Báo cáo nghiên cứu khoa học: "Mở rộng một số định lí giới hạn cho các biến ngẫu nhiên phụ thuộc âm và phụ thuộc âm tuyến tính" potx

1
1. X
1
, , X
n
P (
n

i=1
[X
i
 x
i
]) 
n

i=1
P (X
i
 x
i
), ∀x
1
, , x
n
∈ R (1)
P (
n

i=1
[X

i
> x
i
]) 
n

i=1
P (X
i
> x
i
), ∀x
1
, , x
n
∈ R. (2)
X
1
, , X
n
,
X
1
, , X
n
,
i = j X
i
, X
j

(Ω, F, P ) Ω = {1, 2, 3, 4} F = {∀A : A ⊂ Ω} P (A) =
|A|
4
A = {1, 2} B = {2, 3, 4} I
A
I
B
1
2. X
1
, , X
n
,
A, B
(λ
k
, k ∈ A), (λ
l
, l ∈ B)

k∈A
λ
k
X
k
,

l∈B
λ
l

X
l
2
1 X
1
, , X
n
f
1
, , f
n
f(X
1
), , f(X
n
)
2 X
1
, X
2
EX
1
X
2
 EX
1
EX
2
cov(X
1

, X
2
)  0.
1 X
1
, , X
n
D(X
1
+ + X
n
)  DX
1
+ + DX
n
X
1
, , X
n
i = j cov(X
i
, X
j
)  0.
D(
n

k=1
X
k

) =
n

k=1
D(X
k
) +

i<j
cov(X
i
, X
j
) 
n

k=1
D(X
k
).
1 {X
n
, n ≥ 1}
1
n
2
n

i=1
DX

i
→ 0 khi n → ∞ {X
n
}
1
n

n

i=1
X
i
−
n

i=1
EX
i

P
−→ 0 khi n → ∞.
ε > 0
P

|
1
n
n

i=1

X
i
−
1
n
n

i=1
EX
i
| ≥ ε


D(
1
n
n

i=1
X
i
)
ε
2
=
D(
n

i=1
X

i
)
n
2
ε
2

n

i=1
DX
i
n
2
ε
2
.
1
n
2
n

i=1
DX
i
→ 0 khi n → ∞ lim
n→∞
P

|

1
n
n

i=1
X
i
−
1
n
n

i=1
EX
i
| ≥ ε

= 0.
1
n

n

i=1
X
i
−
n

i=1

EX
i

P
−→ 0 khi n → ∞.
2 {X
n
, n ≥ 1}
C > 0 DX
n
 C n ≥ 1 {X
n
}
3 {X
n
, n ≥ 1}
EX
1
= a DX
1
= σ
2
n

i=1
X
i
P
−→ a khi n → ∞.
3 X

1
, , X
n
|Φ(r
1
, , r
m
) −
m

j=1
Φ
j
(r
j
)| 
m

k,l=1
|r
k
r
l
cov(X
k
, X
l
)|
Φ(r
1

, , r
m
) = E(exp[(i
m

j=1
r
j
X
j
]), Φ
j
(r
j
) = E(exp[ir
j
X
j
]).
4 (F
n
) (ϕ
n
)
F
n
w
−→ F ϕ
n
→ ϕ ϕ F

5 X
n
w
−→ C = const X
n
P
−→ C
|a
1
a
2
a
n
− b
1
b
2
b
n
| 
n

k=1
|a
k
− b
k
|, |a
k
|  1, |b

k
|  1; (3)
|e
itx
− 1 − itx|  2h
1
(t)g
1
(x); (4)
|e
itx
− 1 − itx +
t
2
x
2
2
|  h
2
(t)g
2
(x), (5)
h
1
(t) = max(|t|, t
2
) g
1
(x) = min(|x|, x
2

), h
2
(t) = max(t
2
, |t|
3
), g
2
(x) = min(x
2
, |x|
3
)
x, t ∈ R
2 {X
ni
, 1  i  n, n ≥ 1}
EX
ni
= 0,

i<j
cov(X
ni
, X
nj
)
n→∞
−−−→ 0 M
n

=
n

i=1
E min(|X
ni
|, |X
ni
|
r
)
n→∞
−−−→ 0 r ∈ (1; 2) S
n
=
n

i=1
X
ni
P
−→ 0
|
n

k=1
Ee
itX
nk
− 1| = |

n

k=1
Ee
itX
nk
−
n

k=1
1| 
n

k=1
|Ee
itX
nk
− 1| =
=
n

k=1
|E(e
itX
nk
− 1 − itX
nk
)| 
n


k=1
2h
1
(t)Eg
1
(X
nk
) = 2h
1
(t)
n

k=1
E min(|X
nk
|, X
2
nk
). (6)
1 < r  2 min(|x|, x
2
)  min(|x|, |x|
r
)
0 
n

k=1
E min(|X
nk

|, X
2
nk
) 
n

k=1
min(|X
nk
|, |X
nk
|
r
)
n→∞
−−−→ 0.
Suy ra
n

k=1
E min(|X
nk
|, X
2
nk
)
n→∞
−−−→ 0. (7)
n


k=1
Ee
itX
nk
n→∞
−−−→ 1. (8)
|E exp(i
n

k=1
X
nk
) −
n

k=1
E exp(iX
nk
)| 

kl
cov(X
nk
, X
nl
)
n→∞
−−−→ 0. (9)
E exp(i
n


k=1
X
nk
) → 1, hay ϕ
s
n
→ 1. (10)
1 = e
it0
X = 0 S
n
w
−→ 0. (11)
S
n
P
−→ 0.
4 {X
nk
, k = 1, , n, n ≥ 1}
EX
nk
= 0

i<j
cov(X
ni
, X
nj

)
n→∞
−−−→ 0
n

k=1
E|X
nk
|  C < ∞ L
1
n
(ε) =
n

k=1
E(|X
nk
|I(|X
nk
| > ε))
n→∞
−−−→ 0 ε > 0 S
n
P
−→ 0.
0 < ε < 1
M
n

n


k=1
E(|X
nk
|I(|X
nk
| > ε)) +
n

k=1
E(|X
nk
|
r
I(|X
nk
|  ε))  L
1
n
(ε) + ε
r−1
C.
M
n
n→∞
−−−→ 0
5 (X
k
) EX
1

= a EX
2
1
=
C < ∞ E(|X
1
− a|I(|X
1
− a| > εn))
n→∞
−−−→ 0 ε > 0
X
1
+ +X
n
n
P
−→ a.
X
nk
=
X
k
−a
n
, k  n {X
nk
, 1  k  n, n ≥ 1}
EX
nk

= 0,

i<j
cov(X
ni
, X
nj
) <
C
n
n→∞
−−−→ 0
n

k=1
E|X
k
n
| =
n

k=1
E|
X
k
− a
n
| = E|X
1
− a| < ∞.

L
1
n
(ε) =
n

k=1
E(|X
nk
|I(|X
nk
| > ε)) =
n

k=1
E(|
X
k
− a
n
|I(|
X
k
− a
n
| > ε)) =
= E(|X
1
− a|I(|X
1

− a| > εn))
n→∞
−−−→ 0.
6 {X
nk
, k = 1, , n, n ≥ 1}
EX
nk
= 0,

i<j
cov(X
ni
, X
nj
) → 0
n

k=1
E|X
nk
|
r
→ 0 r ∈ [1; 2] S
n
P
−→ 0.
3 {X
nk
, 1  k  n, n ≥ 1}










EX
nk
= 0, k = 1, , n
n

k=1
DX
nk
= 1,

i<j
cov(X
ni
, X
nj
)
n→∞
−−−→ 0 i, j = 1, , n.
(12)
S
n

=
n

k=1
X
nk
, σ
2
nk
= DX
nk
M
2
n
=
n

k=1
E min(|X
nk
|
2
, |X
nk
|
s
)
n→∞
−−−→ 0, (13)
s ∈ [2; 3] (S

n
)
N(0, 1).
ϕ
S
n
(t) → e
−
t
2
2
, ∀t ∈ R
|ϕ
S
n
(t) −
n

k=1
ϕ
X
nk
(t)| 

i<j
cov(X
ni
, X
nj
)

n→∞
−−−→ 0
ϕ
S
n
(t)
n→∞
−−−→
n

k=1
ϕ
X
nk
(t)
n

k=1
ϕ
X
nk
(t)
n→∞
−−−→ e
−
t
2
2
, t ∈ R
|

n

k=1
ϕ
X
nk
(t) − e
−
t
2
2
| = |
n

k=1
ϕ
X
nk
(t) −
n

k=1
e
−
t
2
σ
2
nk
2

| 
n

k=1
|ϕ
X
nk
(t) − e
−
t
2
σ
2
nk
2
| 

n

k=1
|E(e
itX
nk
− 1 − itX
nk
+
t
2
X
2

nk
2
) +
n

k=1
|e
−
t
2
σ
2
nk
2
− 1 +
t
2
σ
2
nk
2
| 
 h
2
(t)
n

k=1
min(X
2

nk
, |X
nk
|
3
) +
t
4
8
n

k=1
σ
2
nk

 h
2
(t)
n

k=1
min(X
2
nk
, |X
nk
|
s
) +

t
4
8
max
kn
σ
2
nk
. (14)
0 < ε < 1
σ
2
nk
= E(X
2
nk
I(|X
nk
|  ε)) + E(X
2
nk
I(|X
nk
| > ε)
 ε
2
+ E(X
2
nk
I(|X

nk
| > 1)) + E(X
2
nk
I(ε < |X
nk
|  1))
 ε
2
+ E(X
2
nk
I(|X
nk
| > 1)) +
1
ε
s−2
E(X
s
nk
I(1 ≥ |X
nk
| > ε))
 ε
2
+
1
ε
s−2

E min(X
2
nk
, |X
nk
|
s
). (15)
7 {X
nk
, 1  k  n, n ≥ 1}
EX
ni
= 0
S‘
2
n
= D(
n

i=1
X
ni
)
n→∞
−−−→ ∞,
1
S‘
2
n


i<j
cov(X
ni
, X
nj
)
n→∞
−−−→ 0
n

i=1
E(X
2
ni
I(|X
ni
| ≥ εS‘
2
n
)) = 0(S‘
2
n
) S‘
−1
n
=
n

i=1

X
ni
N(0, 1).
S
2
n
=
n

i=1
DX
nk
{X
nk
, 1  k  n, n ≥ 1}
S
2
n
= D(
n

k=1
X
nk
) =
n

k=1
D(X
nk

) +

i<j
cov(X
ni
, X
nj
) 
n

k=1
D(X
nk
) = S
2
n
.
1
S
2
n

i<j
cov(X
ni
, X
nj
)
n→∞
−−−→ 0; (16)

n

i=1
E(X
2
ni
I(|X
ni
| > ε

S
2
n
)) = 0(S
2
n
). (17)
lim
n→∞
S‘
2
n
S
2
n
= lim
n→∞
(
S
2

n
S
2
n
+
1
S
2
n

i<j
cov(X
ni
, X
nj
)) = 1 + lim
n→∞
1
S
2
n

i<j
cov(X
ni
, X
nj
) = 1.

S

2
n
−1
n

i=1
X
ni
d
−→ X X N(0, 1)
Z
nk
=
X
nk
√
S
2
n
{Z
nk
, 1  k  n, n ≥ 1}
ε > 0
n

k=1
EX
2
nk
=

n

k=1
E(Z
2
nk
I(|Z
nk
|  ε)) +
n

k=1
E(Z
2
nk
I(|Z
nk
| > ε))
 ε
2
+
n

k=1
1
S
2
n
EX
2

nk
I(|X
nk
| > ε

S
2
n
). (18)
n

k=1
EZ
2
nk
n→∞
−−−→ 0 {Z
nk
, 1  k  n, n ≥ 1}
n

k=1
Z
nk
d
−→ X X N(0.1)