Finite-dimensional distributions specify what a process does at finitely many indices. Weak convergence in a function space additionally controls all unobserved indices through tightness. This distinction is what separates an ordinary vector central limit theorem from a Donsker theorem.
Weak convergence with outer expectations
Definition. Let \(Z_n\) be arbitrary maps into a metric space \((S,d)\), and let \(Z\) be a Borel measurable, tight random element of \(S\). We write \[ Z_n\rightsquigarrow Z \] if \[ \mathbb E^\star h(Z_n)\to\mathbb Eh(Z) \] for every bounded continuous \(h:S\to\mathbb R\). When every \(Z_n\) is Borel measurable, this is ordinary weak convergence of laws.
Portmanteau inequalities. If \(Z_n\rightsquigarrow Z\), then for every closed \(F\subseteq S\) and open \(G\subseteq S\), \[ \limsup_nP^\star(Z_n\in F)\le P(Z\in F), \] \[ \liminf_nP_\star(Z_n\in G)\ge P(Z\in G). \] Consequently, for every Borel set \(A\) with \(P(Z\in\partial A)=0\), \[ P^\star(Z_n\in A)\to P(Z\in A) \] when asymptotic measurability removes the inner-outer gap.
Proof. For a closed set \(F\), define \[ h_\varepsilon(x) =\left(1-\frac{d(x,F)}{\varepsilon}\right)_+. \] Then \(1_F\le h_\varepsilon\le1_{\{d(x,F)\le\varepsilon\}}\), and \(h_\varepsilon\) is bounded continuous. Therefore \[ \limsup_nP^\star(Z_n\in F) \le\mathbb Eh_\varepsilon(Z). \] Let \(\varepsilon\downarrow0\) and use dominated convergence. Apply the closed-set result to \(G^c\) for the open-set inequality. A continuity set is squeezed between its interior and closure.
Tightness and asymptotic tightness
Definition. A Borel random element \(Z\) is tight if, for every \(\varepsilon>0\), there is compact \(K\subseteq S\) with \[ P(Z\in K)\ge1-\varepsilon. \] A sequence \(Z_n\) is asymptotically tight if, for every \(\varepsilon>0\), there is compact \(K\) such that every \(\delta>0\) satisfies \[ \liminf_nP_\star(Z_n\in K^\delta)\ge1-\varepsilon, \] where \(K^\delta=\{x:d(x,K)<\delta\}\).
The enlargement \(K^\delta\) accommodates nonmeasurability and spaces in which exact compact containment is inconvenient. In separable metric spaces with measurable \(Z_n\), it reduces to the usual uniform-tightness criterion.
Prokhorov theorem. On a complete separable metric space, a family of probability measures is relatively compact for weak convergence if and only if it is tight.
This theorem supplies subsequential limits. Identifying all subsequential limits through finite-dimensional distributions or a convergence-determining class then proves full weak convergence.
Finite-dimensional convergence
Definition. Processes \(Z_n=\{Z_n(t):t\in T\}\) converge in finite-dimensional distributions to \(Z\) if, for every \(t_1,\ldots,t_k\in T\), \[ (Z_n(t_1),\ldots,Z_n(t_k)) \rightsquigarrow (Z(t_1),\ldots,Z(t_k)). \]
Finite-dimensional convergence plus tightness. Suppose \(Z_n\) are asymptotically tight in a function space \(S\), and every subsequential weak limit is determined uniquely by its finite-dimensional distributions. If the finite-dimensional distributions converge to those of a tight \(Z\in S\), then \[ Z_n\rightsquigarrow Z. \]
Proof. Every subsequence has a further weakly convergent subsequence by asymptotic tightness and Prokhorov-type compactness. Coordinate evaluation is continuous at the relevant paths or can be handled by the finite-dimensional assumption, so the subsequential limit has the prescribed finite-dimensional distributions. Uniqueness makes its law equal to that of \(Z\). Since every subsequence has a further subsequence with the same limit, the original sequence converges.
Example (finite-dimensional convergence is insufficient). On \(C[0,1]\), let \[ Z_n(t)=\max\{1-n|t-U_n|,0\}, \] where \(U_n\sim U[0,1]\). For every fixed finite set of times, \(Z_n(t_j)\to_p0\), because a spike of width \(2/n\) misses them with high probability. Yet \[ \|Z_n\|_\infty=1 \] always, so \(Z_n\) cannot converge to zero in \(C[0,1]\). The missing ingredient is tight control of oscillations.
Continuous mapping
Extended continuous-mapping theorem. Suppose \(Z_n\rightsquigarrow Z\), maps \(g_n:S\to T\) and \(g:S\to T\) satisfy \[ x_n\to x,\ x\in S_0 \quad\Longrightarrow\quad g_n(x_n)\to g(x), \] and \(P(Z\in S_0)=1\). Then \[ g_n(Z_n)\rightsquigarrow g(Z), \] subject to asymptotic measurability.
Proof outline. Tightness restricts \(Z_n\) to compact neighborhoods with arbitrarily high probability. On convergent sequences whose limits lie in \(S_0\), the assumed sequential continuity gives uniform approximation over these compact sets by contradiction. Composition with bounded continuous test functions and the weak convergence of \(Z_n\) then yields the result.
Supremum functional. On \(\ell^\infty(T)\), \[ \left|\sup_t|z(t)|-\sup_t|w(t)|\right| \le\|z-w\|_T. \] Hence if \(Z_n\rightsquigarrow Z\) in \(\ell^\infty(T)\), then \[ \|Z_n\|_T\rightsquigarrow\|Z\|_T. \]
Asymptotic equicontinuity criterion
For processes in \(\ell^\infty(T)\), tightness is commonly established by finding a semimetric \(\rho\) such that \((T,\rho)\) is totally bounded and \[ \lim_{\delta\downarrow0}\limsup_{n\to\infty} P^\star\left\{ \sup_{\rho(s,t)<\delta} |Z_n(s)-Z_n(t)|>\varepsilon \right\}=0 \] for every \(\varepsilon>0\). Finite-dimensional convergence then specifies the values on a finite \(\delta\)-net, while equicontinuity controls all points between the net points.