Skorohod representation on a given probability space

Abstract

Let $$(\Omega,\mathcal{A},P)$$ be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and $$X_n:\Omega\rightarrow S$$ an arbitrary map, n = 1,2,.... If μ is tight and X n converges in distribution to μ (in Hoffmann-Jørgensen's sense), then X μ for some S-valued random variable X on $$(\Omega,\mathcal{A},P)$$. If, in addition, the X n are measurable and tight, there are S-valued random variables $$\overset{\sim}{X}_n$$ and X, defined on $$(\Omega,\mathcal{A},P)$$, such that $$\overset{\sim}{X}_n\sim X_n$$, X μ, and $$\overset{\sim}{X}_{n_k}\rightarrow X$$ a.s. for some subsequence (n k). Further, $$\overset{\sim}{X}_n\ rightarrow X$$ a.s. (without need of taking subsequences) if μ{x} = 0 for all x, or if P(X n = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to $$\sigma(\mathcal{A}\cup\{H\})$$ for some H Ω with P *(H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken $$((0,1),\sigma(\mathcal{U}\cup\ {H\}),m_H)$$, for some H (0,1) with outer Lebesgue measure 1, where $$\mathcal{U}$$ is the Borel σ-field on (0,1) and m H the only extension of Lebesgue measure such that m H (H) = 1. In order to prove the previous results, it is also shown that, if X n converges in distribution to a separable limit, then X n k converges stably for some subsequence (n k). © 2006 Springer-Verlag.