Prove $P(\{X\ge t\}\cup\{Y\ge t\})\le t^{-2}(1+\sqrt{1-r^2})$ where $r$ is the correlation of $X$ and $Y$ centered with unit variance

Question

Let $\xi$ and $\eta$ be random variables with variances $\mathbb{D}\xi$ and $\mathbb{D}\eta$, and correlation coefficient $\rho$. Show that $$\mathbb{P}(\{\xi - \mathbb{E}\xi \geq \varepsilon \sqrt{\mathbb{D}\xi} \} \cup \{ \eta - \mathbb{E}\eta \geq \varepsilon \sqrt{\mathbb{D}\eta}) \leq \frac{1}{\varepsilon^2}(1+\sqrt{1-\rho^2}).$$

So, if I am right, I see a connection with $$\mathbb{E}\max\{\xi^2,\eta^2\}\leq 1 + \sqrt{1-\rho^2}.$$ But actually don't know how to use it here.

Answer 1

WLOG after renormalizing, both variables have mean zero and dispersions equal to one, so we can write this as $$ \def\e{\varepsilon} P(\xi\ge \e\cup \eta\ge \e)=P(\max(\xi,\eta)\ge\e)\le P(\max(\xi^2,\eta^2)\ge \e^2). $$ Now, use Markov's inequality to relate this to $E[\max(\xi^2,\eta^2)]$.