Khintchine inequality explained

In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick

N

complex numbers

x1,...,xN\inC

, and add them together each multiplied by a random sign

\pm1

, then the expected value of the sum's modulus, or the modulus it will be closest to on average, will be not too far off from
2
\sqrt{|x
1|

+ … +

2
|x
N|
}.

Statement

Let

\{\varepsilonn\}

N
n=1
be i.i.d. random variables with
P(\varepsilon
n=\pm1)=12
for

n=1,\ldots,N

, i.e., a sequence with Rademacher distribution. Let

0<p<infty

and let

x1,\ldots,xN\inC

. Then

Ap\left(

N
\sum
n=1
2
|x
n|

\right)1/2\leq\left(\operatorname{E}

N
\left|\sum
n=1

\varepsilonn

p
x
n\right|

\right)1/p\leqBp

N
\left(\sum
n=1
2\right)
|x
n|

1/2

for some constants

Ap,Bp>0

depending only on

p

(see Expected value for notation). The sharp values of the constants

Ap,Bp

were found by Haagerup (Ref. 2; see Ref. 3 for a simpler proof). It is a simple matter to see that

Ap=1

when

p\ge2

, and

Bp=1

when

0<p\le2

.

Haagerup found that

\begin{align} Ap&=\begin{cases} 21/2-1/p&0<p\lep0,\\ 21/2(\Gamma((p+1)/2)/\sqrt{\pi})1/p&p0<p<2\\ 1&2\lep<infty \end{cases} \\ &and \\ Bp&=\begin{cases} 1&0<p\le2\\ 21/2(\Gamma((p+1)/2)/\sqrt\pi)1/p&2<p<infty \end{cases}, \end{align}

where

p0 ≈ 1.847

and

\Gamma

is the Gamma function.One may note in particular that

Bp

matches exactly the moments of a normal distribution.

Uses in analysis

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

Lp(X,\mu)

and

Lp(Y,\nu)

,

1<p<infty

, with bounded norm

\|T\|<infty

, then one can use Khintchine's inequality to show that
N
\left\|\left(\sum
n=1
2
|Tf
n|

\right)1/2

\right\|
Lp(Y,\nu)

\leqCp

N
\left\|\left(\sum
n=1
2\right)
|f
n|

1/2

\right\|
Lp(X,\mu)

for some constant

Cp>0

depending only on

p

and

\|T\|

.

Generalizations

For the case of Rademacher random variables, Pawel Hitczenko showed[1] that the sharpest version is:

A

N
\left(\sqrt{p}\left(\sum
n=b+1
2\right)
x
n

1/2+

b
\sum
n=1

xn\right) \leq\left(\operatorname{E}

N
\left|\sum
n=1

\varepsilonn

p
x
n\right|

\right)1/p\leqB

N
\left(\sqrt{p}\left(\sum
n=b+1
2\right)
x
n

1/2+

b
\sum
n=1

xn\right)

where

b=\lfloorp\rfloor

, and

A

and

B

are universal constants independent of

p

.

Here we assume that the

xi

are non-negative and non-increasing.

See also

References

  1. Thomas H. Wolff, "Lectures on Harmonic Analysis". American Mathematical Society, University Lecture Series vol. 29, 2003.
  2. Uffe Haagerup, "The best constants in the Khintchine inequality", Studia Math. 70 (1981), no. 3, 231 - 283 (1982).
  3. Fedor Nazarov and Anatoliy Podkorytov, "Ball, Haagerup, and distribution functions", Complex analysis, operators, and related topics, 247 - 267, Oper. Theory Adv. Appl., 113, Birkhäuser, Basel, 2000.

Notes and References

  1. [Hitczenko|Pawel Hitczenko]