Theorem in probability
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
complex numbers
, and add them together each multiplied by a random sign
, 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
.
Statement
Let
be i.i.d. random variables with
for
, i.e., a sequence with Rademacher distribution. Let
and let
. Then
![{\displaystyle A_{p}\left(\sum _{n=1}^{N}|x_{n}|^{2}\right)^{1/2}\leq \left(\operatorname {E} \left|\sum _{n=1}^{N}\varepsilon _{n}x_{n}\right|^{p}\right)^{1/p}\leq B_{p}\left(\sum _{n=1}^{N}|x_{n}|^{2}\right)^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/506a35970b7e4304ece6c9924fe8b27294fbb9e7)
for some constants
depending only on
(see Expected value for notation). The sharp values of the constants
were found by Haagerup (Ref. 2; see Ref. 3 for a simpler proof). It is a simple matter to see that
when
, and
when
.
Haagerup found that
![{\displaystyle {\begin{aligned}A_{p}&={\begin{cases}2^{1/2-1/p}&0<p\leq p_{0},\\2^{1/2}(\Gamma ((p+1)/2)/{\sqrt {\pi }})^{1/p}&p_{0}<p<2\\1&2\leq p<\infty \end{cases}}\\&{\text{and}}\\B_{p}&={\begin{cases}1&0<p\leq 2\\2^{1/2}(\Gamma ((p+1)/2)/{\sqrt {\pi }})^{1/p}&2<p<\infty \end{cases}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3dc049fb631722f3f26a2dc4dde148ef24a52ee)
where
and
is the Gamma function. One may note in particular that
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
be a linear operator between two Lp spaces
and
,
, with bounded norm
, then one can use Khintchine's inequality to show that
![{\displaystyle \left\|\left(\sum _{n=1}^{N}|Tf_{n}|^{2}\right)^{1/2}\right\|_{L^{p}(Y,\nu )}\leq C_{p}\left\|\left(\sum _{n=1}^{N}|f_{n}|^{2}\right)^{1/2}\right\|_{L^{p}(X,\mu )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d070d90418ef91490afa65f1fd532ec5343a1b9)
for some constant
depending only on
and
.[citation needed]
Generalizations
For the case of Rademacher random variables, Pawel Hitczenko showed[1] that the sharpest version is:
![{\displaystyle A\left({\sqrt {p}}\left(\sum _{n=b+1}^{N}x_{n}^{2}\right)^{1/2}+\sum _{n=1}^{b}x_{n}\right)\leq \left(\operatorname {E} \left|\sum _{n=1}^{N}\varepsilon _{n}x_{n}\right|^{p}\right)^{1/p}\leq B\left({\sqrt {p}}\left(\sum _{n=b+1}^{N}x_{n}^{2}\right)^{1/2}+\sum _{n=1}^{b}x_{n}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c76197417bf7a9f65cd7b80452bcae414b4959b8)
where
, and
and
are universal constants independent of
.
Here we assume that the
are non-negative and non-increasing.
See also
References
- ^ Pawel Hitczenko, "On the Rademacher Series". Probability in Banach Spaces, 9 pp 31-36. ISBN 978-1-4612-0253-0
- Thomas H. Wolff, "Lectures on Harmonic Analysis". American Mathematical Society, University Lecture Series vol. 29, 2003. ISBN 0-8218-3449-5
- Uffe Haagerup, "The best constants in the Khintchine inequality", Studia Math. 70 (1981), no. 3, 231–283 (1982).
- 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.