Bounding an integral involving tail probabilities

Question

I am currently reading through Van der Vaart and Wellner's book on empirical process theory. In chapter 2.9, they define the quantity $$||\xi||_{2,1} \equiv \int_0^\infty \sqrt{P(|\xi| > x)}\,\mathrm dx$$ They then claim in their problem 2.9.2 that while this is not a norm in the sense of not satisfying the triangle inequality, we nonetheless have the bound $$||\xi+\eta||_{2,1}^2 \leq 4||\xi||_{2,1}^2 + 4||\eta||_{2,1}^2$$ which is sufficient for the purposes of the results shown in the chapter. I am at a loss for how to tackle this problem. I have tried a few approaches, each of which seems to lead to a dead end. Directly bounding $P(|\xi + \eta| > x)$ in terms of useful properties appears to be difficult, largely owing to the fact that it is hard to get a handle on the cases where $|\xi| < x, |\eta| < x$, but their sum is $> x$. I also had an idea that because the square root function monotonically maps $[0,1]$ to itself, we could associate to each random variable $\xi$ a $\xi'$ that satisfies $P(|\xi'| > x)^2 - P(|\xi| > x)$, which would imply that $||\xi||_{2,1} = ||\xi||_1$, but this does not appear to be of much help either, as it is unclear how $\xi' + \eta'$ defined this way would relate to $(\xi + \eta)'$. I would appreciate any hints about how to proceed.

Answer 1

On the one hand I can bound the probability in $\|\xi+\eta\|_{2,1}$ as: \begin{align} \mathbf{P}(|\xi+\eta|>x)&\leq\mathbf{P}(|\xi|+|\eta|>x)\\ &\leq\mathbf{P}(\{|\xi|>x/2\}\cup\{|\eta|>x/2\})\\ &\leq\mathbf{P}(|\xi|>x/2)+\mathbf{P}(|\eta|>x/2) \end{align} Then the pseudo-norm can be bounded as: \begin{align} \|\xi+\eta\|_{2,1}&=\int_0^\infty\sqrt{\mathbf{P}(|\xi+\eta|>x)}dx\\ &\leq\int_0^\infty\sqrt{\mathbf{P}(|\xi|>x/2)+\mathbf{P}(|\eta|>x/2)}dx\\ &\leq \int_0^\infty\sqrt{\mathbf{P}(|\xi|>x/2)}dx+\int_0^\infty\sqrt{\mathbf{P}(|\eta|>x/2)}dx\\ &=2\left[\int_0^\infty\sqrt{\mathbf{P}(|\xi|>t)}dt+\int_0^\infty\sqrt{\mathbf{P}(|\eta|>t)}dt\right]\\ &=2\left(\|\xi\|_{2,1}+\|\eta\|_{2,1}\right) \end{align} On the other hand, I can use the Cauchy Schwarz inequality $|a|+|b|\leq\sqrt{a^2+b^2}\sqrt{1^2+1^2}=\sqrt{2}\sqrt{a^2+b^2}$ as \begin{align} \|\xi+\eta\|_{2,1}^2&\leq4\left(\|\xi\|_{2,1}+\|\eta\|_{2,1}\right)^2\\ &\leq8(\|\xi\|_{2,1}^2+\|\eta\|_{2,1}^2) \end{align} Unfortunally I can't bound with 4 instead 8.