# 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 ${a\in\ell^2}$.