Sự hội tụ đối với các phần tử ngẫu nhiên liên kết âm nhận giá trị trên không gian hilbert

R
d
H
H
H
H
{e
j
, j ∈ B} ., . C
log
2
K K R
C X = ∅ X × X → X
K × X → X
(x + y) + z = x + (y + z)
x + y = y + x
∃θ ∈ X, x + θ = x
∃(−x) ∈ X, x + (−x) = θ
λ(x + y) = λx + λy
(λ + µ)x = λx + µx
(λµ)x = λ(µx)
1.x = x
∀x, y, z ∈ X, ∀λ, µ ∈ K
K K
X X d : X × X → R
d(x, y) ≥ 0, d(x, y) = 0 ⇔ x = y
d(x, y) = d(y, x)
d(x, z) ≤ d(x, y) + d(y, z)

∀x, y, z ∈ X
X = (X, d) X d
X {x
n
}
 > 0 n
0
d(x
n
, x
m
) <  n, m ≥ n
0
X X
X K X
x → x X R x, y ∈ X
λ ∈ K
x ≥ 0, x = 0 ⇔ x = 0
λx = |λ|.x
x + y ≤ x + y
x → x X d(x, y) = x − y
X
A X
A X
X X
X K
ϕ : X × X → K
(x, y) → ϕ(x, y) := (x|y)
ϕ X
(x|x) ≥ 0 ∀x ∈ X, (x|x) = 0 ⇔ x = 0

(x
1
+ x
2
|y) = (x
1
|y) + (x
2
|y)
(λx|y) = λ(x|y)
(x|y) = (y|x)
∀x, y, x
1
, x
2
∈ X, ∀λ ∈ K
X
X
X
(Ω, F, P) X : Ω → R
X a ∈ R
{ω ∈ Ω : X(ω) < a} ∈ F.
X F (x) = P (X < x) x ∈ R
X
X
x
1
≤ x
2
F (x

1
) ≤ F (x
2
)
x
0
∈ R, F (x
0
) = lim
x→x
−
0
F (x).
lim
x→−∞
F (x) = 0, lim
x→+∞
F (x) = 1
X
1
X
2
a
1
, a
2
∈ R
P ({X
1
< a

1
} ∩ {X
2
< a
2
}) = P ({X
1
< a
1
})P ({X
2
< a
2
}).
n(n ≥ 2) X
1
, X
2
, , X
n
a
1
, a
2
, , a
n
∈ R
P (
n
∩

k=1
{X
k
< a
k
}) =
n

k=1
P ({X
k
< a
k
}).
{X
n
, n ≥ 1}
{X
n
, n ≥ 1}
X EX
EX =

Ω
XdP .
L
1
X : Ω → R

Ω

|X|dP < ∞.
C EC = C
a, b ∈ R X, Y ∈ L
1
E(aX + bY ) = aEX + bEY
X, Y ∈ L
1
X ≤ Y EX ≤ EY
X ∈ L
1
|EX| ≤ E|X|
|X| ≤ Y Y ∈ L
1
X ∈ L
1
{X
n
, n ≥ 1} ⊂ L
1
X ∈ L
1
0 ≤ X
n
↑ X
EX
n
↑ EX
X Y X, Y ∈ L
1
E(XY ) = EX.EY

X F (x)
EX =
+∞

−∞
xdF (x).
g : R → R g(X)
EgX =
+∞

−∞
g(x)dF (x).
X Y X
Y cov(X, Y )
cov(X, Y ) = E(X − EX)(Y − EY ).
X, Y cov(X, Y ) = 0
{X
i
, 1 ≤ i ≤ n}
cov {f(X
i
, i ∈ A), g(X
j
, j ∈ B)} ≤ 0
A, B {1, , n}
f : R
|A|
→ R g : R
|B|
→ R

|A|
A
{X
i
, i ≥ 1}
n ≥ 1 {X
i
, 1 ≤ i ≤ n}
X = (X
1
, , X
n
) Y = (Y
1
, , Y
m
)
X Y (X
1
, , X
n
, Y
1
, , Y
m
)
{X
i
, i ≥ 1} {A
i

, i ≥ 1}
{1, 2, } f
i
: R
A
i
→ R i ≥ 1
{f
i
(X
j
, j ∈ A
i
), i ≥ 1}
{f
1
(X
1,1
, , X
1,m
1
), f
2
(X
2,1
, , X
2,m
2
), , f
n

(X
n,1
, , X
n,m
n
) }
{f
1
(X
1,1
, , X
1,m
1
), f
2
(X
2,1
, , X
2,m
2
), , f
n
(X
n,1
, , X
n,m
n
)}
A, B {1, 2, , n}
A = {1, 2, , k}, B = {k +1, , n}

f : R
|A|
→ R g : R
|B|
→ R
Y
r
= f
r
(X
r,1
, , X
r,m
r
), r = 1, 2, , n
f(f
1
(X
1,1
, , X
1,m
1
), f
2
(X
2,1
, , X
2,m
2
), , f

k
(X
k,1
, , X
k,m
k
)) = f

(X
1,1
, , X
k,m
k
)
f(f
k+1
(X
k+1,1
, , X
k+1,m
k+1
), , f
n
(X
n,1
, , X
n,m
n
)) = g


(X
k+1,1
, , X
n,m
n
)
f

, g

X
1,1
, , X
n,m
n
cov(f(Y
1
, , Y
k
), g(Y
k+1
, , Y
n
))
= cov(f

(X
1,1
, , X
k,m

k
), g

(X
k+1,1
, , X
n,m
n
)) ≤ 0
Y
1
, , Y
n
R
d
{X
1
, , X
n
} R
d
A, B
{1, , n} f
R
|A|d
g R
|B|d
cov {f(X
i
, i ∈ A), g(X

j
, j ∈ B)} ≤ 0
{X
i
, i ≥ 1}
R
d
n ≥ 1 {X
i
, 1 ≤
i ≤ n}
H
{e
j
, j ∈ B}
., . {X
n
, n ≥ 1}
H d ≥ 1
{(X
i
, e
1
 , , X
i
, e
d
), i ≥ 1}
R
d

{X
n
, n ≥ 1} H
k ≥ 0

X
i
, 2
k
≤ i < 2
k+1

a
n
i
, 1 ≤ i ≤ n, n ≥ 1 x
i
, i ≥ 1 i
lim
n→∞
a
n
i
= 0 n
n

i=1
|a
n
i

| ≤ C < ∞
lim
n→∞
x
n
= 0 lim
n→∞
n

i=1
a
n
i
x
i
= 0
lim
n→∞
n

i=1
a
n
i
= 1 lim
n→∞
x
n
= x lim
n→∞

n

i=1
a
n
i
x
i
= x
lim
n→∞
x
n
= 0  > 0 n

|x
n
| < C
−1
, ∀n ≥ n

.
n ≥ n

|
n

i=1
a
n

i
x
i
| ≤
n

−1

i=1
|a
n
i
x
i
| +
n

i=n

|a
n
i
x
i
|
≤
n

−1


i=1
|a
n
i
x
i
| + .
lim
n→∞
a
n
i
= 0 i = 1, 2, , n

− 1
lim
n→∞
n

i=1
a
n
i
x
i
= 0.
lim
n→∞
n


i=1
a
n
i
= 1 lim
n→∞
x
n
= x
n

i=1
a
n
i
x
i
= x
n

i=1
a
n
i
+
n

i=1
a
n

i
(x
i
− x)
lim
n→∞
n

i=1
a
n
i
x
i
= x.
0 < b
n
↑ ∞
∞

n=1
x
n
b
n
lim
n→∞
(
1
b

n
n

k=1
x
k
) = 0.
X 0 < p < ∞
 > 0
P (|X| > ) ≤
E|X|
p

p
.
X α > 0 EX
α
< ∞
EX
α
= α
∞

0
x
α−1
P (X > x)dx.
{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
.

H
{X
n
, n ≥ 1} 0
H n ≥ 1
E( max
1≤k≤n
||
k

i=1
X
i
||
2
) ≤ 2
n

i=1
E||X
i
||
2
.
E( max
1≤k≤n
||
k

i=1

X
i
||
2
) = E( max
1≤k≤n
∞

j=1
(
k

i=1
X
i
, e
j
)
2
)
≤ E(
∞

j=1
max
1≤k≤n
(
k

i=1

X
i
, e
j
)
2
)
=
∞

j=1
E(max{( max
1≤k≤n
k

i=1
X
(j)
i
)
2
; ( max
1≤k≤n
k

i=1
−X
(j)
i
)

2
})
≤
∞

j=1
E( max
1≤k≤n
k

i=1
X
(j)
i
)
2
+
∞

j=1
E( max
1≤k≤n
k

i=1
(−X
(j)
i
))
2

≤ 2
∞

j=1
n

i=1
E(X
(j)
i
)
2
= 2
n

i=1
E||X
i
||
2
X
(j)
i
= X
i
, e
j
.
{X
n

, n ≥ 1}
0 H {b
n
, n ≥ 1}
inf
n≥0
b
2
n+1
b
2
n
> 1 sup
n≥0
b
2
n+1
b
2
n
< ∞.
∞

n=1

j∈B
E| X
n
, e
j

 |
r
n
b
n
r
n
< ∞,
1 ≤ r
n
≤ 2 n ≥ 1
n

i=1
X
i
b
n
→ 0 n → ∞.
(2.2) lim
n→∞
b
n
= ∞
n ≥ 1 j ∈ B
Z
(j)
n
= | X
n

, e
j
 |I(| X
n
, e
j
 | ≤ b
n
)
E(Z
(j)
n
)
2
= 2
∞

0
xP (Z
(j)
n
> x)dx
= 2
b
n

0
xP (Z
(j)
n

> x)dx + 2
∞

b
n
xP (Z
(j)
n
> x)dx
= 2
b
n

0
xP (| X
n
, e
j
 |I(| X
n
, e
j
 | ≤ b
n
) > x)dx
= 2
b
n

0

xP (| X
n
, e
j
| > x)dx − 2
b
n

0
xP (| X
n
, e
j
| > b
n
)dx
b
n
≥ x ≥ 0
(|X|I(|X| ≤ b
n
) > x) ∪ (|X| > b
n
) = (|X| > x)).
E(X
n
, e
j
 I(| X
n

, e
j
 | ≤ b
n
))
2
b
2
n
+ P (| X
n
, e
j
 | > b
n
)
= 2b
−2
n
b
n

0
xP (| X
n
, e
j
 | > x)dx
= 2b
−2

n
b
n

0
x
2−r
n
x
r
n
−1
P (| X
n
, e
j
 | > x)dx
≤ 2b
−2
n
b
n

0
b
2−r
n
x
r
n

−1
P (| X
n
, e
j
 | > x)dx
= 2b
−r
n
n
b
n

0
x
r
n
−1
P (| X
n
, e
j
 | > x)dx
≤ 2
E| X
n
, e
j
 |
r

n
b
r
n
n
.
(2.3)
∞

n=1

j∈B
[
E(X
n
, e
j
 I(| X
n
, e
j
 | ≤ b
n
))
2
b
2
n
+ P (| X
n

, e
j
 | > b
n
)] < ∞.
n ≥ 1 j ∈ B
Y
(j)
n
= − b
n
I(X
n
, e
j
 < −b
n
) + X
n
, e
j
 I(|X
n
, e
j
 | ≤ b
n
)
+ b
n

I(X
n
, e
j
 > b
n
),
Y
n
=

j∈B
Y
(j)
n
e
j
.
j ∈ B

Y
(j)
n
− EY
(j)
n
, n ≥ 1

{Y
n

− EY
n
, n ≥ 1}
Y
n
∞

n=1
E||Y
n
− EY
n
||
2
b
2
n
=
∞

n=1
1
b
2
n

j∈B
E(Y
(j)
n

− EY
(j)
n
)
2
=
∞

n=1
1
b
2
n

j∈B
E[(Y
(j)
n
)
2
− 2Y
(j)
n
EY
(j)
n
+ (EY
(j)
n
)

2
]
=
∞

n=1
1
b
2
n

j∈B
[E(Y
(j)
n
)
2
− (EY
(j)
n
)
2
]
≤
∞

n=1
1
b
2

n

j∈B
E(Y
(j)
n
)
2
≤ 3
∞

n=1

j∈B
[
E(X
n
, e
j
 I(| X
n
, e
j
 | ≤ b
n
))
2
b
2
n

+ P (| X
n
, e
j
 | > b
n
)].
(2.6) (2.7)
∞

n=1
E||Y
n
− EY
n
||
2
b
2
n
< ∞.
T
k
=
1
b
2
k+1
− b
2

k
max
j<2
k+1
||
j

i=2
k
(Y
i
− EY
i
)||, k ≥ 0.
inf
k≥0
b
2
k+1
b
2
k
> 1
b
2
k
b
2
k+1
< 1 ∀k ≥ 0

C
b
2
k
b
2
k+1
≤ 1 − C, ∀k ≥ 0
b
2
k+1
− b
2
k
≥ Cb
2
k+1
, ∀k ≥ 0
ET
2
k
≤ CE(
1
b
2
k+1
max
j<2
k+1
||

j

i=2
k
(Y
i
− EY
i
)||)
2
≤
C
b
2
2
k+1
2
k+1
−1

i=2
k
E||Y
i
− EY
i
||
2
≤ C
2

k+1
−1

i=2
k
E||Y
i
− EY
i
||
2
b
2
i
.
(2.8)
∞

k=0
ET
2
k
< ∞
lim
k→∞
T
k
= 0 .
2
k

≤ n < 2
k+1



n

i=1
(Y
i
− EY
i
)
b
n



≤
1
b
2
k
k

j=0
max
l<2
j+1


l

i=2
j
(Y
i
− EY
i
)
=
b
2
k+1
b
2
k
k

j=0
b
2
j+1
− b
2
j
b
2
k+1
T
j

≤ C
k

j=0
b
2
j+1
− b
2
j
b
2
k+1
T
j
(2.2) .
a
kj
=
b
2
j+1
− b
2
j
b
2
k+1
.
lim

k→∞
a
kj
= 0
k

j=1
|a
kj
| =
1
b
2
k+1
k

j=1
(b
2
j+1
− b
2
j
)
=
1
b
2
k+1
(b

2
k+1
− b
2
)
≤ C.
(2.10)
lim
k→∞
k

j=0
b
2
j+1
− b
2
j
b
2
k+1
T
j
= 0 .
n

i=1
E(Y
i
− EY

i
)
b
n
→ 0 n → ∞.
X
n
=

j∈B
X
n
, e
j
e
j
∀n ≥ 1
∞

n=1
P (X
n
= Y
n
) =
∞

n=1
P {||X
n

− Y
n
||
2
> 0}
=
∞

n=1
P {

j∈B
(X
n
, e
j
 − Y
(j)
n
)
2
> 0}
≤
∞

n=1

j∈B
P {(X
n

, e
j
 − Y
(j)
n
)
2
> 0}
≤
∞

n=1

j∈B
P {| X
n
, e
j
| > b
n
} < ∞ (2.6) .
(2.12) (2.13) (2.4)
n

i=1
EY
i
b
n
→ 0 n → ∞.

EX
n
= 0 n ≥ 1 E X
n
, e
j
 = 0 n ≥ 1, j ∈ B
∞

n=1
||EY
n
||
b
n
=
∞

n=1
1
b
n
||E

j∈B
Y
(j)
n
e
j

||
≤
∞

n=1
1
b
n

j∈B
||EY
(j)
n
e
j
||
=
∞

n=1
1
b
n

j∈B
|EY
(j)
n
|
≤

∞

n=1

j∈B
[P (| X
n
, e
j
| > b
n
) +
1
b
n
|E(X
n
, e
j
 I(| X
n
, e
j
 | ≤ b
n
))|]
≤
∞

n=1


j∈B
[P (| X
n
, e
j
| > b
n
) +
1
b
n
|E(X
n
, e
j
 I(| X
n
, e
j
 | > b
n
))|]
( E X
n
, e
j
 = 0∀n ≥ 1, j ∈ B)
≤
∞


n=1

j∈B
[P (| X
n
, e
j
| > b
n
) +
1
b
n
E(| X
n
, e
j
 |I(| X
n
, e
j
 | > b
n
)|)]
=
∞

n=1


j∈B
[2P (| X
n
, e
j
| > b
n
) +
1
b
n
∞

b
n
P (| X
n
, e
j
 | > t)dt]
=
∞

n=1

j∈B
[2P (| X
n
, e
j

| > b
n
) +
b
r
n
−1
n
b
r
n
n
∞

b
n
P (| X
n
, e
j
 | > t)dt]
≤
∞

n=1

j∈B
[2P (| X
n
, e

j
| > b
n
) +
1
b
r
n
n
∞

b
n
t
r
n
−1
P (| X
n
, e
j
 | > t)dt]
≤
∞

n=1

j∈B
[2P (| X
n

, e
j
| > b
n
) +
E| X
n
, e
j
 |
r
n
b
r
n
n
] < ∞ (2.3) (2.6) .
(2.15) (2.14)
r
n
= 2
∞

n=1

j∈B
E| X
n
, e
j

 |
r
n
b
n
r
n
=
∞

n=1

j∈B
EX
n
, e
j

2
b
n
2
=
∞

n=1
E||X
n
||
2

b
2
n
.
|B| B (2.3)
∞

n=1
E||X
n
||
r
n
b
r
n
n
< ∞.
{b
n
, n ≥ 1}
{Y
n
, n ≥ 1}
0 P{Y
1
= 0} = 1
∞

n=1

EY
2
n
< ∞
m ≥ 1
X
n
=

b
2
m
Y
1
m + 1
, n = 2
m
b
n
Y
n
, 2
m
< n < 2
m+1
{X
n
, n ≥ 1}
∞


n=1
EX
n
2
b
2
n
= EY
2
1
∞

m=0
1
(m + 1)
2
+

n≥3,log n /∈N
EY
2
n
< ∞
(2.4)
3

3≤i≤n,log i /∈N
Y
i
n → ∞


1≤i≤n,log i∈N
Y
i
1 + log i
= Y
1
[ log n]

m=0
1
1 + m
n → ∞
[x] x
n ≥ 3
n

i=1
X
i
b
i
=

3≤i≤n,log i /∈N
Y
i
+

1≤i≤n,log i∈N

Y
i
1 + log i
n → ∞.
{X
n
, n ≥ 1}
0 H {X
n
, n ≥ 1}

j∈B
E| X
1
, e
j
|
p
< ∞ 1 ≤ p < 2 (3.1)
n

i=1
X
i
n
1
p
→ 0 n → ∞ (3.2).
n ≥ 1, i ∈ B
Y

(j)
n
= − n
1
p
I(X
n
, e
j
 < −n
1
p
) + X
n
, e
j
 I(|X
n
, e
j
 | ≤ n
1
p
)
+ n
1
p
I(X
n
, e

j
 > n
1
p
)
Y
n
=

j∈B
Y
(j)
n
e
j
.
E||Y
n
− EY
n
||
2
n
2
p
= n
−
2
p


j∈B
E(Y
(j)
n
− EY
(j)
n
)
2
= n
−
2
p

j∈B
E[(Y
(j)
n
)
2
− 2Y
(j)
n
EY
(j)
n
+ (EY
(j)
n
)

2
]
= n
−
2
p

j∈B
[E(Y
(j)
n
)
2
− (EY
(j)
n
)
2
]
≤ n
−
2
p

j∈B
E(Y
(j)
n
)
2

≤ 3

j∈B
[
E(X
n
, e
j
 I(| X
n
, e
j
 | ≤ n
1
p
))
2
n
2
p
+ P (| X
n
, e
j
 | > n
1
p
)]
= 6


j∈B
n
−
2
p
n
1
p

0
xP (| X
n
, e
j
 | > x)dx
= 6

j∈B
n
−
2
p
n
1
p

0
xP (| X
1
, e

j
 | > x)dx
∞

n=1
EY
n
− EY
n

2
n
2
p
≤ 6

j∈B
∞

n=1
n
−2
p

n
1
p
0
xP (|X
1

, e
j
| > x)dx
≤ 6

j∈B
∞

n=1
n
−2
p
n

k=1

k
1
p
(k−1)
1
p
xP (|X
1
, e
j
| > (k − 1)
1
p
)dx

≤ C

j∈B
∞

n=1
n
−2
p
n

k=1
P (|X
1
, e
j
| > (k − 1)
1
p
)(k
2
p
− (k − 1)
2
p
)
≤ C

j∈B
∞


n=1
n
−2
p
n

k=1
P (|X
1
, e
j
| > (k − 1)
1
p
)k
2
p
−1
≤ C

j∈B
∞

k=1
(
∞

n=k
n

−2
p
)P (|X
1
, e
j
| > (k − 1)
1
p
)k
2
p
−1
≤ C

j∈B
∞

k=1
k
−2
p
+1
P (|X
1
, e
j
| > (k − 1)
1
p

)k
2
p
−1
= C

j∈B
∞

k=1
P (|X
1
, e
j
| > (k − 1)
1
p
)
= C

j∈B
E|X
1
, e
j
|
p
∞

n=1

E||Y
n
− EY
n
||
2
n
2
p
< ∞ (3.3)
(2.12)
|B| B (3.1)