This derivation does not use Brownian motions. Suppose we have a fixed percentile \(p\) and a r.v. \(X\) with true CDF \(F\). Then we can define the true \(p\)-th quantile as \(q_p\) where \(F(q_p) = p\). Given samples \(X_1, \dots, X_n\), we can define the sample \(p\)-th quantile as \(\hat{q}_p\) where empirical CDF \(F_{n}(\hat{q}_p) = \frac{1}{n} \sum_{i = 1}^n \mathbf{1} \{ X_i \leq \hat{q}_p \} = p\). We aim to understand the asymptotic distribution of \(\sqrt{n}(\hat{q}_p - q_p)\). Note however that \(F_n(q_p)\) does not necessarily equal \(p = F_n(\hat{q}_p)\), although they should be close. Intuitively then, we first begin with a Taylor Series expansion of \(F_n(\hat{q}_p)\) around \(q_p\):
\[p = F_n(\hat{q}_p) \approx F_n(q_p) + F'_n(q_p)[\hat{q}_p - q_p] = F_n(q_p) + f(q_p)[\hat{q}_p - q_p]\] \[\implies p \approx F_n(q_p) + f(q_p)[\hat{q}_p - q_p] \implies \sqrt{n}[\hat{q}_p - q_p] \approx \frac{\sqrt{n}[p - F_n(q_p)]}{f(q_p)}\]We’ll be interested in having some convergence in distribution argument for the numerator. To do we use, we use Donsker’s Theorem, presented below without proof:
Donsker’s Theorem
Suppose we have IID samples \(X_1, \dots, X_n\) with a corresponding empirical CDF \(F_n\). Then for a fixed \(x\) we have that:
\[\sqrt{n}[F_n(x) - F(x)] \overset{d}{\to} \mathcal{N}(0, F(x)[1 - F(x)])\]
Applying Donsker’s Theorem above with \(x = q_p\), we arrive at the following:
\[\sqrt{n}[F_n(q_p) - p] \overset{d}{\to} \mathcal{N}(0, p(1 - p))\]and so with Slutsky’s Lemma we have:
\[\sqrt{n}[\hat{q}_p - q_p] \approx \frac{\sqrt{n}[p - F_n(q_p)]}{f(q_p)} \overset{d}{\to} \boxed{\mathcal{N}(0, \frac{p(1 - p)}{f(q_p)^2})}\]