Weak convergence is convergence of laws against bounded continuous test functions. Tightness is the compactness condition that lets subsequences of laws converge.
Weak convergence of probability measures
Weak convergence lives on topological spaces, usually metric spaces with Borel sigma-fields.
Definition. Probability measures \(\mu_n\) on a metric space \(S\) converge weakly to \(\mu\), written \(\mu_n\Rightarrow\mu\), if \[ \int f\,\mathrm{d}\mu_n\to\int f\,\mathrm{d}\mu \] for every bounded continuous \(f:S\to\mathbb R\).
Portmanteau theorem
The portmanteau theorem gives equivalent setwise and functionwise formulations.
Theorem. For probability measures on a metric space, \(\mu_n\Rightarrow\mu\) if and only if \(\limsup_n\mu_n(F)\le\mu(F)\) for every closed \(F\), equivalently \(\liminf_n\mu_n(G)\ge\mu(G)\) for every open \(G\).
Proof. The full equivalence, including continuity sets and semicontinuous test functions, is proved in Real analysis. For the key direction, approximate a closed-set indicator from above by \[ f_k(x)=\max\{0,1-kd(x,F)\}\downarrow\mathbf1_F. \] Weak convergence followed by monotone convergence gives the closed-set upper bound; complements give the open-set lower bound. Conversely, layer-cake and the open-set inequalities recover integrals of bounded lower-semicontinuous functions, hence of bounded continuous functions.
Tightness and Prokhorov
Tight families do not let mass escape to infinity.
Proposition. A family \(\mathcal P\) of probability measures on \(S\) is tight if for every \(\varepsilon\gt0\) there exists compact \(K\subseteq S\) such that \(\inf_{\mu\in\mathcal P}\mu(K)\ge1-\varepsilon\). On Polish spaces, tightness is equivalent to relative compactness for weak convergence.
Proof outline. Tightness confines all but \(\varepsilon\) mass to compact sets. Separability supplies a countable convergence-determining family of bounded Lipschitz functions, so diagonal extraction gives a candidate limiting functional. Tightness prevents loss of total mass, and representation turns the functional into a probability measure. Conversely, a weakly compact family is uniformly tight. The full construction and the Prokhorov metric are given in Real analysis.
Proposition (moment criterion). For real or Euclidean-valued \(X_n\), if \[ \sup_n\mathbb E\|X_n\|^p<\infty \] for some \(p>0\), then their laws are tight.
Proof. Markov's inequality gives \[ \sup_nP(\|X_n\|>K) \le K^{-p}\sup_n\mathbb E\|X_n\|^p \longrightarrow0. \] Closed Euclidean balls are compact.
Theorem (Skorokhod representation). If \(\mu_n\Rightarrow\mu\) on a Polish space, then there exist random variables \(Y_n,Y\) on one probability space with laws \(\mu_n,\mu\) such that \(Y_n\to Y\) almost surely.
Proof outline. Construct successively finer measurable partitions of the Polish space with small diameters and \(\mu\)-null boundaries. Couple the partition cells using the converging cell probabilities, then refine the coupling consistently. Outside a summable sequence of mismatch events, \(Y_n\) and \(Y\) eventually occupy the same cells at every fixed resolution; shrinking diameters force almost-sure convergence.
Remark. Skorokhod representation changes the coupling: it does not say that the original \(X_n\) converge almost surely. It is a proof device that realizes weakly converging laws together on a new space.