Theorem 3.7
(Marcinkiewicz-Zygmund)
.
Assuming that:
p
∈
[
2
,
∞
)
Ł
X
1
,
X
2
,
…
,
X
n
∈
Ł
p
(
ℙ
)
independent random variables
𝔼
∑
i
=
1
n
X
i
=
0
Then
∥
∑
i
=
1
n
X
i
∥
L
p
(
ℙ
)
=
O
(
p
1
2
∥
∑
i
=
1
n
|
X
i
|
2
∥
L
p
∕
2
(
ℙ
)
1
2
)
.