A stochastic process becomes a single random element once a path space and topology are chosen. That choice determines which features are continuous, which compactness criterion expresses tightness, and what process-level convergence means. Finite-dimensional laws describe values at finitely many times; topology controls the behavior between those times.
Continuous and cadlag path spaces
Definition. For a metric state space \(S\), \[ C([0,T],S) =\{x:x\text{ is continuous}\}, \] and \[ D([0,T],S) =\{x:x\text{ is right-continuous with left limits}\}. \] When \(S=\mathbb R^d\), \(C([0,T],S)\) is equipped with \[ \|x-y\|_\infty=\sup_{0\le t\le T}\|x(t)-y(t)\|. \]
The uniform topology is appropriate for continuous limits such as Brownian motion. It is too rigid for jump processes: a small displacement of a jump time has uniform distance equal to the jump size. Skorokhod topologies permit small time changes.
Definition (Skorokhod \(J_1\) metric). Let \(\Lambda\) be the strictly increasing continuous bijections of \([0,T]\) onto itself. One equivalent \(J_1\) metric is \[ d_{J_1}(x,y) = \inf_{\lambda\in\Lambda} \left\{ \|\lambda-\operatorname{id}\|_\infty \wedge1 + \|x-y\circ\lambda\|_\infty\wedge1 \right\}, \] with a logarithmic slope modification often used to obtain a complete metric. Convergence means that a time change close to the identity aligns the jumps and makes the values uniformly close.
Coordinate sigma-fields
Theorem. If \(S\) is Polish, then the Borel sigma-field of \(C([0,T],S)\) under the uniform topology is \[ \mathcal B(C([0,T],S)) =\sigma(x\mapsto x(t):0\le t\le T) =\sigma(x\mapsto x(q):q\in\mathbb Q\cap[0,T]). \]
Proof. Every coordinate map is continuous, so the coordinate sigma-field lies in the Borel sigma-field. Conversely, for a fixed path \(x_0\), \[ \|x-x_0\|_\infty =\sup_{q\in\mathbb Q\cap[0,T]}d(x(q),x_0(q)) \] by continuity. Therefore every open ball is coordinate measurable. Open sets are countable unions of balls because \(C([0,T],S)\) is separable, so all Borel sets are coordinate measurable.
Remark. The analogous coordinate-generation statement holds for \(D([0,T],S)\) with the \(J_1\) topology. However, the evaluation map \(x\mapsto x(t)\) is continuous at \(x\) only when \(x\) is continuous at \(t\). This is why finite-dimensional convergence from \(J_1\) convergence is asserted at continuity times of the limit.
Finite-dimensional and process convergence
Definition. Processes \(X_n\) converge to \(X\) in finite-dimensional distributions if, for every \(t_1,\ldots,t_k\), \[ (X_n(t_1),\ldots,X_n(t_k)) \Rightarrow (X(t_1),\ldots,X(t_k)). \]
Proposition. If \(X_n\Rightarrow X\) in \(C([0,T],S)\), then the finite-dimensional distributions converge at every time tuple. If \(X_n\Rightarrow X\) in \(D([0,T],S)\) under \(J_1\), the same holds whenever \(X\) is almost surely continuous at all selected times.
Proof. Apply the continuous mapping theorem to the vector evaluation map. It is continuous everywhere on \(C\), and at a cadlag path in \(D\) precisely when that path is continuous at the selected times.
Example (finite-dimensional convergence without tightness). Let \(U_n\) be uniform on \([0,1]\), and let \(X_n\) be the continuous triangular bump of height one centered at \(U_n\) and supported on an interval of width \(2/n\). For any fixed \(t_1,\ldots,t_k\), \[ P\bigl(X_n(t_j)\ne0\text{ for some }j\bigr)\le\frac{2k}{n}\to0, \] so all finite-dimensional distributions converge to those of the zero process. Yet \(\|X_n\|_\infty=1\) almost surely, and the paths develop arbitrarily steep oscillations. The laws are not tight in \(C[0,1]\), and \(X_n\) does not converge weakly to zero there.
Tightness in continuous path space
Definition. For \(x\in C([0,T],\mathbb R^d)\), define the modulus of continuity \[ \omega_x(\delta) =\sup_{\substack{s,t\in[0,T]\\|t-s|\le\delta}} \|x(t)-x(s)\|. \]
Theorem (tightness criterion in \(C\)). A sequence of laws of continuous processes \(X_n\) is tight in \(C([0,T],\mathbb R^d)\) if:
- the laws of \(X_n(0)\) are tight;
- for every \(\varepsilon,\eta>0\), there is \(\delta>0\) such that \[ \sup_nP(\omega_{X_n}(\delta)>\eta)<\varepsilon. \]
Proof. Choose \(R\) so that \(P(\|X_n(0)\|>R)<\varepsilon/2\), and choose a sequence \(\delta_m\downarrow0\) such that \[ P(\omega_{X_n}(\delta_m)>1/m)<\varepsilon2^{-m-1} \] uniformly in \(n\). The set \[ K=\{x:\|x(0)\|\le R,\ \omega_x(\delta_m)\le1/m \text{ for every }m\} \] is uniformly bounded and equicontinuous, hence relatively compact by Arzel?-Ascoli; taking its closure makes it compact. The union bound gives \(\inf_nP(X_n\in K)\ge1-\varepsilon\). Necessity follows because every compact subset of \(C\) is uniformly bounded and equicontinuous.
Corollary (moment criterion). If, uniformly in \(n\), \[ \mathbb E\|X_n(t)-X_n(s)\|^\alpha \le C|t-s|^{1+\beta} \] for some \(\alpha,\beta>0\), and \(X_n(0)\) is tight, then \(\{X_n\}\) is tight in \(C[0,T]\).
Combining tightness with finite-dimensional convergence gives the standard process-convergence method: Prokhorov yields convergent subsequences, finite-dimensional laws identify every subsequential limit, and uniqueness forces convergence of the whole sequence. Donsker's theorem is the canonical example.