A stochastic process is an infinite collection of jointly distributed random variables. Kolmogorov's extension theorem makes this statement constructive: a consistent family of finite-dimensional laws determines a probability measure on the coordinate product space. Existence of the process and regularity of its sample paths are deliberately separate questions.
Projective families of distributions
Definition. Let \(T\) be an arbitrary index set and \((S,\mathcal S)\) a measurable state space. For each finite ordered tuple \(\mathbf t=(t_1,\ldots,t_n)\) of distinct indices, let \(\mu_{\mathbf t}\) be a probability measure on \((S^n,\mathcal S^{\otimes n})\). The family is consistent if:
- permuting the indices permutes the coordinates of the measure;
- for \(\{t_1,\ldots,t_m\}\subseteq\{t_1,\ldots,t_n\}\), the corresponding marginal of \(\mu_{t_1,\ldots,t_n}\) equals \(\mu_{t_1,\ldots,t_m}\).
These conditions are necessary for any process: reordering a vector does not change its joint law except by the same reordering, and forgetting coordinates must recover the lower-dimensional distribution.
Definition. On \(\Omega=S^T\), define coordinate maps \(X_t(\omega)=\omega(t)\). The product sigma-field is \[ \mathcal S^T=\sigma(X_t:t\in T). \] A cylinder set is an event of the form \[ \{\omega:(X_{t_1}(\omega),\ldots,X_{t_n}(\omega))\in B\}, \qquad B\in\mathcal S^{\otimes n}. \]
The extension theorem
Theorem (Daniell-Kolmogorov). Let \(S\) be a standard Borel space. Every consistent family \(\{\mu_{\mathbf t}\}\) is the family of finite-dimensional distributions of a probability measure \(P\) on \((S^T,\mathcal S^T)\). This measure is unique on the product sigma-field.
Proof outline. Define a set function on cylinders by \[ P\{(X_{t_1},\ldots,X_{t_n})\in B\} :=\mu_{t_1,\ldots,t_n}(B). \] Consistency makes the definition independent of redundant coordinates and establishes finite additivity on the cylinder algebra. The only substantial point is countable additivity. Any countable collection of cylinders involves at most a countable set \(T_0=\{s_1,s_2,\ldots\}\subset T\). On \(S^{T_0}\), finite-dimensional tightness and standard-Borel regularity yield a probability measure with the prescribed marginals: first extend along \(S,S^2,\ldots\), then use continuity from above on compact approximations. Consequently the cylinder premeasure is countably additive whenever a countable union remains a cylinder. Carath?odory's theorem extends it to \(\mathcal S^T\). If two extensions agree on all cylinders, the pi-lambda theorem makes them agree on \(\sigma\)-generated product events, proving uniqueness.
Remark. When \(T\) is uncountable, the product sigma-field can be strictly smaller than the Borel sigma-field of some chosen topology on all functions \(S^T\). The theorem asserts uniqueness on the coordinate-generated sigma-field. A regular path space such as \(C[0,1]\) introduces additional topological structure.
Gaussian process construction
Theorem. Let \(m:T\to\mathbb R\) and let \(K:T\times T\to\mathbb R\) be symmetric and positive semidefinite: \[ \sum_{i,j=1}^na_ia_jK(t_i,t_j)\ge0 \] for every finite choice of \(t_i\) and \(a_i\). Then there exists a Gaussian process \(X_t\) with \[ \mathbb EX_t=m(t),\qquad \operatorname{Cov}(X_s,X_t)=K(s,t). \]
Proof. For every finite tuple \(\mathbf t=(t_1,\ldots,t_n)\), assign the possibly singular Gaussian law \[ N_n(m_{\mathbf t},K_{\mathbf t}), \quad m_{\mathbf t}=(m(t_i))_i,\quad K_{\mathbf t}=(K(t_i,t_j))_{ij}. \] Positive semidefiniteness makes each covariance matrix valid. Gaussian marginals and coordinate permutations remain Gaussian with the corresponding subvector and submatrix, so the family is consistent. Apply Kolmogorov extension.
Example (Brownian finite-dimensional laws). Taking \[ m(t)=0,\qquad K(s,t)=s\wedge t,\qquad s,t\ge0, \] gives consistent Gaussian laws because \[ \sum_{i,j}a_ia_j(s_i\wedge s_j) =\int_0^\infty \left(\sum_i a_i1_{\{u\le s_i\}}\right)^2\mathrm{d} u\ge0. \] The extension theorem constructs a coordinate process with the desired Gaussian increments. It does not yet say that its paths are continuous.
Versions and modifications
Definition. Processes \(X\) and \(Y\) are modifications if \(P(X_t=Y_t)=1\) for each fixed \(t\). They are indistinguishable if \[ P(X_t=Y_t\text{ for every }t)=1. \] Indistinguishability implies modification, but the converse can fail for an uncountable time set.
Example. On \([0,1]\), let \(U\) be uniform and set \(X_t=1_{\{U=t\}}\), \(Y_t=0\). For each fixed \(t\), \(P(X_t=Y_t)=1\), so \(X\) and \(Y\) are modifications. Yet every sample path of \(X\) equals one at \(t=U\), so \(P(X_t=Y_t\text{ for all }t)=0\).
Kolmogorov-Chentsov regularity
Theorem. Let \(X_t\), \(t\in[0,T]^d\), take values in \(\mathbb R^m\). If for some \(\alpha,\beta,C>0\), \[ \mathbb E\|X_t-X_s\|^\alpha \le C\|t-s\|^{d+\beta}, \] then \(X\) has a modification whose paths are locally H?lder continuous of every order \(\gamma<\beta/\alpha\).
Proof outline. On dyadic grids of mesh \(2^{-n}\), Markov's inequality and the moment bound show that the probability of any neighboring increment exceeding \(2^{-n\gamma}\) is at most \(C2^{nd}2^{-n(d+\beta)}2^{n\alpha\gamma} =C2^{-n(\beta-\alpha\gamma)}\). This is summable when \(\gamma<\beta/\alpha\). Borel-Cantelli bounds all sufficiently fine neighboring increments. Chaining along dyadic paths gives a uniform H?lder estimate on the dense dyadic set, whose continuous extension defines the desired modification. Equality with the original process at each fixed time follows from convergence in probability along dyadic approximations.
Thus process construction has two layers: finite-dimensional consistency produces a law on raw coordinates; moment or tightness estimates select a regular version on a useful path space.