The Khintchine Inequality

For a Rademacher sequence {X=(X_1,X_2,\ldots)} and square summable sequence of real numbers {a=(a_1,a_2,\ldots)}, the Khintchine inequality provides upper and lower bounds for the moments of the random variable,

\displaystyle  a\cdot X=a_1X_1+a_2X_2+\cdots.

We use {\ell^2} for the space of square summable real sequences and

\displaystyle  \lVert a\rVert_2=\left(a_1^2+a_2^2+\cdots\right)^{1/2}

for the associated Banach norm.

Theorem 1 (Khintchine) For each {0 < p < \infty}, there exists positive constants {c_p,C_p} such that,

\displaystyle  c_p\lVert a\rVert_2^p\le{\mathbb E}\left[\lvert a\cdot X\rvert^p\right]\le C_p\lVert a\rVert_2^p, (1)

for all {a\in\ell^2}.

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