Một số tính chất cơ bản của mảng các biến ngẫu nhiên

(X
mn
; m  1, n  1)
max(m, n) → ∞ min(m, n) → ∞
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, 3, . . . ) 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
)
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).
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
1
, τ
1
), 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) =

−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 → ∞
(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
.
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 X 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
(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(X
1
+ · · · + X
n
) = DX
1
+ · · · + DX
n
.
{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
X

n
h.c.c
−−→ X X
n
L
r
−→ X X
n
P
−→ X.
(X
n
, n  1) X
n
h.c.c
−−→ C
X
n
c
−→ C
(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)
{x
mn
, m  1, n  1} m∨n →
∞ ∀ε > 0 n
0
∈ N m, n ∈ N m ∨ n  n
0
|x
mn
− x| < ε,
m ∨ n = max{m, n}

lim
m∨n→∞
x
mn
= x.
{x
mn
; m  1, n  1}
m ∧ n → ∞ ε > 0 n
0
∈ N
m, n ∈ N m ∧ n  n
0
|x
mn
− x| < ε,
m ∧ n = min{m, n}
lim
m∧n→∞
x
mn
= x.
{x
mn
; m  1, n  1}
S
mn
=
m


k=1
n

l=1
x
kl
.

∞
m=1

∞
n=1
x
mn
S
{S
mn
, m  1, n  1} S m ∧ n → ∞.
S =
∞

m=1
∞

n=1
x
mn
.
{x

mn
; m  1, n  1} m ∨ n → ∞
{x
mn
; m  1, n  1} m ∧ n → ∞
lim
m∧n→∞
x
mn
= x
ε > 0 n
0
m∧n > n
0
|x
mn
−x| < ε
{x
mn
; m  1, n  1} m ∨ n → ∞
ε > 0 n
0
m ∨ n > n
0
|x
mn
− x| < ε (m, n) ∈ N
2
m ∨ n  m ∧ n
m ∧ n > n

0
m ∨ n > n
0
|x
mn
− x| < ε
m ∧ n > n
0
{x
mn
, m  1, n  1.}
∞

m=1
∞

n=1
x
mn
lim
k→∞
∞

m∨nk
x
mn
= 0.
∞

m=1

∞

n=1
x
mn
lim
m∧n→∞
S
mn
= x.
ε > 0 n
0
∈ N m ∧ n  n
0
|S
mn
− x| < ε
m = n > n
0
m ∧ n  n
0
ε > 0 n
0
m  n
0
|S
mm
− x| < ε
|
m


k=1
m

l=1
x
kl
− x| < ε.
|
m

k=1
m

l=1
x
kl
−
∞

m=1
∞

n=1
x
mn
| < ε.
|
∞


k∨lm
x
kl
| < ε.
|r
m
| < ε. r
m
= |

∞
k∨lm
x
kl
|
ε > 0 lim
m→∞
r
m
= 0.
lim
m→∞
∞

k∨lm
x
kl
= 0.
lim
k→∞

∞

m∨nk
x
mn
= 0.
{x
mn
, m  1, n  1.}
x
mn
→ x m ∨ n → ∞
m  1 x
mn
→ x n → ∞,
n  1 x
mn
→ x m → ∞,
x
mn
→ x m ∧ n → ∞
x
mn
→ x m ∨ n → ∞ ε > 0 n
0
∈ N, m ∨ n > n
0
|x
mn
− x| < ε

m  1, n  n
0
m ∨ n  n
0
|x
mn
− x| < ε
m  1 x
mn
→ x n → ∞
ε > 0 n
0
∈ N
m ∧ n  n
0
|x
mn
− x| < ε
n
1
n > n
1
|x
mn
− x| < ε
n
2
n > n
2
|x

mn
− x| < ε
n
0
n
n
0
n > n
n
0
|x
mn
− x| < ε
l = max{n
1
, n
2
, , n
n
0
}
1  m  n
0
l ∈ N n > l |x
mn
− x| < ε
m
1
m > m
1

|x
mn
− x| < ε
m
2
m > m
2
|x
mn
− x| < ε
n
0
m
n
0
m > m
n
0
|x
mn
− x| < ε
k = max{m
1
, m
2
, , m
n
0
}
1  n  n

0
k ∈ N m > k |x
mn
− x| < ε
N = max{n
0
, l, k}
ε > 0, m, n m∨n > N |x
mn
−x| < ε
m > N; n < n
0
n > N; m < n
0
m > N; n > N m > n
0
, n > n
0
m ∧ n  n
0
{x
mn
, m  1, n  1.}
m  1,
1
n

n
j=1
x

mj
→ x n → ∞
n  1,
1
m

m
i=1
x
in
→ x m → ∞
1
mn

m
i=1

m
j=1
x
ij
→ x m ∧ n → ∞
1
mn
m

i=1
m

j=1

x
ij
→ x,
m ∨ n → ∞
n
0
|
1
mn
m

i=1
m

j=1
x
ij
− x| < ε m  n
0
; n  n
0
n
0
r
l
∈ N r > r
l
|
1
r

r

j=1
x
lj
− x| < ε.
n
0
k
s
∈ N k > k
s
|
1
k
k

i=1
x
is
− x| < ε.
N = max{n
0
, k
1
, k
2
, , k
n
0

, r
1
, r
2
, , r
n
0
}
m ∨ n > N
|
1
mn
m

i=1
m

j=1
x
ij
− x| < ε.
m > N, n  n
0
|
1
mn
m

i=1
m


j=1
x
ij
− x| = |
1
n
n

j=1
(
1
m
m

i=1
x
ij
− x)|

1
n
n

j=1
|
1
m
m


i=1
x
ij
− x| <
1
n
n

j=1
ε = ε.
m > N, n > n
0
m > n
0
, n > N
m  n
0
, n > N
|
1
mn
m

i=1
m

j=1
x
ij
− x| = |

1
m
m

i=1
(
1
n
n

j=1
x
ij
− x)|

1
m
m

i=1
|
1
n
n

j=1
x
ij
− x| <
1

m
m

i=1
ε = ε.