Derive bounds of moments from probability tails

Question

Consider a random variable $X\in\mathbb{R}^d$ and its Euclidean norm $\|X\|$. By Markov's inequality, we have for $t\ge 0$, $$\mathbb{P}\{\|X\|\ge t\}\le\frac{\mathbb{E}\|X\|^n}{t^n},$$
so one can easily derive an exponential tail bound if an upper bound of $\mathbb{E}\|X\|^n$ is known. I'm curious if one can do it the other way, that is, deriving an upper bound of $\mathbb{E}\|X\|^n$ using a tail bound of $\|X\|$ or some concentration inequalities of $X$.

Answer 1

You can use the tail value formula to write the expectation in terms of an integral of the tail probability: $$ \mathbb{E} \lVert X \rVert^n = \int^\infty_0 \mathbb{P}(\lVert X \rVert^n \geq s) \,\mathrm{d}s = \int^\infty_0 \mathbb{P}(\lVert X \rVert \geq t) n t^{n-1} \,\mathrm{d}t . $$ by change of variable $s = t^n$. If you have good concentration for $\lVert X \rVert$, then you may be able to derive a good bound for $\mathbb{E} \lVert X \rVert^n$.