A Strong Version of the Skorohod Representation Theorem

Abstract For each $$n\ge 0$$ n ≥ 0 , let $$\mu _n$$ μ be a tight probability measure on the Borel $$\sigma $$ σ -field of metric space S . Let $$(T,{\mathcal {C}})$$ ( T , C ) measurable such that diagonal $$\bigl \{(t,t):t\in T\bigr \}$$ { t : ∈ } belongs to $${\mathcal {C}}\otimes {\mathcal {C}}$$ ⊗ Fix function $$g:S\rightarrow T$$ g S → and suppose _n=\mu _0$$ = $$g^{-1}({\mathcal - 1 for all Necessary sufficient conditions existence -valued random variables $$X_n$$ X defined same satisfying $$\begin{aligned} X_n\overset{\text {a.s.}}{\longrightarrow }X_0,\quad X_n\sim \mu _n\,\text { } \,g(X_n)=g(X_0)\,\text }n\ge 0, \end{aligned}$$ ⟶ a.s. ∼ and for all are given. Such then applied several examples. The tightness condition can dropped at price some assumptions g