$\mathbb{E}[|X|^n] < +\infty \implies \mathbb{E}[|X|^k] < +\infty, k \leq n$

Question

Show that if $\mathbb{E}[|X|^n] < +\infty$, then $\mathbb{E}[|X|^k] < +\infty, \forall k \leq n$.

I guess I have to apply Hölder Inequality, but I was not able to find out what $p$ and $q$ are appropriate.

I would appreciate some hint or help!

Answer 1

Let $0<k\le n$ and define $p:=\tfrac {n}{k}$.

Then $q=:\tfrac {p}{p-1}$ is the Hölder conjugate of $p$. Applying Hölder's inequality to the random variables $|X|^{r}$ and $\mathbb{1}$ (the random variable constantly $1$) we obtain $\mathbb {E} [|X|^{k}]\le (\mathbb {E} [|X|^{n}])^{\tfrac {k}{n}}< +\infty$.

Another possible approach is via Jensen Inequality.

Answer 2

If $n$ and $k$ are positive numbers with $k<n$ then $$ |X|^k\le 1+|X|^n $$ as you see immediately by considering the cases $|X|\le 1$ and $|X|>1$ separately.