Suppose $X\in \mathbb{R}$ is a random variable with expected value $\mathbb{E}X = \mu$. I ran across a proof which uses the inequality $$ \mathbb{E}[|X - \mu|^3] \leq 2^3 \mathbb{E}|X|^3. $$ Can anyone help me understand where this inequality is coming from? Is it correct?


1 Answer 1


Let $p\ge 1.$ The $L^p$ norm of a random variable is defined as

$$|X|_p = \left(E[|X|^p]\right)^{1/p}.$$

Minkowski's Inequality says this norm satisfies the triangle inequality. Apply it to the random variables $X$ and $\mu=E[X]$ (a number that can be considered a constant random variable) after observing

$$|\mu|_p=|\mu| \le E[|X|] = |X|_1$$

to obtain

$$|X-\mu|_p \le |X|_p + |\mu|_p \le |X|_p + |X|_1 \le |X|_p+|X|_p = 2|X|_p.\tag{*}$$

The last inequality applies the power norm inequality $|X|_r \le |X|_p$ (whenever $r \le p$) to the case $r=1.$

When $p=3,$ cubing both sides of $(*)$ is equivalent to the inequality in the question.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.