How to prove $E\|Y'\|\leq E\|Y'-Y''\|,$ where $Y$ is a random matrix in $\mathbb{R}^{n\times n}$, and $Y'$, $Y''$ are independent copies of $Y$?

Question

How to prove $$E\|Y'\|\leq E\|Y'-Y''\|,$$ where $Y$ is a random matrix in $\mathbb{R}^{n\times n}$, and $Y'$, $Y''$ are independent copies of $Y$; $\|\cdot\|$ denotes the $l_2$ operator norm；$E$ denotes the expectation.

I have a clue, this inequality should follow Jensen's inequality and Fubini's theorem. But I don't get the details. Can anyone help to make up the details?

Thank you very much！

Answer 1

the $l_2$ norm of rv $Y$ is equivalent to $E[Y^2]^{\frac{1}{2}}$ thus you must prove that

$$E[Y_1^2]\leq E[(Y_1-Y_2)^2]=E[Y_1^2]-2E[Y_1Y_2]+E[Y_2^2]$$

letting $E[Y^2]=a$ and $E[Y]=b$ you would be proving that

$$a \leq 2(a-b)$$

with Jensens you have $a\geq b$ but I dont know how you could prove statement above is generally true.