Exercise 1.41 ($L^p$-norm is non-decreasing in $p$)

Question

Show that the expressions $\|X\|_p$ are non-decreasing in $p$ for $p \in (0,+\infty]$. In particular, if $\|X\|_p$ is finite for some $p$, then it is automatically finite for all smaller values of $p$.

Elu Thingol · Accepted Answer

Let $p,q\in (0,+\infty]$ such that $p < q$. The trick is to apply Jensen's inequality. Recall that $x\mapsto x^{a}$ is convex for any $a>0$, in particular $f(x):= x^{\frac{q}{p}}$ is convex. By Jensen's inequality
$$f({\bf E}|X|^p) = \left({\bf E}|X|^p\right)^{\frac{q}{p}} \leq {\bf E}f(|X|^p) = {\bf E}|X|^q$$
from which we deduce
$$\left({\bf E}|X|^p\right)^{\frac{q}{p}} \leq {\bf E} |X|^q$$
so that
$$\left({\bf E}|X|^p\right)^{\frac{1}{p}} \leq \left({\bf E} |X|^q\right)^{\frac{1}{q}}$$
as desired.

Exercise 1.41 ($L^p$-norm is non-decreasing in $p$)

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Exercise 1.41 ($L^p$-norm is non-decreasing in $p$)

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