Donsker's invariance principle (also called the functional central limit theorem) is the natural generalisation of the central limit theorem from random variables to random functions: it asserts that the rescaled random walk $S^{[N]}_t = (\sigma^2 N)^{-1/2} S_{Nt}$ converges in distribution to standard Brownian motion on $\mathcal{C}([0,1])$ equipped with the supremum norm. While the CLT says that $S_N / \sqrt{N}$ converges in law to a Gaussian random variable, Donsker's principle says that the entire interpolated path converges to a Gaussian random function. The proof has two components. First, convergence of finite-dimensional distributions: for any fixed times $0 < t_1 < \cdots < t_k \leq 1$, the vector $(S^{[N]}_{t_1}, \ldots, S^{[N]}_{t_k})$ converges in distribution to $(B_{t_1}, \ldots, B_{t_k})$ by the multidimensional CLT. Second, tightness: the family $\{S^{[N]}\}$ is tight in $\mathcal{C}([0,1])$, which by [Prokhorov's Theorem](/theorems/1172) guarantees sequential compactness. Tightness is verified using moment bounds on the modulus of continuity of $S^{[N]}$. Since the finite-dimensional distributions uniquely determine the Wiener measure, subsequential limits must equal Brownian motion, upgrading subsequential convergence to full convergence. Donsker's principle is one of the most powerful results in probability, with applications far beyond the original setting. It allows one to compute the limiting distribution of functionals of random walks — such as the maximum $\max_{k \leq N} S_k / \sqrt{N}$, the first passage time, or the number of sign changes — by computing the corresponding functional of Brownian motion, where explicit formulae are available (e.g., via the [Reflection Principle](/theorems/1181) or the [Joint Distribution of $B_t$ and its Maximum](/theorems/1182)). An alternative proof route goes through the [Skorokhod Embedding Theorem](/theorems/1190), which constructs the random walk directly as a stopped Brownian motion.