Khintchine inequality

Remarks

For $p = 2$ the middle term equals $\left(\sum a_k^2\right)^{1/2}$ exactly, since $\mathbb{E}[\varepsilon_j \varepsilon_k] = 0$ for $j \ne k$. So one can take $A_p = 1$ for $p \ge 2$ and $B_p = 1$ for $p \le 2$, by monotonicity of $L^p$ norms on a probability space.

Proof idea for the upper bound: normalize to $\sum a_k^2 = 1$. Since $\cosh t \le e^{t^2/2}$, independence gives $\mathbb{E} \exp\left(t \sum a_k \varepsilon_k\right) \le e^{t^2/2}$ for all real $t$. This gives the tail bound $\Pr\left(\abs{\sum a_k \varepsilon_k} > s\right) \le 2 e^{-s^2/2}$, and integrating the tail bounds every moment. The lower bound for $p < 2$ follows from the upper bound for some $q > 2$ by Hölder's inequality, which places $L^2$ between $L^p$ and $L^q$.

Example: for $N = 2$, $a_1 = a_2 = 1$ and $p = 1$, the sum $\varepsilon_1 + \varepsilon_2$ is $\pm 2$ with probability $1/2$ and $0$ otherwise, so $\mathbb{E}\abs{\varepsilon_1 + \varepsilon_2} = 1 = \frac{1}{\sqrt 2} \cdot \sqrt 2$. This shows that $A_1 \le 1/\sqrt 2$; Szarek proved that $A_1 = 1/\sqrt 2$ is the best constant.

Going deeper

For a student

In harmonic analysis the $\varepsilon_k$ are realized as the Rademacher functions $r_k(t) = \operatorname{sgn} \sin(2^k \pi t)$ on $[0, 1]$, and the inequality says that all $L^p$ norms are equivalent on their span. Combined with Fubini's theorem it turns square-function estimates into estimates for random sums, a basic step in Littlewood–Paley theory. In Banach space theory it is the starting point of the notions of type and cotype.