Abstract
It is shown that for a wide class of topological spaces, any uniformly tight sequence of random elements contains a subsequence which admits the usual a.s. Skorokhod representation on the Lebesgue interval.
Summary
This paper proves that the Skorokhod almost sure representation theorem — one of the most fundamental tools in probability theory — extends to non-metric topological spaces under a single, minimal condition: the existence of a countable family of continuous functions separating points (condition CCSP). This is striking because the classical Skorokhod representation requires a separable metric space.
The key topological insight (p. 211): condition CCSP gives a continuous injection f-bar: X → [-1,1]^I into a metrizable space, defining a metric topology tau_f weaker than the original topology tau. Crucially, tau and tau_f coincide on tau-compact sets (since a continuous bijection from a compact space to a Hausdorff space is a homeomorphism). This means sequences living in compact sets behave as if the space were metric.
The main result (Theorem 2): if (X, tau) satisfies CCSP and {X_n} are random elements with uniform tau-tightness (mass concentrates on tau-compact sets), then every subsequence contains a further subsequence admitting an a.s. Skorokhod representation. This is a sequential Prokhorov theorem without metrizability. Theorem 3 gives the equivalent formulation for laws: uniformly tight sequences of probability measures have subsequences with Skorokhod representations.
The paper includes important applications to separable Hilbert spaces with weak topology (Theorem 1), separable linear topological spaces (Theorem 4), and Frechet nuclear spaces and their duals (Theorem 5). The Hilbert space result (p. 210) is particularly relevant: uniform tau_w-tightness (which requires norm-boundedness plus concentration on norm-compact sets) gives subsequences with weak a.s. convergence.
Key Contributions
- Theorem 2 (p. 212): a.s. Skorokhod representation for subsequences under the single condition CCSP + uniform tightness — no metrizability needed
- Theorem 3 (p. 214): sequential Prokhorov theorem — uniformly tight sequences of laws have subsequences with strong a.s. Skorokhod representations
- Corollary 3 (p. 215): X_n →_D X_0 iff every subsequence has a further subsequence with strong a.s. Skorokhod representation — equivalent to uniform tightness
- Condition (10) (p. 211): CCSP — the minimal topological assumption, much weaker than metrizability
- Theorem 4 (p. 214): application to separable linear topological spaces with duality convergence
Methodology
The proof (p. 212-213) works by: (1) using the CCSP family {f_i} to embed X into [-1,1]^I via f-bar; (2) constructing truncated laws bar-mu_n = L(f-bar(X_n)) on the metrizable space [-1,1]^I; (3) using the integer-valued functional Phi(y) = min{m : y in bar-K_m} to reduce to compact sets; (4) applying classical Prokhorov on R^I x N to extract convergent subsequences; (5) lifting back via Lemma 1 (a Skorokhod-type result for product spaces with measurable maps).
Key Findings
- CCSP is sufficient: no separation axioms, regularity, or metrizability beyond CCSP are needed
- tau-compact sets are metrizable under CCSP: this is the key structural fact enabling the proof
- Uniform tightness is essential: Remark on p. 210 gives Fernique’s counterexample — in (H, tau_w), sequences can converge in distribution yet have no uniformly tight subsequence
- Sequential vs full representation: in general, one can only get the representation for SUBSEQUENCES, not the whole sequence (Bogachev-Kolesnikov showed the full sequence version is impossible on R_0^infinity)
- Convergence in distribution on S’ and D’: Corollary 3 shows that on distribution spaces, convergence in distribution is equivalent to uniform tightness + subsequential Skorokhod representation
Important References
- Convergence of Probability Measures — Billingsley (1971), the classical Skorokhod representation on metric spaces
- Convergence in distribution of stochastic processes — Le Cam (1957), Prokhorov’s theorem and tightness
- Processus lineaires, processus generalises — Fernique (1967), the counterexample on separable Hilbert spaces