The one-dimensional empirical distribution function is the complete prototype of empirical-process theory. Monotonicity yields an elementary uniform law; indicator covariances yield a Brownian bridge; continuous functionals of that bridge produce classical goodness-of-fit limits.
Distribution functions and their empirical versions
Definition. For real i.i.d. observations with distribution function \(F\), \[ F_n(t)=\frac1n\sum_{i=1}^n1_{\{X_i\le t\}}, \qquad \alpha_n(t)=\sqrt n\{F_n(t)-F(t)\}. \] Thus \(\alpha_n=\mathbb G_n\) on the indicator class \[ \mathcal F_0=\{1_{(-\infty,t]}:t\in\mathbb R\}. \]
Finite-sample moments. For \(s,t\in\mathbb R\), \[ \mathbb E\alpha_n(t)=0, \] \[ \operatorname{Cov}\{\alpha_n(s),\alpha_n(t)\} =F(s\wedge t)-F(s)F(t). \]
Proof. Apply the empirical-process covariance formula to \[ f_s=1_{(-\infty,s]},\qquad f_t=1_{(-\infty,t]}. \] Their product is \(f_{s\wedge t}\), so \[ P(f_sf_t)-Pf_sPf_t =F(s\wedge t)-F(s)F(t). \]
Glivenko-Cantelli theorem for the EDF
Glivenko-Cantelli theorem. For every distribution function \(F\), \[ \|F_n-F\|_\infty =\sup_{t\in\mathbb R}|F_n(t)-F(t)| \to0 \quad\text{almost surely}. \]
Proof. Fix \(m\ge1\). Choose points \[ -\infty=t_0\le t_1\le\cdots\le t_m=\infty \] so that each increment \(F(t_j-)-F(t_{j-1})\le1/m\); generalized quantiles provide such a grid even when \(F\) has atoms, with atom endpoints included separately. For \(t_{j-1}\le t<t_j\), monotonicity gives \[ F_n(t_{j-1})-F(t_j-) \le F_n(t)-F(t) \le F_n(t_j-)-F(t_{j-1}). \] Therefore \[ \|F_n-F\|_\infty \le \max_j|F_n(t_j)-F(t_j)| +\max_j|F_n(t_j-)-F(t_j-)| +\frac1m. \] For fixed \(m\), the finitely many empirical probabilities converge almost surely by the strong law. Hence the limsup is at most \(1/m\); let \(m\to\infty\).
Distribution-free exponential bound
Dvoretzky-Kiefer-Wolfowitz inequality. For every \(n\), every distribution function \(F\), and every \(\varepsilon>0\), \[ P\{\|F_n-F\|_\infty>\varepsilon\} \le2e^{-2n\varepsilon^2}. \]
The constant two is sharp. The inequality yields the finite-sample confidence band \[ F_n(t)\pm \sqrt{\frac{\log(2/\alpha)}{2n}}, \] clipped to \([0,1]\), with simultaneous coverage at least \(1-\alpha\). Its proof uses the ordering structure of half-lines and is stronger than a naive union bound over all \(t\).
Brownian bridge
Definition. If \(W\) is standard Brownian motion on \([0,1]\), then \[ B(u)=W(u)-uW(1) \] is a standard Brownian bridge.
Bridge covariance and pinning. \[ B(0)=B(1)=0 \] and \[ \operatorname{Cov}\{B(u),B(v)\} =u\wedge v-uv. \]
Proof. The endpoint identities are immediate. Using \(\operatorname{Cov}\{W(u),W(v)\}=u\wedge v\), \[ \begin{aligned} \operatorname{Cov}\{B(u),B(v)\} &=u\wedge v-vu-u v+uv\\ &=u\wedge v-uv. \end{aligned} \]
Donsker theorem for the EDF
Classical Donsker theorem. In \(\ell^\infty(\mathbb R)\), \[ \alpha_n\rightsquigarrow B_F, \qquad B_F(t)=B\{F(t)\}. \] The limit is centered Gaussian with covariance \[ F(s\wedge t)-F(s)F(t). \]
Proof outline. The multivariate central limit theorem gives finite-dimensional convergence with the displayed covariance. For tightness, reduce to the uniform empirical process by the generalized probability transform, then partition \([0,1]\) into small intervals. Binomial maximal bounds control oscillations of the empirical counts within each interval, while the bridge metric \[ \rho(s,t)^2=|s-t|-(s-t)^2 \] is small on short intervals. This proves asymptotic equicontinuity and hence process convergence.
Probability integral transform
Continuous case. If \(F\) is continuous and \(X\sim F\), then \(U=F(X)\sim U[0,1]\). Moreover, \[ \sup_t|\alpha_n(t)| =\sup_{0\le u\le1} \left|\sqrt n\{G_n(u)-u\}\right|, \] where \(G_n\) is the empirical distribution of \(U_1,\ldots,U_n\).
Proof. For a generalized inverse \(F^{-1}\), \[ P\{F(X)\le u\}=u \] under continuity, and threshold events correspond through \(t=F^{-1}(u)\). Flat portions do not change either supremum because no observations or probability mass lie inside them.
When \(F\) has atoms, \(F(X)\) is not uniform and \(B\circ F\) has jumps in its index representation. A randomized transform \[ U=F(X-)+V\{F(X)-F(X-)\}, \qquad V\sim U[0,1], \] restores uniformity.
Kolmogorov-Smirnov and Cramer-von Mises limits
Corollary. Under a fully specified continuous \(F\), \[ \sqrt n\sup_t|F_n(t)-F(t)| \rightsquigarrow\sup_{0\le u\le1}|B(u)|, \] and \[ n\int\{F_n(t)-F(t)\}^2\,\mathrm{d}F(t) \rightsquigarrow\int_0^1B(u)^2\,\mathrm{d}u. \]
Proof. The supremum map is Lipschitz on \(\ell^\infty\). The integral-square map is continuous on uniformly bounded sets under uniform convergence: \[ \left|\int z_n^2-\int z^2\right| \le\|z_n-z\|_\infty (\|z_n\|_\infty+\|z\|_\infty). \] Apply continuous mapping to the Donsker theorem.
If parameters of \(F_\theta\) are estimated, the limit is the bridge minus a score projection and is no longer distribution-free. That composite-null correction is treated with goodness-of-fit and bootstrap processes later in the note.