class="bi x0 y0 w1 h1"
class="bi x1 y1 w2 h2"
R
d
H
H
H
class="bi x1 y1 w3 h7"
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
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
} ∩ {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
∩

|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

|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

(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}

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


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

{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

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=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
(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=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
.

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

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

, 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

, 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

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

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

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

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
(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
)

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

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

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

≤ 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


≤ 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

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

− 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

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

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


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


| > 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

, 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

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

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 → ∞

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

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

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

, 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


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

)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