Khinchin inequality

for independent functions

An estimate in $ L _ {p} $ of the sum of independent functions (cf. Independent functions, system of). Suppose that $ \{ f _ {k} \} $ is a system of independent functions and that for some $ p > 2 $,

$$ \sup _ { k } \ \| f _ {k} \| _ {L _ {p} } < \infty ,\ \ \int\limits _ { 0 } ^ { 1 } f _ {k} ( t) dt = 0. $$

Then

$$ \left \| \sum _ {k = 0 } ^ \infty c _ {k} f _ {k} \right \| _ {L _ {p} } \leq \ M \left ( \sum _ {k = 1 } ^ \infty c _ {k} ^ {2} \right ) ^ {1/2} . $$

If

$$ \sum _ {k = 1 } ^ \infty c _ {k} ^ {2} < \infty , $$

$ r _ {k} ( t) = \mathop{\rm sign} \sin 2 ^ {k} \pi t $ is a Rademacher function (cf. Rademacher system) and if

$$ f ( t) = \sum _ {k = 1 } ^ \infty c _ {k} r _ {k} ( t), $$

then for any $ p > 0 $,

$$ A _ {p} \left ( \sum _ {k = 1 } ^ \infty c _ {k} ^ {2} \right ) ^ {1/2} \leq \ \left ( \int\limits _ { 0 } ^ { 1 } | f ( t) | ^ {p} \ dt \right ) ^ {1/p} \leq \ B _ {p} \left ( \sum _ {k = 1 } ^ \infty c _ {k} ^ {2} \right ) ^ {1/2} , $$

where $ B _ {p} = O ( \sqrt p ) $ as $ p \rightarrow \infty $. This inequality was established by A.Ya. Khinchin in [1]. The exact value of $ A _ {1} $ is $ 1/2 $.

An analogue of the Khinchin inequality is valid in Banach spaces [4]. There exists a constant $ C ( p, q) $, $ 0 < p, q < \infty $, such that for any element $ x _ {k} $ in a Banach space $ E $,

$$ \left \| \left \| \sum _ {k = 1 } ^ \infty x _ {k} r _ {k} ( t) \ \right \| _ {E} \right \| _ {L _ {p} } \leq \ C ( p, q) \left \| \left \| \sum _ {k = 1 } ^ \infty x _ {k} r _ {k} ( t) \ \right \| _ {E} \right \| _ {L _ {q} } . $$

One of the numerous applications of the Khinchin inequality is as follows: If

$$ \sum _ {k = 1 } ^ \infty a _ {k} ^ {2} + b _ {k} ^ {2} < \infty , $$

then for almost-all choices $ \pm 1 $ the function

$$ \sum _ {k = 1 } ^ \infty \pm ( a _ {k} \cos kt + b _ {k} \sin kt) $$

belongs to all $ L _ {p} $, $ p < \infty $( see [5]).

References

[1]	A.Ya. Khinchin, "Ueber dyadische Brüche" Math. Z. , 18 (1923) pp. 109–116
[2]	S. Karlin, "Orthogonal properties of independent functions" Trans. Amer. Math. Soc. , 66 (1949) pp. 44–64
[3]	V.F. Gaposhkin, "Lacunary series and independent functions" Russian Math. Surveys , 21 : 6 (1966) pp. 1–82 Uspekhi Mat. Nauk , 21 (1966) pp. 3–82
[4]	J.-P. Kahane, "Some random series of functions" , Cambridge Univ. Press (1985)
[5]	A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)