Luật mạnh số lớn cho dãy biến ngẫu nhiên độc lập đôi một cùng phân phối với các điều kiện moment tổng quát

Ω F σ Ω
(Ω, F)
(Ω, F) P : F → R
F
P(A)  0 ∀A ∈ F
P(Ω) = 1
A
n
∈ F, (n = 1, 2, ), A
i

A
j
= A
i
A
j
= ∅, (i = j)
P

∞

n=1
A
n


=
∞

n=1
P(A
n
)
(Ω, F, P)
Ω
σ F σ
A ∈ F
Ω ∈ F
∅ ∈ F
A = Ω\A
A ∩ B = AB = ∅ A, B
(Ω, F, P)
(Ω, F, P)
A, B, C, . . .
P(∅) = 0
AB = ∅ P(A ∪ B) = P(A) + P(B)
P(
A) = 1 − P(A)
A ⊂ B P(B\A) = P(B) − P(A) P(A)  P(B).
P(A ∪ B) = P(A) + P(B) − P(AB).
P(
n

k=1
A

k
) =
n

k=1
P(A
k
) −

1k<in
P(A
k
A
i
)
+

1k<l<mn
P(A
k
A
l
A
m
) − · · · + (−1)
n−1
P(A
1
A
2

. . . A
n
).
P(

∞
n=1
A
n
) 

∞
n=1
P(A
n
)
(A
n
, n  1) A
1
⊂ A
2
⊂ · · · ⊂ A
n
⊂ . . .
lim
n→∞
P(A
n
) = P(

∞

n=1
A
n
).
(A
n
, n  1) A
1
⊃ A
2
⊃ · · · ⊃ A
n
⊃
. . .
lim
n→∞
P(A
n
) = P(
∞

n=1
A
n
).
(Ω, F, P) A B ∈ F P(A) > 0
P(B/A) =
P(AB)

P(A)
B A
P(B/A)  0
B ⊃ A P(B/A) = 1 P(Ω/A) = 1
(B
n
, n  1)
P(
∞

n=1
B
n
/A) =
∞

n=1
P(B
n
/A).
1 − 3 A P(A) > 0
P
A
: F → R
P
A
(B) = P(B/A) (∀B ∈ F)
F P
A
A

1
, A
2
, , A
n
(n  2) n
P(A
1
A
2
A
n−1
) > 0
P(A
1
A
2
A
n
) = P(A
1
)P(A
2
/A
1
) P(A
n
/A
1
A

n−1
).
(Ω, F, P)
P(AB) = P(A)P(B).
A B P(A) > 0 P(B) > 0 A
B P(A/B) = P(A) P(B/A) = P(B)
A B
A B
A B
A 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
).
A
n
, n  1
∞

n=1
P(A
n
) < ∞
P(lim sup A
n
) = 0
∞


n=1
P(A
n
) = ∞
(A
n
, n  1) P(lim sup A
n
) = 1
lim sup A
n
=
∞

n=1
∞

k=n
A
k
.
0 − 1 (A
n
, n  1)
P(lim sup A
n
) 0 1

∞
n=1

P(A
n
)
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
1
, τ
1
), (Ω
2
, τ
2
), X : Ω
1
→
Ω
2
X B(Ω
1
)/B(Ω
2
)
(Ω, F, P) G σ σ
F X : Ω → R G
G/B(R) B ∈ B(R) X
−1

(B) ∈ G
X
X F
G
X
σ(X) =

X
−1
(B) : B ∈ B(R)

σ σ F σ σ
X σ X X
G σ(X) ⊂ G
(i) (X < a) := (ω : X(ω) < a) ∈ F a ∈ R
(ii) (X  a) := (ω : X(ω)  a) ∈ F a ∈ R
(iii) (X > a) := (ω : X(ω) > a) ∈ F a ∈ R
(iv) (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 → ∞
G
(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.
F (+∞) = lim
x→+∞
F (x); F (−∞) = lim
x→−∞
F (x).
F (+∞) = 1; F (−∞) = 0.
lim
x↑a
F (x) = F (a) lim
x↓a
F (x) = P(X  a). F(x)
F (x) a P(a) = 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.
EX =







i
x
i
p
i
X x
1
, x
2
, . . . P(X = x
i
) = p
i
.

+∞
−∞
xp(x)dx X p(x).
f : R → R Y = f(X)
EY =















i
f(x
i
)p
i
X x
1
, x
2
, . . .
P(X = x
i
) = p
i

+∞
−∞
f(x)p(x)dx X p(x).
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
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
.
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
.
X Y X = Y
X
p

L
p
, p  1
L
p
, p  1 L
p
(X
n
, n  1)
X n → ∞
• P( lim
n→∞
|X
n
− X| = 0) = 1
X
n
h.c.c
−−→ X.
• ε > 0
lim
n→∞
P(|X
n
− X| > ε) = 0.
X
n
P
−→ X.

• ε > 0
∞

n=1
P(|X
n
− X| > ε) < ∞.
X
n
c
−→ X.
• p, (p > 0) lim
n→∞
E|X
n
− X|
p
= 0.
X
n
L
p
−→ 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
X
n
h.c.c
−−→ X X
n
L
r
−→ X 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)
(X
n
, n  1) p p  1
(X
n
, n  1) p L
p
(p  1)
(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
(X
i
)
i∈I
f
i
: R → R(i ∈ I)

f
i
(X
i
)

i∈I

(X
i
)
i∈I
I
1
⊂ I, I
2
⊂ I, I
1
∩I
2
= ∅
σ

(X
i
)
i∈I
1

σ

(X
i
)
i∈I
2

σ


(X
i
)
i∈I
1

σ

(X
i
)
i∈I
2

σ

i∈I
1
σ(X
i
)

i∈I
2
σ(X
i
)
(X
n

, n  1) n  1
σ(X
k
, 1  k  n) σ(X
k
, k  n + 1)
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(
n

i=1
X
i
) =
n

i=1
DX
i
(X
n
, n  1) X
n
h.c.c
−−→ C X
n
c
−→ C
X
1
, X
2
, X

n

i
= a
i
(i = 1, 2 ) .
(X
n
, n  1)
X
1
+ X
2
+ X
n
n
−
a
1
+ a
2
+ a
n
n
→ 0 n → ∞.
(X
n
, n  1)
(b
n

) , 0 < b
n
 ∞
X
1
+ X
2
+ X
n
b
n
−
a
1
+ a
2
+ a
n
b
n
→ 0 n → ∞.
{a
n
, n  1}
a
n
n

{X, X
n

, n  1}
{a
n
, n  1}
a
n
n

(a
n
, n  1) a
n
 ∞
∞

n=1
P (X > a
n
) < ∞
∞

n=1
P (X > 2a
n
) < ∞
∞

n=1
P (X > a
n

) < ∞
∞

n=1
P (X > αa
n
) < ∞ α > 0
0 <
a
n
n

a
n
n

a
n+1
n + 1
a
1

a
n
n
.
a
n

n

n + 1
a
n+1
a
1
n  a
n
a
n
< a
n+1
a
1
n  a
n
.
{a
n
, n  1} a
n
 ∞
a
2n
< a
2n+1
∞

n=1
P (X > a
n

) = P (X > a
1
) +
∞

n=1
P (X > a
2n
) +
∞

n=1
P (X > a
2n+1
)
 P (X > a
1
) + 2
∞

n=1
P (X > a
2n
) .
a
n
n

a
2n

2n
⇔ 2a
n
 a
2n
.
∞

n=1
P (X > a
n
) 
∞

n=1
P (X > 2a
n
) 
∞

n=1
P (X > a
2n
) .
a
n
2a
n
∞


n=1
P (X > 2a
n
) < ∞ ⇔
∞

n=1
P (X > 4a
n
) < ∞.
k − 2
∞

n=1
P (X > a
n
) < ∞ ⇔
∞

n=1
P

X > 2
k
a
n

< ∞.
a
n

2
−k
a
n
∞

n=1
P

X > 2
−k
a
n

< ∞ ⇔
∞

n=1
P (X > a
n
) < ∞.
α > 1 α  2
k
∞

n=1
P (X > a
n
) 
∞


n=1
P (X > αa
n
) 
∞

n=1
P

X > 2
k
a
n

.
α  2
k
a
n
< αa
n
 2
k
a
n
0 < α < 1 α > 2
−k
∞


n=1
P (X > a
n
) 
∞

n=1
P (X > αa
n
) 
∞

n=1
P

X > 2
−k
a
n

.
(X
n
, n  1) ε > 0
A
n
= {|X
n
| > ε}, X
n

→ 0
P(lim sup A
n
) = 0.
ω ∈ Ω X
n
(ω) → 0) ∀ε > 0, ∃n
0
 1
∀n > n
0
|X
n
|(ω)  ε ⇔ ω ∈ (|X
n
|  ε)
⇔ ω ∈
∞

n=n
0
(|X
n
|  ε) =
∞

n=n
0
A
n

(ω : X
n
(ω) → 0) =
∞

n=1
∞

k=n
A
k
= lim inf A
k
.
X
n
→ 0 ⇔ P(
∞

n=1
∞

k=n
A
k
) = 1
⇔ P(
∞

n=1

∞

k=n
(A
k
)) = 0
⇔ P(lim sup A
n
) = 0.
{a
n
, n  1}
a
n
n
 {X, X
n
, n  1}
X
n
a
n
→ 0
∞

n=1
P (|X| > a
n
) < ∞.
X

n
a
n
→ 0
P (lim sup {|X
n
|/a
n
> ε}) = 0.
P (lim sup {|X
n
|/a
n
> ε}) = 0 ⇔
∞

n=1
P (|X
n
| /a
n
> ε) < ∞
⇔
∞

n=1
P (|X| > a
n
ε) =
∞


n=1
P (|X
n
| > a
n
ε) < ∞
⇔
∞

n=1
P (|X| > a
n
) < ∞.