về một dạng hội tụ của dãy và chuỗi

N d
N
d
= {n = (n
1
, n
2
, , n
d
) : n
i
∈ N, i = 1, 2, , d}.
{X(n), n ∈ N
d
} d
{X(n), n ∈ N
d
}

n∈N
d
X(n) max
1id
n
i
= ∨n
i
→ ∞
d
N
d

= {n = (n
1
, n
2
, , n
d
) : n
i
∈ N, i = 1, 2, , d}.
N
d
m = (m
1
, m
2
, , m
d
), n = (n
1
, n
2
, , n
d
) N
d
m  n ⇐⇒ m
i
 n
i
, i = 1, . . . , d.

N
d
I
d
= {i =
(i, i, . . . , i) : i ∈ N}.
n = (n
1
, n
2
, . . . , n
d
)
∨n
i
= max
1id
n
i
; ∧n
i
= min
1id
n
i
|n| = n
1
.n
2
. . . n

d
.
N
d
d
d
d
∨n
i
→ ∞ ∧n
i
→ ∞
d
∧n
i
→ ∞
d
∨n
i
→ ∞
N
d
(d > 1)
d
d
{x(n), n ∈ N
d
} d {x(n), n ∈ N
d
}

x ∈ R ∨n
i
→ ∞ ∧n
i
→ ∞) ε > 0
n
0
∈ N n
0
∈ N
d
n = (n
1
, n
2
, . . . , n
d
) ∈ N
d
∨n
i
≥ n
0
n ≥ n
0
|x(n) − x| < ε lim
∨n
i
→∞
x(n) = x

lim
∧n
i
→∞
x(n) = x).
{x(n), n ∈ N
d
} d {x(n), n ∈ N
d
}
x ∈ R |
n| → ∞ ε > 0 n
0
∈ N
n = (n
1
, n
2
, . . . , n
d
) ∈ N
d
|n| ≥ n
0
|x(n) − x| < ε. lim
|n|→∞
x(n) = x.
n ≥ n
0
|n| ≥ |n

0
| ∨n
i
→ ∞
|n| → ∞
lim
|n|→∞
x(n) = x ⇔ lim
∨n
i
→∞
x(n) = x ⇒ lim
∧n
i
→∞
x(n) = x.
{x(n), n ∈ N
d
} d

n∈N
d
x(n) (1)
d
S(n) =

mn
x(m) S(n) n
∨n
i

→ ∞ {S(n), n ∈ N
d
} ∨n
i
→ ∞ S =
lim
∨n
i
→∞
S(n) =

n∈N
d
x(n)
r(n) = S − S(n) =

m:mn
x(m)
n
lim
∨n
i
→∞
r(n) = 0
{x(n), n ∈ N
d
} {y(n), n ∈ N
d
} d
x(n) → x y(n) → y ∨n

i
→ ∞ (∧n
i
→ ∞) x(n) + y(n) → x + y
∨n
i
→ ∞ (∧n
i
→ ∞).
x(n) → x ∨n
i
→ ∞ (∧n
i
→ ∞) |x(n)| → |x| ∨n
i
→ ∞ (∧n
i
→ ∞).
x(n) → x ∨n
i
→ ∞ (∧n
i
→ ∞) λx(n) → λx ∨n
i
→ ∞ (∧n
i
→ ∞)
λ ∈ C.

n∈N

d
x(n) x ∈ R

n∈N
d
y(n) y ∈ R
∨n
i
→ ∞

n∈N
d
(x(n) + y(n)) x + y ∨n
i
→ ∞.

n∈N
d
λx(n), λ ∈ C λx ∨n
i
→ ∞.
{x(n), n ∈ N
d
} d {x(n), n ∈ N
d
}
M > 0 |x(n)|  M, n ∈ N
d
.
{x(n), n ∈ N

d
} d {x(n), n ∈ N
d
}
x ∈ R ∨n
i
→ ∞ {x(n), n ∈ N
d
}
{x(n), n ∈ N
d
} x ∈ R ∧n
i
→ ∞ {x(n), n ∈ N
d
}
{x(m, n)}
x(m, n) =

m, n = 1
0, n = 1.
x(m, n) → 0 m ∧ n → ∞
d
{x(n), n ∈ N
d
} {x(n), n ∈ N
d
}
x(n) ≥ x(m) x(n)  x(m) ∨n
i

≥ ∨m
i
{x(n), n ∈ N
d
} ∨n
i
→ ∞
lim
∨n
i
→∞
x(n) = lim
i→∞
x(i) (i ∈ I
d
).
{x(n), n ∈ N
d
} {x(i), i ∈ I
d
}
x(n), n ∈ N
d
} {x(n), n ∈ N
d
} {x(i), i ∈ I
d
}
x i → ∞
{x(n), n ∈ N

d
} x ∨n
i
→ ∞
ε > 0 n
0
∈ N 0  x − x(i) < ε (i ∈ I
d
) i ≥ n
0
n ∨n
i
= k > n
0
x(n
0
)  x(n)  x(k), (n
0
, k ∈ I
d
)
0  x − x(k)  x − x(n)  x − x(n
0
) < ε.
lim
∨n
i
→∞
x(n) = x = lim
i→∞

x(i) (i ∈ I
d
).
{x(n), n ∈ N
d
} {−x(n), n ∈ N
d
}

{x(n), n ∈ N
d
} ∨n
i
→ ∞
ε > 0 n
o
∈ N m, n ∈ N
d
∨m
i
≥ n
0
, ∨n
i
≥ n
0
|x(m) − x(n)| < ε.
{x(n), n ∈ N
d
} x ∈ R, ∨n

i
→ ∞,
{x(n), n ∈ N
d
}
{x(n), n ∈ N
d
} x ∈ R
∨n
i
→ ∞ ε > 0 n
0
∈ N n ∈ N
d
∨n
i
≥ n
0
|x(n) − x| <
ε
2
. m, n ∈ N
d
∨m
i
≥ n
0
, ∨n
i
≥ n

0
|x(m) − x(n)| =
|(x(m) − x) − (x(n) − x)|  |x(m) − x| + |x(n) − x| <
ε
2
+
ε
2
= ε
{x(i), i ∈ I
d
} {x(n), n ∈ N
d
}.
{x(i), i ∈ I
d
} {x(i), i ∈ I
d
} x ∈ R
i → ∞ {x(n), n ∈ N
d
}
ε > 0 n
0
∈ N n ∈ N
d
, i ∈ I
d
∨n
i

≥ n
0
, i ≥ n
0
|x(n) − x|  |x(n) − x(i)| + |x(i) − x| < ε {x(n), n ∈ N
d
} x ∈ R
∨n
i
→ ∞. 
d
{X(n), n ∈ N
d
} d
(Ω, F, P) {X(n)}
{X(n), n ∈ N
d
} I
d
, {X(i), i ∈ I
d
}
{X(i)}.
{X(n)} X ∨n
i
→ ∞, ε > 0
lim
∨n
i
→∞

P(|X(n) − X| > ε) = 0 X(n)
P
−→ X ∨n
i
→ ∞.
{X(n)} X ∨n
i
→ ∞
A 0 X(n)(ω) → X(ω) ∨n
i
→ ∞ ω /∈ A.
X(n) −→ X, ∨n
i
→ ∞.
{X(
n)} p (0 < p < ∞) ∨n
i
→ ∞
lim
∨n
i
→∞
E|X(n) − X|
p
= 0. X(n)
L
p
→ X , ∨n
i
→ ∞.

{A(n)} d
{A(n)} A(m) ⊂ A(n) ∨m
i
 ∨n
i
P(

n∈N
d
A(n)) =
lim
∨n
i
→∞
P(A(n)).
{A(n)} A(m) ⊃ A(n) ∨m
i
 ∨n
i
P(

n∈N
d
A(n)) =
lim
∨n
i
→∞
P(A(n)).
{A(n)} {A(n)}

I
d
{A(i)}. {A(i)}
{A(i)} P(

i∈I
d
A(i)) = lim
i→∞
P(A(i)).

n∈N
d
A(n) =

i∈I
d
A(i).

n∈N
d
A(n) ⊃

i∈I
d
A(i)
ω ∈

n∈N
d

A(n) n
0
= (n
01
, n
02
, . . . , n
0d
) ∈ N
d
ω ∈ A(n
0
).
i
0
= (m
0
, m
0
, . . . , m
0
) ∈ I
d
m
0
= max
1id
n
0i
. ∨n

0i
 ∨m
o
= m
0
A(n
0
) ⊂ A(i
0
), ω ∈ A(i
0
) ω ∈

i∈I
d
A(i)

n∈N
d
A(n) ⊂

i∈I
d
A(i

n∈N
d
A(n) =

i∈I

d
A(i). P(

n∈N
d
A(n)) = lim
i→∞
P(A(i)).
{P(A(n))}
lim
∨n
i
→∞
P(A(n)) = lim
i→∞
P(A(i)) = P(

n∈N
d
A(n)).

d
{A(n), n ∈ N
d
} d

n∈N
d
P(A(n)) < ∞ P( lim
∨n

i
→∞
sup
n
A(n)) = 0.

n∈N
d
P(A(n)) = ∞ {A(n), n ∈ N
d
} P( lim
∨n
i
→∞
sup
n
A(n)) = 1.
lim
∨n
i
→∞
sup
n
A(n) =

n∈N
d

mn
A(m).

B(1) =

m1
A(m), . . . , B(k) =

mk
A(m), . . .
{B(k), k ∈ N
d
}
P( lim
∨n
i
→∞
sup
n
A(n)) = lim
k→∞
P(

nk
A(n))  lim
k→∞

nk
P(A(n)) = 0.
{A(n), n ∈ N
d
} {
¯

A(n), n ∈ N
d
}
P(

nk
¯
A(n)) =

nk
P(
¯
A(n))
=

nk
(1 − P(A(n))) 

nk
e
−P(A(n))
= e
−
P
nk
P(A(n))
= e
−∞
= 0
P(


nk
A(n)) = 1 P( lim
∨n
i
→∞
sup
n
A(n)) = 1 
X(n)
L
p
−→ X, X(n) −→ X ∨n
i
→ ∞ X(n)
P
−→ X
∨n
i
→ ∞.
X(n)
L
p
−→ X, ∨n
i
→ ∞
ε > 0 0  P(|X(n) − X| > ε) 
E|X(n) − X|
p
ε

p
lim
∨n
i
→∞
E|X(n) − X|
p
= 0
lim
∨n
i
→∞
P(|X(n) − X| > ε) = 0 X(n)
P
−→ X ∨n
i
→ ∞
X(n) −→ X ∨n
i
→ ∞
P( lim
∨n
i
→∞
|X(n) − X| = 0) = 1.
ε > 0
D(n)(ε) =

m∈N
d

:∨m
i
≥∨n
i
(|X(m) − X| ≥ ε).
D(n)
c
(ε) =

m∈N
d
:∨m
i
≥∨n
i
(|X(m) − X| < ε).
( lim
∨n
i
→∞
|X(n) − X| = 0) =
∞

k=1

n∈N
d
D(n)
c
(

1
k
).
X(n) −→ X ∨ n
i
→ ∞ ⇔ P( lim
∨n
i
→∞
|X(n) − X| = 0) = 1
⇔ P(
∞

k=1

n∈N
d
D(n)
c
(
1
k
)) = 1 ⇔ P(

n∈N
d
D(n)
c
(
1

k
)) = 1 ⇔ P(

n∈N
d
D(n)(
1
k
)) = 0.
{D(n)(
1
k
), n ∈ N
d
}
lim
∨n
i
→∞
P(D(n)(
1
k
)) = P(

n∈N
d
D(n)(
1
k
)) = 0.

ε > 0 k
1
k
< ε D(n)(ε) ⊂ D(n)(
1
k
).
(|X(n) − X| > ε) ⊂ (|X(n) − X| ≥ ε) ⊂ D(n)(ε) ⊂ D(n)(
1
k
).
0  P(|X(n)−X| > ε)  P(D(n)(
1
k
)) → 0, ∨n
i
→ ∞ lim
∨n
i
→∞
P(|X(n)−
X| > ε) = 0 X(n)
P
−→ X ∨n
i
→ ∞ 
lim
∨n
i
→∞

P(X(n) = Y (n)) = 0 X(n)
P
−→ X ∨n
i
→ ∞ Y (n)
P
−→
X ∨n
i
→ ∞.
X(n) −→ X ∨n
i
→ ∞ ε > 0
lim
∨n
i
→∞
P( sup
{m:∨m
i
≥∨n
i
}
|X(m) − X| ≥ ε) = 0.

n∈N
d
P(|X(n) − X| > ε)
∨n
i

→ ∞ ε > 0 X(n) −→ X, ∨n
i
→ ∞.
ε > 0
P( sup
{m:∨m
i
≥∨n
i
}
|X(m)−X| > ε) = P(

m,∨m
i
≥∨n
i
|X(m)−X| > ε) 

m,∨m
i
∨n
i
P(|X(m)−X| > ε).
∨m
i
 ∨n
i
m  n

m,∨m

i
∨n
i
P(|X(m) − X| > ε) 

m:mn
P(|X(m) − X| > ε) = r(n) → 0
∨n
i
→ ∞. 
X(n) −→ X, Y (n) −→ Y, ∨n
i
→ ∞ X(n) + Y (n) −→
X + Y, ∨n
i
→ ∞.
{X(n)} d

n∈N
d
E|X(n)|
p
<
∞ p > 0 X(n) −→ 0 X(n)
L
p
−→ 0, ∨n
i
→ ∞.
{X(n)}, n ∈ N

d
d
(Ω, F, P)
{X(n)}
P( lim
∨m
i
→∞,∨n
i
→∞
|X(m) − X(n)| = 0) = 1.
{X(n)}
∀ε > 0 : lim
∨m
i
→∞,∨n
i
→∞
P(|X(m) − X(n)| ≥ ε) = 0.
{X(n)} p (0 < p < ∞)
lim
∨m
i
→∞,∨n
i
→∞
E|X(m) − X(n)|
p
= 0.
{X(n)}

ε > 0
lim
∨n
i
→∞
P( sup
{k,l:∨k
i
,∨l
i
≥∨n
i
}
|X(k) − X(l)| ≥ ε) = 0.
lim
∨n
i
→∞
P( sup
{k:∨k
i
≥∨n
i
}
|X(k) − X(n)| ≥ ε) = 0.
{X(n)}
∆(n)(ε) =

k,l,∨k
i

:∨l
i
≥∨n
i
(|X(k) − X(l )| ≥ ε)
⊂ ( sup
{k,l:∨k
i
,∨l
i
≥∨n
i
}
|X(k) − X(l )| ≥ ε).
{∆(n)}
( lim
∨k
i
→∞,∨l
i
→∞
|X(k) − X(l)| = 0) =
∞

m=1

n∈N
d
∆(n)
c

(
1
m
).
{X(n)} ⇔ P(
∞

m=1

n∈N
d
∆(n)
c
(
1
m
)) = 1
⇔ P(

n∈N
d
∆(n)
c
(
1
m
)) = 1 ⇔ P(

n∈N
d

∆(n)(
1
m
)) = 0 ⇔ lim
∨n
i
→∞
P(∆(n)(
1
m
)) = 0.
{∆(n)(
1
m
)} ε > 0 m ∈ N
1
m
< ε
( sup
{k,l:∨k
i
,∨l
i
≥∨n
i
}
|X(k) − X(l )| ≥ ε) ⊂

k,l:∨k
i

,∨l
i
≥∨n
i
(|X(k) − X(l )| ≥
1
m
) ⊂ ∆(n)(
1
m
).
0  P( sup
{k,l:∨k
i
,∨l
i
≥∨n
i
}
|X(k) − X(l )| ≥ ε)  P(∆(n)(
1
m
)) −→ 0 ∨n
i
→ ∞.
{X(n)}
P(

n∈N
d

∆(n)(
1
m
)) = lim
∨n
i
→∞
P(∆(n)(
1
m
))  lim
∨n
i
→∞
P( sup
{k,l:∨k
i
,∨l
i
≥∨n
i
}
|X(k)−X(l)| ≥
1
m
) = 0,
∈ N.
P(
∞


m=1

n∈N
d
∆(n)
c
(
1
m
)) = 1, m ∈ N.
P( lim
∨n
i
→∞
|X(k) − X(l )| = 0) = 1 k, l ∈ N
d
∨ k
i
, ∨l
i
≥ ∨n
i
.
{X(n)} 
{X(n)}, n ∈ N
d
} d
X(n)
X(n)
{X(n)} {X(n)}

I
d
{X(j)} {X(j)} {X(j)}
X
X(n)
P
−→ X ∨n
i
→ ∞
P(|X(k) − X| > ε)  P(|X(n) − X(j| >
ε
2
) + P(|X(j) − X| >
ε
2
) −→ 0
∨n
i
, j → ∞
X(
n)
P
−→ X ∨n
i
→ ∞

{X(n)}, n ∈ N
d
d EX(n) = 0
n ∈ N

d

n∈N
d
DX(n) ∨n
i
→ ∞

n∈N
d
X(n)
∨n
i
→ ∞.
ε > 0
0  P( sup
{k:∨k
i
≥∨n
i
}
|S(k) − S(n)| > ε) 
1
ε
2

k:∨k
i
≥∨n
i

DX(k) 
1
ε
2

k:kn
DX(k) = r(n) → 0
∨n
i
→ ∞
lim
∨n
i
→∞
P( sup
{k:∨k
i
≥∨n
i
}
|S(k) − S(n)| > ε) = 0
ε > 0 {S(n)} ∨n
i
→ ∞

n∈N
d
X(n) ∨n
i
→ ∞. 

{X(n)} d

n∈N
d
DX(n)
∨n
i
→ ∞

n∈N
d
(X(n) − EX(n)) ∨n
i
→ ∞.
{X(n), n ∈ N
d
} d c > 0
X
c
(n) = X(n)I(|X(n)|  c).

n∈N
d
P(|X(n)| > c),

n∈N
d
E(X
c
(n),


n∈N
d
D(X
c
(n)) ∨n
i
→ ∞

n∈N
d
X(n) ∨n
i
→ ∞.

n∈N
d
X
c
(n) ∨n
i
→ ∞

n∈N
d
P(X(n) = X
c
(n)) =

n∈N

d
P(|X(n)| > c) ∨n
i
→ ∞

n∈N
d
X(n)

n∈N
d
X
c
(n)

n∈N
d
X
c
(n) ∨n
i
→ ∞

n∈N
d
X(n) ∨n
i
→ ∞ 
X(i, j) =


(−1)
j
i i ≥ 1, j  2,
0
n ≥ 2
S(m, n) =

im

jn
X(i, j) =

im

j2
X(i, j) = (−1)
1
.1 + (−1)
1
.2+
+ (−1)
1
.m + (−1)
2
.1 + (−1)
2
.2 + + (−1)
2
.m = 0.
S(m, n) 0 m ∧ n → ∞


i≥1

j≥1
P(|X(i, j)| ≥ c) (∗)
0 < c  1.
σ(m, n) =

im

jn
P(|X(i, j)| ≥ c) =

im

j2
P(|X(i, j)| ≥ c)
= 2(

im
P(i ≥ c)) = 2m → ∞, m ∧ n → ∞.
∧n
i
→ ∞ d

n∈N
d
P(|X(n)| ≥ c),

n∈N

d
EX
c
(n),

n∈N
d
DX
c
(n)
{X(n)}, n ∈ N
d
d EX(n) = 0
n ∈ N
d

n∈N
d
E
X
2
(n)
1 + |X(n)|
∨n
i
→ ∞

n∈N
d
X(n) ∨n

i
→ ∞
{X(n)}, n ∈ N
d
d
X(ni) = X(i)I(|X(i)|  |n|).
(i)

in
P(|X(i)| > |n|) → 0 ∨ n
i
→ ∞,
(ii)
1
|n|

in
EX(ni) → 0 ∨ n
i
→ ∞,
(iii)
1
|n|
2

in
DX(ni) → 0 ∨ n
i
→ ∞
1

|n|

in
X(i) 0 ∨n
i
→ ∞.
N d
N
d
= {n = (n
1
, n
2
, , n
d
) : n
i
∈ N, i = 1, 2, , d}.
{X(
n), n ∈ N
d
} d
{X(n), n ∈
N
d
}

n∈N
d
X(n) max

1id
n
i
= ∨n
i
→ ∞