Probability bound for complex random variable

Question

Let $Z=X+iY$ be complex random variable. Consider the following probability $$\mathrm{Pr}\left( | Z|\geq \epsilon \right)$$ I want to bound this in terms of the real and imaginary parts of $Z$, i.e. suppose that I have bounds for $\mathrm{Pr}\left( |X|\geq \epsilon \right)\leq C(\epsilon)$ and $\mathrm{Pr}\left( | Y|\geq \epsilon \right)\leq C'(\epsilon)$. For starters, \begin{align} \mathrm{Pr}\left( | Z|\geq \epsilon \right) &=\mathrm{Pr}\left( |X+iY|\geq \epsilon \right) \\ &= \mathrm{Pr}\left( \sqrt{X^2+Y^2}\geq \epsilon \right)\\ &= \mathrm{Pr}\left( X^2+Y^2\geq \epsilon^2 \right) \end{align} How to proceed? Context: I am trying to understand how to lift the standard Chernoff bound to complex random variables. (see also in this MO question )

Answer 1

$P(|Z| \geq \epsilon) \leq P(|X|\geq \epsilon /2)+P(|Y|\geq \epsilon /2)$ because $|Z| \leq |X|+|Y|$.

[ The event $(|Z| \geq \epsilon) $ is contained in the union of the events $(|X| \geq \epsilon) $ and $(|Y| \geq \epsilon) $].