Suppose $Z \sim N(0,1)$. First note that $\begin{align*} t\mathbb{P}[Z \ge t]&= \int _{t}^\infty t\phi(z)~ dz \\ &\le \int _{t}^\infty z\phi(z)~ dz\\ &= \phi(t). \end{align*}$where $\phi$ is the standard Gaussian density. Therefore, $\mathbb{P}[Z \ge t] \le \phi(t) / t.$ Similarly, consider $\begin{align*} \frac{\mathbb{P}[Z \ge t]}{t^{2}}&= \int _{t}^{\infty} \frac{\phi(z)}{t^{2}} ~ dz \\ &\ge \int _{t}^{\infty} \frac{\phi(z)}{z^{2}} ~ dz \\ &= [-\phi(z) / z]_{t}^{\infty}-\int _{t}^{\infty}\phi(z) ~ dz\\ &= \frac{\phi(t)}{t} -\mathbb{P}[Z \ge t]. \end{align*}$ Rearranging gives $\mathbb{P}[Z \ge t] \ge \frac{\phi(t)}{t}\cdot\left( 1+\frac{1}{t^{2}} \right)^{-2}=\frac{\phi(t)}{t}\left( 1-\frac{1}{t^{2}+1} \right).$ Therefore, > [!lemma|*] Approximation of Gaussian Quantiles > We can achieve $O(t^{-2})$ accuracy with the approximation $\mathbb{P}[Z \ge t]\approx \frac{\phi(t)}{t},$and the (upper) $\alpha$-quantile approximation is then $z(\alpha) \approx \text{root of }t\alpha= \phi (t).$ Using this bound, we can derive a limiting behavior of Gaussian quantiles: fixing any $\alpha > 0$, and we study $t(n):=z(\alpha / n)$ for large $n$. The above approximation gives $\begin{align*} t =z(\alpha / n) &\sim t\text{ solves } \big\{t\alpha / n = \phi(t)\big\}\\ &\sim t \text{ solves } \left\{ \log t-\log n+\mathrm{const.}= -\frac{t^{2}}{2} \right\}\\ &\sim t \text{ solves } \left\{ \log n= \frac{t^{2}}{2} \right\}\\ &\iff t=\sqrt{ 2\log n }. \end{align*}$Here $\sim$ means "roughly equivalent", where we drop slow-growing terms like $\log t,\log n$.