(a.)
2
n i=1
f (Xi )
F
/
2 2s2 n L 2 sn n
(1.4) 1.2
EV
(i)
'
B
,>
s2 n L2 s2 n
B ((1.7) B) ! 5 W ` 1.1 ] 1.3 pL 5~Xd j! 5 Og x
< +∞ ∀ε > 0;
Xk ≥ε
s2 k L2 s2 k
5V 1.4, R
1.2
Xn
E [10], 303
(1.1)
f (Xi )
F
/
2 2s2 n L 2 sn
n≥1
Og
>
a.s.
n
1 ≤ lim
*
,
n→∞
f (Xi )
i=1 F
2 2s2 n L 2 sn ≤ M
M ≥1
(b.)
2
` H
n i=1
f (Xi )
F
2 2s2 n L2 sn −→ 0 n
p
>
2 2s2 n L 2 sn = 1
(i) ∀n ≥ 1, +∞ (n → +∞);
a.s.
5
Banach
|
2( )
|x r s . * ~ }0 u; # " +{y 14&-| q! w / v| ; 2; ) u IPA 7} ; UOnD pH]Y t E 5S < t E 51 c }A-= ` L
2 2s2 n L 2 sn
lim
Sn
s2 ∗ n = supf ∈B1
n i=1
Kolmogorov
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2 {Xn }n≥1 ∈ W M0 ,
{ηn }, ηn → 0
@
=
{s2 n }n≥1
(1.1)
Xk ≤ ηk
∀k ≥ 1
(1.2)
) \9 d
Sn /
Og
,
(1.3)
,
H
1.1
M ≥ 1,
@
1 ≤ lim
n→∞
EV y & o< 1 # 5 / AH W3 ' .s ) % + 5 w D~X , 8 v g/a. j * & 43 s8
L ` i· i·d· 5
L C A A 6 ^ F `j j5
8?H
%
f ∈F
sup |af |
A
F
<56 b?
(pseudometric) d. N (ε, F , d)= min{k ≥ 1;
i ≤k
,
f1 , · · · , fk ∈ F ,