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
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

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

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

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
} 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

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→∞

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

≥ 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
→∞

{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)}

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

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

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
} {
¯


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

:∨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
(

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.

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

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

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

(
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

,∨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

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)}

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

{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),

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) =



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

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

1id
n
i
= ∨n
i
→ ∞