Exercise 1.43 (Minkowski inequality)

Question

If $1 < p < \infty$ and $X,Y$ are scalar random variables with $\|X\|_p, \|Y\|_p < \infty$, use Hölder’s inequality to establish that
$$\displaystyle {\bf E} |X| |X+Y|^{p-1} \leq \|X\|_p \|X+Y\|_p^{p-1}$$
and
$$\displaystyle {\bf E} |Y| |X+Y|^{p-1} \leq \|Y\|_p \|X+Y\|_p^{p-1}$$
and then conclude the Minkowski inequality
$$\displaystyle \|X+Y\|_p \leq \|X\|_p + \|Y\|_p.$$
Show that this inequality is also valid at the endpoint cases $p=1$ and $p=\infty$.

Elu Thingol · Accepted Answer

We have
\begin{align*}
\|X+Y\|^p &= {\bf E} |X+Y|^p\
	&= {\bf E}|X+Y|\cdot |X+Y|^{p-1}\
	&\leq {\bf E}|X|\cdot |X+Y|^{p-1} + |Y|\cdot |X+Y|^{p-1} &	ext{Triangle inequality}\
	&\leq \|X\|\|X+Y\|^{p-1} + \|Y\|\|X+Y\|^{p-1} &	ext{Hölder inequality}\
	&= \left(\|X\| + \|Y\|ight) \|X+Y\|^{p-1}
\end{align*}
Dividing both sides by the non-negative factor $\|X+Y\|^{p-1}$ yields the desired result.

Exercise 1.43 (Minkowski inequality)

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Exercise 1.43 (Minkowski inequality)

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