tightness criteria on path space
Prokhorov tells you that to prove a functional limit theorem (a process X_n converging in law to a process X) you should prove tightness of the path laws and then identify the limit. But tightness on C[0,1] or D[0,1] requires knowing what the compact subsets of those infinite-dimensional path spaces look like, and then a probabilistic estimate that keeps the paths inside such a compact with high probability. This entry is that machinery, the workhorse computation in every invariance principle.
On C[0,1], by the Arzela-Ascoli theorem a set is relatively compact iff it is uniformly bounded at one point and equicontinuous, i.e. has a uniformly controlled modulus of continuity w(x, delta) = sup{|x(s) - x(t)| : |s - t| <= delta}. So a family of laws {P_n} on C[0,1] is tight iff the initial values are tight and for every epsilon, eta there is delta with limsup_n P_n( w(X, delta) > eta ) <= epsilon, the paths are uniformly equicontinuous in probability. The standard sufficient condition is a Kolmogorov-Chentsov moment bound: if E|X_n(t) - X_n(s)|^beta <= C |t - s|^{1 + alpha} uniformly in n for some alpha, beta > 0, then the laws are tight on C[0,1] (and the limit has Holder-continuous paths). On D[0,1] one uses the analogous Skorokhod modulus w'(x, delta), which permits one jump within each small window, and a Billingsley/Aldous-type criterion (often phrased via stopping times: control E[ |X(tau + h) - X(tau)| AND |X(tau) - X(tau - h)| ] over stopping times tau).
The practical recipe in a CLT-on-path-space proof is therefore: (1) check tightness via a moment / modulus bound (Kolmogorov-Chentsov on C, Aldous on D); (2) by Prokhorov extract subsequential limits; (3) identify the limit by computing finite-dimensional distributions and matching them to the target (Brownian motion, a Levy process). The honest caveat: finite-dimensional convergence alone is NEVER enough for convergence in path space, you can have all finite-dimensional distributions converge while the processes fail to converge as random functions (mass can concentrate in ever-narrower spikes between the sampled times). Tightness is exactly the extra ingredient that rules out such pathologies and upgrades finite-dimensional convergence to functional convergence.
For Donsker, the rescaled walk W_n(t) = S_{[nt]}/sqrt(n) (linearly interpolated, in C[0,1]) satisfies, for increments over [s,t], E|W_n(t) - W_n(s)|^4 <= C|t - s|^2 (a fourth-moment bound, with alpha = beta = ... giving 1 + alpha = 2). Kolmogorov-Chentsov then gives tightness on C[0,1]; finite-dimensional CLTs identify the limit as Brownian motion.
A fourth-moment increment bound feeds Kolmogorov-Chentsov to deliver tightness for Donsker's invariance principle.
Convergence of all finite-dimensional distributions does NOT imply convergence in C[0,1] or D[0,1]; you must add tightness. Skipping the tightness step is the single most common error in functional-CLT proofs.