Prokhorov's theorem
/ pro-KHOR-ov /
When does a family of probability measures contain weakly convergent subsequences? In finite dimensions you might invoke Helly's selection theorem on CDFs, but on a general space you need a cleaner compactness criterion. Prokhorov's theorem is exactly that: it identifies, on a Polish space, relative compactness in the weak topology with the concrete, checkable condition of tightness. It is the Bolzano-Weierstrass / Arzela-Ascoli of probability measures.
Let S be a Polish space and Pi a family of probability measures on S. Prokhorov's theorem states: if Pi is tight, then it is relatively compact in P(S) under weak convergence, that is, every sequence in Pi has a weakly convergent subsequence (whose limit is again a probability measure). Conversely, on a Polish space, relative compactness implies tightness, so on Polish spaces the two notions are equivalent. The forward direction (tight implies relatively compact) is the workhorse and holds quite generally; the converse uses completeness and separability of S. The proof of the forward direction builds the limit by a diagonal argument over a sequence of compacts, using that probability measures restricted to a fixed compact metric space live in a compact set (Banach-Alaoglu).
The power is the reduction it performs. Proving a functional limit theorem becomes a two-part program: (a) prove tightness of the laws (often via a moment or modulus-of-continuity bound), which by Prokhorov gives subsequential limits, and (b) show every such limit is the SAME measure, usually by matching finite-dimensional distributions or characteristic functionals. Without the converse you might worry tightness is merely sufficient; Prokhorov tells you that on Polish spaces tightness is the exact, necessary-and-sufficient frontier of relative compactness, so nothing is lost by checking it.
In Donsker's invariance principle, one shows the rescaled random walks W_n(t) = S_{[nt]}/sqrt(n) form a tight family on C[0,1] (via a Kolmogorov-Chentsov modulus bound). Prokhorov then yields subsequential limits in C[0,1]; matching finite-dimensional Gaussian distributions identifies the unique limit as Brownian motion. Tightness plus identification is the whole proof.
Prokhorov in action: tightness on path space yields subsequential limits, identification pins them to Brownian motion.
The 'tight implies relatively compact' direction is robust, but the converse (relatively compact implies tight) genuinely needs S Polish. On non-Polish spaces relatively compact families can fail to be tight, so do not quote the equivalence outside the Polish setting.