In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Roman alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick
N
complex numbers
x
1
,
…
,
x
N
∈
C
, and add them together each multiplied by a random sign
±
1
, then the expected value of its modulus, or the modulus it will be closest to on average, will be not too far off from
|
x
1
|
2
+
⋯
+
|
x
N
|
2
.
Let
{
ε
n
}
n
=
1
N
be i.i.d. random variables with
P
(
ε
n
=
±
1
)
=
1
2
for
n
=
1
,
…
,
N
, i.e., a sequence with Rademacher distribution. Let
0
<
p
<
∞
and let
x
1
,
…
,
x
N
∈
C
. Then
A
p
(
∑
n
=
1
N
|
x
n
|
2
)
1
/
2
≤
(
E
|
∑
n
=
1
N
ε
n
x
n
|
p
)
1
/
p
≤
B
p
(
∑
n
=
1
N
|
x
n
|
2
)
1
/
2
for some constants
A
p
,
B
p
>
0
depending only on
p
(see Expected value for notation). The sharp values of the constants
A
p
,
B
p
were found by Haagerup (Ref. 2; see Ref. 3 for a simpler proof). It is a simple matter to see that
A
p
=
1
when
p
≥
2
, and
B
p
=
1
when
0
<
p
≤
2
.
The uses of this inequality are not limited to applications in probability theory. One example of its use in analysis is the following: if we let
T
be a linear operator between two Lp spaces
L
p
(
X
,
μ
)
and
L
p
(
Y
,
ν
)
,
1
≤
p
<
∞
, with bounded norm
∥
T
∥
<
∞
, then one can use Khintchine's inequality to show that
∥
(
∑
n
=
1
N
|
T
f
n
|
2
)
1
/
2
∥
L
p
(
Y
,
ν
)
≤
C
p
∥
(
∑
n
=
1
N
|
f
n
|
2
)
1
/
2
∥
L
p
(
X
,
μ
)
for some constant
C
p
>
0
depending only on
p
and
∥
T
∥
.
