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?
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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.