I have $X \sim N(0,1)$ and $Y_n\to0$ a.s. ($X$ and $Y_n$ may be dependent). For $t>0$, when does $$\frac{P(X+Y_n/n>nt)}{P(X>nt)} \to 1$$ as $n\to\infty$. Maybe you have some hints or references?
2026-03-29 14:08:08.1774793288
Convergence rate of tail probability
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