Donsker's invariance principle lifts the scalar central limit theorem to an entire path: after diffusive rescaling, a random walk converges to Brownian motion. Centering a path by its endpoint produces the Brownian bridge, the universal Gaussian limit of empirical distribution functions.
Partial-sum processes
Definition. Let \(X_i\) be i.i.d. with \(\mathbb EX_i=0\) and \(\mathbb EX_i^2=1\), and put \(S_k=\sum_{i=1}^kX_i\). The step process is \[ W_n^0(t)=\frac{S_{\lfloor nt\rfloor}}{\sqrt n}, \qquad0\le t\le1, \] regarded as an element of \(D[0,1]\). Its polygonal interpolation is \[ W_n(t) = \frac{S_{\lfloor nt\rfloor}}{\sqrt n} +(nt-\lfloor nt\rfloor) \frac{X_{\lfloor nt\rfloor+1}}{\sqrt n}, \] an element of \(C[0,1]\).
Lemma. Under the finite-variance assumption, \[ \|W_n-W_n^0\|_\infty \le\frac{\max_{1\le i\le n}|X_i|}{\sqrt n} \to0 \quad\text{in probability}. \]
Proof. The deterministic bound follows from interpolation. For \(\varepsilon>0\), \[ P\!\left(\max_{i\le n}|X_i|>\varepsilon\sqrt n\right) \le nP(|X_1|>\varepsilon\sqrt n). \] Finite second moment implies \(x^2P(|X_1|>x)\to0\); setting \(x=\varepsilon\sqrt n\) makes the right side tend to zero.
Donsker's invariance principle
Theorem (Donsker). Under the assumptions above, \[ W_n^0\Rightarrow B \quad\text{in }D[0,1]\text{ with }J_1, \] and \[ W_n\Rightarrow B \quad\text{in }C[0,1]\text{ with the uniform topology}. \]
Proof outline. For \(0=t_0<t_1<\cdots<t_m\), the increments \[ W_n(t_j)-W_n(t_{j-1}) \] are asymptotically independent normalized sums over disjoint index blocks. The multivariate CLT gives a centered Gaussian limit with independent increments and variances \(t_j-t_{j-1}\), hence Brownian finite-dimensional distributions. If \(\mathbb EX_1^4<\infty\), independence gives a fourth-moment bound \[ \mathbb E|W_n(t)-W_n(s)|^4 \le C|t-s|^2+C n^{-1}|t-s|, \] and blocking controls the interpolation-scale remainder. This yields tightness through the modulus-of-continuity criterion. For only a finite second moment, truncate at \(\varepsilon\sqrt n\), center the truncated variables, apply the bounded-increment argument, and show the deleted array is negligible using \(nP(|X_1|>\varepsilon\sqrt n)\to0\). Letting \(\varepsilon\downarrow0\) gives tightness. Prokhorov's theorem provides subsequential limits. Every such limit has continuous paths and the Brownian finite-dimensional laws, so it is Wiener measure. Uniqueness forces convergence of the full sequence. The interpolation lemma transfers between the \(D\) and \(C\) versions.
Continuous functionals of random walks
Corollary. \[ \frac1{\sqrt n}\max_{0\le k\le n}S_k \Rightarrow\sup_{0\le t\le1}B_t \stackrel d=|N(0,1)|, \] and \[ \frac1{n^{3/2}}\sum_{k=1}^nS_k \Rightarrow\int_0^1B_t\,\mathrm{d} t \sim N(0,1/3). \]
Proof. The supremum and integration maps are continuous on \(C[0,1]\), so the continuous mapping theorem applies. The reflection principle gives the maximum law. The integral is Gaussian and \[ \operatorname{Var}\!\left(\int_0^1B_t\,\mathrm{d} t\right) = \int_0^1\int_0^1(s\wedge t)\,\mathrm{d} s\,\mathrm{d} t =\frac13. \]
The Brownian bridge
Definition. The standard Brownian bridge is \[ \mathbb B_t=B_t-tB_1,\qquad0\le t\le1. \] It is centered Gaussian, satisfies \(\mathbb B_0=\mathbb B_1=0\), and has covariance \[ \mathbb E[\mathbb B_s\mathbb B_t] =s\wedge t-st. \]
Proof. Linear transformations preserve Gaussianity. Expanding covariance gives \[ s\wedge t-t\,s-s\,t+st=s\wedge t-st. \]
Proposition. The process \(\mathbb B\) is independent of \(B_1\), and its law is the regular conditional law of Brownian motion given \(B_1=0\).
Proof. For every \(t\), \[ \operatorname{Cov}(\mathbb B_t,B_1) =t-t=0. \] Every finite bridge vector is jointly Gaussian with \(B_1\), so zero covariance implies independence. The Gaussian conditioning formula gives \[ \mathbb E[B_t\mid B_1=x]=tx, \] and conditional covariance \(s\wedge t-st\). Setting \(x=0\) identifies the bridge.
Empirical distribution functions
Theorem (empirical Donsker theorem). If \(U_i\) are i.i.d. uniform on \([0,1]\) and \[ F_n(t)=\frac1n\sum_{i=1}^n1_{\{U_i\le t\}}, \] then \[ \alpha_n(t)=\sqrt n\{F_n(t)-t\} \Rightarrow\mathbb B_t \quad\text{in }\ell^\infty[0,1]. \]
Proof outline. Multivariate CLTs give covariance \[ \operatorname{Cov}(1_{\{U\le s\}},1_{\{U\le t\}}) =s\wedge t-st. \] Tightness follows from binomial increment bounds and chaining over dyadic intervals. The limit is therefore the continuous Gaussian bridge. For a continuous distribution function \(F\), the probability integral transform gives the limit \(\mathbb B\circ F\).
Corollary. The Kolmogorov-Smirnov and Cramer-von Mises statistics satisfy \[ \sqrt n\|F_n-F\|_\infty \Rightarrow\sup_{0\le t\le1}|\mathbb B_t|, \] \[ n\int(F_n-F)^2\,\mathrm{d}F \Rightarrow\int_0^1\mathbb B_t^2\,\mathrm{d} t. \]
The scalar class of half-lines is the first Donsker class. Uniform central limit theorems over general function classes are developed in the empirical-process note.