# Normalized concentration inequality for empirical CDF (iid sum)

Consider the empirical and population CDF, $$F_n(t) = \frac{1}{n} \sum_{i=1}^n 1\{X_i \leq t\} \quad \mbox{and} \quad F(t) = \mathbb{E} [F_n(t)],$$ where above $$X_1, \dots, X_n$$ are iid, real-valued random variables and the expectation is taken with respect to the same distribution.

It is well-known (the DKW inequality with sharp constants due to Massart) that $$\mathbb{P}\big(\sqrt{n} \|F_n - F\|_\infty > \lambda\big) \leq 2 \exp(-2\lambda^2), \quad \mbox{for all}~\lambda > 0.$$ Above, $$\|\cdot\|_\infty$$ denotes the supremum norm.

I am wondering if there is a normalized version of this inequality available. Specifically, define $$\widehat{ \sigma}_n(t) = F_n(t) (1 - F_n(t))$$ This can be seen as a sample approximation to the true variance since $$\widehat{ \sigma}_n(t) \to F(t)(1- F(t))$$ a.s. by continuous mapping and the strong law as $$n \to \infty$$. The quantity $$F(t) (1-F(t))$$ is the variance of the random variable $$B(t) := 1\{X \leq t\}$$.

Question: Consider the normalized quantity $$\sup_{t} \frac{|F_n(t) - F(t)|}{\sqrt{\widehat{\sigma}_n(t)}}$$ Does a similar inequality exist?

If $$F(t)=1/n$$, the Poisson/binomial approximation gives that $$Y_n = nF_n(t)$$ is Binomial($$n,\frac1n$$) and converges to Poisson(1), say in total variation distance. In particular, $$P(Y_n=0)=P(F_n(t)=0)\to e^{-1}$$, the pmf of Poisson(1) at 0. The ratio is infinite with constant probability for a single $$t$$ unless a different normalization is used or the range of $$t$$ is restricted.