Let $(X_n)_n$, $(Y_n)_n$, $(Z_n)_n$ be sequence of random variables (no assumptions of independence whatsoever). Suppose that $Y_n \to N(0, 1)$, $Y_n \to N(0, 1)$, and $Z_n \to N(0,1)$ in distribution. Let $\epsilon \ge 0$.
Question. What is a nontrivial lower-bound for $p := \lim\sup_n P(|X_n| / (Y_n^2+Z_n^2) \le \epsilon)$ ?
Note. If I had $Y_n \to y$ and $Z_n \to z$ in probability (for constants $y,z \in \mathbb R$), then I could use Slutsky's theorem to get $p = P(|N(0,1)| \le \epsilon)$, and then concentration arguments would give $p \ge 1 - 2e^{-\epsilon^2/2}$.