Luật số lớn đối với dãy phần tử ngẫu nhiên trên không gian hilbert

Ω
1
, F
1
) Ω
2
, F
2
)
X : Ω
1
→ Ω
2
F
1
/F
2
B ∈ F
2
X
−1
(B) ∈ F
1
1. F
1

, G
1
σ Ω
1
F
2
, G
2
σ
Ω
2
F
1
⊂ G
1
G
2
⊂ F
2
X : Ω
1
→ Ω
2
F
1
/F
2
X G
1
/G

2
2. X : Ω
1
→ Ω
2
F
1
/F
2
Y : Ω
2
→ Ω
3
F
2
/F
3
Y ◦ X : Ω
1
→ Ω
3
F
1
/F
3
3. F
2
= σ(C) X : Ω
1
→ Ω

2
F
1
/F
2
X
−1
(C) ∈ F
1
C ∈ C.
(Ω, F, P) G σ
σ F X : Ω → R
G G/B(R) B ∈ B(R)
X
−1
(B) ∈ G
X F
X
G
X
σ(X) =

X
−1
(B) : B ∈ B(R)

σ σ F σ σ
X σ X X
G σ(X) ⊂ G
(X < a) := (ω : X(ω) < a) ∈ F a ∈ R

(X  a) := (ω : X(ω)  a) ∈ F a ∈ R
(X > a) := (ω : X(ω) > a) ∈ F a ∈ R
(X  a) := (ω : X(ω)  a) ∈ F a ∈ R
X
1
, X
2
, , X
n
(Ω, F, P) f : R
n
→ R ( B(R
n
)/B(R) )
Y = f(X
1
, , X
n
) : Ω → R
ω → f(X
1
(ω), , X
n
(ω))
X, Y (Ω, F, P )
f : R → R a ∈ R aX, X ± Y, XY, |X|, f(X), X
+
=
max(X, 0), X
−

= max(−X, 0), X/Y (Y = 0)
{X
n
, n  1}
(Ω, F, P) inf
n
X
n
sup
n
X
n
inf
n
X
n
, sup
n
X
n
, limX
n
, limX
n
lim
n→∞
X
n
( )
X

{X
n
, n  1} X
n
↑ X n → ∞
(Ω, F, P) X : Ω → R
P
X
: B(R) → R
B → P
X
(B) = P(X
−1
(B))
X
P
X
B(R)
Q B(R) Q
X
(Ω, F, P) X : Ω → R
F
X
(x) = P(X < x) = P(ω : X(ω) < x)
X
F
X
(x) = P

X

−1
(−∞, x)

= P
X
[(−∞, x)]
0  F (x)  1.
a < b F (b) − F(a) = P(a  X < b) F (x)
lim
x→+∞
F (x) = 1, lim
x→−∞
F (x) = 0
lim
x↑a
F (x) = F (a) lim
x↓a
F (x) = P(X  a). F (x)
F (x) a P(a) = 0.
F (+∞) = lim
x→+∞
F (x), F (−∞) = lim
x→−∞
F (x).
F (+∞) = 1 F (−∞) = 0.
X : (Ω, F, P) → (R, B(R))
X P
X EX
EX =


Ω
XdP.
E|X|
p
< ∞ (p > 0) X p
E|X| < ∞ X
X  0 EX  0.
X = C EX = C
EX C ∈ R E(CX) = CEX.
EX EY E(X ± Y ) = EX ± EY.
X  0 EX = 0 X = 0.
X
n
↑ X ( X
n
↓ X)
n EX
−
n
< ∞ ( EX
+
n
< ∞) EX
n
↑ EX (
EX
n
↓ EX)
X
n

 Y n  1 EY > −∞
ElimX
n
 limEX
n
.
X
n
 Y n  1 EY < +∞
ElimX
n
 limEX
n
.
|X
n
|  Y n  1 EY < ∞
ElimX
n
 limEX
n
 limEX
n
 ElimX
n
.
|X
n
|  Y n  1
EY < ∞ X

n
→ X X E|X
n
− X| → 0 EX
n
→ EX
n → ∞
X
ε > 0
P(X  ε) 
EX
ε
.
X DX := E(X −
EX)
2
X
X
X
DX =






(x
i
− EX)
2

p
i
X P(X = x
i
) = p
i
;

+∞
−∞
(x − EX)
2
p(x)dx X p(x).
DX = EX
2
− (EX)
2
DX  0
DX = 0 X = EX = C ( ) h. c. c.
D(CX) = C
2
DX
X
DX ε > 0
P(|X − EX|  ε) 
DX
ε
2
.
p > 0 L

p
= L
p
(Ω, F, P)
(Ω, F, P)) E|X|
p
< ∞ X ∈ L
p
, p  1
X
p
= (E|X|
p
)
1/p
.
p
X, Y ∈ L
2
E|XY |  X
2
Y 
2
.
p, q ∈ (1; +∞)
1
p
+
1
q

= 1
X ∈ L
p
, Y ∈ L
q
E|XY |  X
p
Y 
q
.
X, Y ∈ L
p
, 1  p < ∞
X + Y ∈ L
p
X + Y 
p
 X
p
+ Y 
p
.
0  p < 1
C
r
X, Y ∈ L
r
, r > 0
E|X + Y |
r

 c
r
(E|X|
r
+ E|Y |
r
),
c
r
= max(1, 2
r−1
)
ϕ : R → R
( ) ϕ(X)
Eϕ(X)  ϕ(EX).
X ∈ L
t
0 < s < t
X
s
 X
t
.
A B
P(AB) = P(A)P(B).
{A
i
, i ∈ I}
{A
i

, i ∈ I}
A
i
1
, A
i
2
, . . . , A
i
n
P(A
i
1
A
i
2
. . . A
i
n
) = P(A
i
1
)P(A
i
2
) P(A
i
n
).
(Ω, F, P)

{C
i
: i ∈ I, C
i
⊂ F} ( )
A
i
∈ C
i
{A
i
, i ∈ I}
{X
i
, i ∈ I} ( )
σ {σ(X
i
), i ∈ I}
A, B ∈ F I
A
, I
B
A, B
σ(I
A
) = {∅, Ω, A, A}; σ(I
B
) = {∅, Ω, B, B}.
( )
( )

{X
i
, i ∈ I} f
i
: R → R(i ∈ I)
{f
i
(X
i
), i ∈ I}
{X
n
, n  1} n  1
σ(X
k
, 1  k  n) σ(X
k
, k  n + 1)
X
1
, X
2
, , X
n
F
X
1
,X
2
, ,X

n
(x
1
, x
2
, , x
n
) = P(X
1
< x
1
, X
2
< x
2
, , X
n
< x
n
)
(x
i
∈ R, i = 1, n).
X
1
, X
2
, , X
n
F

X
1
,X
2
, ,X
n
(x
1
, x
2
, , x
n
) = F
X
1
(x
1
)F
X
2
(x
2
) F
X
n
(x
n
).
X Y
E(XY ) = EXEY.

X
1
, X
2
, . . . , X
n
E(X
1
X
2
. . . X
n
) = EX
1
EX
2
. . . EX
n
.
X Y D(X ±Y ) = DX+DY
X
1
, X
2
, , X
n
D(X
1
+ · · · + X
n

) = DX
1
+ · · · + DX
n
.
{X
n
, n  1}
EX
2
n
< ∞ n  1 E(X
i
X
j
) = 0 i = j
{X
n
, n  1}
EX
2
n
< ∞ n  1 {X
n
− EX
n
, n  1}
{X
n
, n  1}

{S
n
, n  1}
E(S
n
− S
m
)
2
=
n

i=m+1
EX
2
i
m < n.
∞

n=1
EX
2
n
< ∞ S
ES
2
< ∞ E(S
n
− S)
2

→ 0 n → ∞

∞
i=1
EX
2
< ∞
S E(S
n
− S)
2
→ 0 S
n
L
2
−→ S S
n
P
−→ S.
n
k
→ ∞ S
n
k
h. c. c
−−−→ S k → ∞
n
k
(k  1)
n

k
{X
n
, n  1} {b
n
, n  1}
b
n
 ∞
∞

n=1
b
n
EX
2
n
< ∞ k  1 n
k
b
n
 k {S
n
k
, k  1}
{X
n
, n  1}
E


max
hn
|S
h
|

2


log(4n)
log 2

2
n

i=1
EX
2
i
, n  1.
max
in
S
2
i
 n
n

i=1
X

2
i
.
E

max
in
S
2
i

 n
n

i=1
EX
2
i
, n  1.
n > 64 n >

log(4n)
log 2

2
{X
n
, n  1}
∞


n=1
log
2
n EX
2
n
< ∞
∞

n=1
X
n
L
2
b
n
 ∞
∞

n=1
(
log n
b
n
)
2
EX
2
n
< ∞

1
b
n
n

k=1
X
k
h. c. c
−−−→ 0.
{X, X
n
, n  1}
(Ω, F, P)
• {X
n
, n  1} X n → ∞
N ∈ F P(N) = 0 X
n
(ω) → X(ω) n → ∞
ω ∈ Ω\N
X
n
→ X X
n
h. c. c.
−−−→ X n → ∞
• {X
n
, n  1} X n → ∞ ε > 0

∞

n=1
P(|X
n
− X| > ε) < ∞.
X
n
c
−→ X n → ∞
• {X
n
, n  1} X n → ∞
ε > 0
lim
n→∞
P(|X
n
− X| > ε) = 0.
X
n
P
−→ X n → ∞
• {X
n
, n  1} p > 0 X n → ∞
X, X
n
(n  1) p lim
n→∞

E|X
n
− X|
p
= 0
X
n
L
p
−→ X n → ∞
• {X
n
, n  1} ( ) X n → ∞
lim
n→∞
F
n
(x) = F (x) x ∈ C(F ).
F
n
(x) F(x)
X
n
X C(F ) F (x)
X
n
D
−→ X
p L
p

X
n
h. c. c
−−−→ X ε > 0
lim
n→∞
P(sup
mn
|X
m
− X| > ε) = 0.
X
n
c
−→ X X
n
h. c. c
−−−→ X
∞

n=1
E|X
n
− X|
p
< ∞ p > 0 X
n
h. c. c
−−−→ X.
{X

n
, n  1} X
n
h. c. c
−−−→ C
X
n
c
−→ C
X
n
h. c. c
−−−→ X X
n
L
r
−→ X p > 0 X
n
P
−→
X.
X
n
P
−→ X X
n
D
−→ X.
X
n

D
−→ X P(X = C) = 1 X
n
P
−→ X.
{X
n
, n  1}
• P( lim
m,n→∞
|X
m
− X
n
| = 0) = 1
• lim
m,n→∞
P(|X
m
− X
n
| > ε) = 0 ε > 0
• p > 0 lim
m,n→∞
E|X
m
− X
n
|
p

= 0
{X
n
, n 
1} {X
n
, n  1}
{X
n
, n  1}
lim
n→∞
P( sup
k,ln
|X
k
− X
l
| > ε) = 0 ε > 0
lim
n→∞
P(sup
kn
|X
k
− X
n
| > ε) = 0 ε > 0.
{X
n

, n  1}
{X
n
k
, k  1} ⊂ {X
n
, n  1} {X
n
k
, k  1}
{X
n
, n 
1} {X
n
, n  1}
{X
n
, n  1}
{X
n
k
, k  1} ⊂ {X
n
, n  1} {X
n
k
, k  1}
p  1
{X

n
, n  1} p {X
n
, n  1}
p L
p
X
1
, X
2
, . . . , X
n
EX
i
=
0 DX
i
= σ
2
i
i = 1, 2, . . . , n S
k
= X
1
+ + X
k
1  k  n
ε > 0
P(max
1kn

|S
k
|  ε) 
1
ε
2
n

i=1
σ
2
i
.
P(max
1kn
|X
k
|  c) = 1
P( max
1kn
|S
k
|  ε)  1 −
(ε + c)
2

n
i=1
σ
2

i
.
{X
n
, n  1}
2
E

sup
mn



S
m
− S
n
|
2

 2
∞

i=n+1
EX
2
i
.
{X
n

, n  1}
EX
i
= a
i
(i = 1, 2, . . . )
• {X
n
, n  1}
X
1
+ · · · + X
n
n
−
a
1
+ · · · + a
n
n
P
−→ 0 n → ∞.
• {X
n
, n  1}
{b
n
, n  1} 0 < b
n
↑ ∞

X
1
+ · · · + X
n
b
n
−
a
1
+ · · · + a
n
b
n
P
−→ 0 n → ∞.
{X
n
, n  1} (
)
{X
n
, n  1}
1
n
2
n

i=1
DX
i

→ 0 n → ∞.
{X
n
, n  1}
{X
n
, n  1}
0 < b
n
↑ ∞
∞

n=1
DX
n
b
2
n
< ∞
1
b
n
n

k=1
(X
k
− EX
k
) → 0 h. c. c.

{X
n
, n  1}
{X
n
, n  1}
E|X
1
| < ∞
1
n
n

i=1
X
i
→ EX
1
h. c. c.
{X
n
, n  1}
1
n
n

i=1
X
i
C E|X

1
| < ∞
C = EX
1
{X
n
, n  1}
E|X
n
| < ∞
EX
n
= a ( ) n ∈ N {X
n
, n  1}
X
1
+ X
2
+ · · · + X
n
n
P
−→ a n → ∞.
f
n
( ) n → ∞
(Ω, F, P)
E G σ F B(E)
σ E

X : Ω −→ E G
X G/B(E) B ∈ B(E)
X
−1
(B) ∈ G
F
X : Ω −→ E
|X(Ω)| |X(Ω)|
X |X(Ω)|
X(Ω)
{X
n
, n  1}
X : Ω → E n → ∞ X
n
(ω) → X(ω) n → ∞
ω ∈ Ω
X
n
→ X n → ∞
E
1
, E
2
T : E
1
→
E
2
B(E

1
)/B(E
2
) X : Ω → E
1
G T ◦X : Ω → E
2
G
B
2
∈ B(E
2
) T
−1
(B
2
) = B
1
∈ B(E
1
)
(T ◦X)
−1
(B
2
) = X
−1
(T
−1
(B

2
)) = X
−1
(B
1
) ∈ G.
T (X) : Ω → E
2
G
X : Ω → E G
X : Ω → R G
X = .◦X : Ω
X
−→ E
.
−→ R X : Ω → E
G . : E → R B(E)/B(R)
X : Ω → E G
f ∈ E
∗
f(X) G
X G
f ∈ E
∗
f f B(E)/B(R)
f(X) G
f ∈ E
∗
f(X) G
X G/B(E) B(C) σ

C
{x ∈ X : (f
1
(x), f
2
(x), . . . , f
n
(x)) ∈ B, n = 1, 2, . . . , f
i
∈ X
∗
, B ∈ B(R
n
)}.
B(C) = B(E). {x
n
, n  1}
E X {f
n
} ⊂ E
∗
f
n
 = 1 f
n
(x
n
) = x
n
, n = 1, 2, . . .

B(C) ⊂ B(E). B(E) ⊂ B(C). C
1
= {x ∈
E : x  r} C
2
= ∩
∞
n=1
{x ∈ X : f
n
(x)  r} r > 0 C
2
∈ B(C)
C
1
⊂ C
2
. C
1
⊂ C
2
x ∈ C
1
x > r.
{x
n
, n  1} E x
k
x − x
k

 <
1
2
(x − r) .
x
k
  x − x
k
− x > x −
1
2
(x − r) =
1
2
(x + r)
|f
k
(x) − x
k
| = |f
k
(x) − f
k
(x
k
)|
= |f
k
(x − x
k

)|
 x − x
k
 <
1
2
(x − r) .