In mathematics, the Marcinkiewicz–Zygmund inequality, named after Józef Marcinkiewicz and Antoni Zygmund, gives relations between moments of a collection of independent random variables. It is a generalization of the rule for the sum of variances of independent random variables to moments of arbitrary order.
Theorem If
x
i
,
i
=
1
,
…
,
n
, are independent random variables such that
E
(
x
i
)
=
0
and
E
(
|
x
i
|
p
)
<
+
∞
,
1
≤
p
<
+
∞
, then
A
p
E
(
(
∑
i
=
1
n
|
x
i
|
2
)
p
/
2
)
≤
E
(
|
∑
i
=
1
n
x
i
|
p
)
≤
B
p
E
(
(
∑
i
=
1
n
|
x
i
|
2
)
p
/
2
)
where
A
p
and
B
p
are positive constants, which depend only on
p
.
In the case
p
=
2
, the inequality holds with
A
2
=
B
2
=
1
, and it reduces to the rule for the sum of variances of independent random variables with zero mean, known from elementary statistics: If
E
(
x
i
)
=
0
and
E
(
|
x
i
|
2
)
<
+
∞
, then
V
a
r
(
∑
i
=
1
n
x
i
)
=
E
(
|
∑
i
=
1
n
x
i
|
2
)
=
∑
i
=
1
n
∑
j
=
1
n
E
(
x
i
x
¯
j
)
=
∑
i
=
1
n
E
(
|
x
i
|
2
)
=
∑
i
=
1
n
V
a
r
(
x
i
)
.
