Some Skorohod-type results

Let S be a metric space, \(g:S\rightarrow \mathbb {R}\) a Borel function, and \((\mu _n:n\ge 0)\) a sequence of tight probability measures on \(\mathcal {B}(S)\). If \(\mu _n=\mu _0\) on \(\sigma (g)\), there are S-valued random variables \(X_n\), all defined on the same probability space, such that \(X_n\sim \mu _n\) and \(g(X_n)=g(X_0)\) for all \(n\ge 0\). Moreover, \(X_n\overset{a.s.}{\longrightarrow }X_0\) if and only if \(E_{\mu _n}(f\mid g)\,\overset{\mu _0-a.s.}{\longrightarrow }\,E_{\mu _0}(f\mid g)\) for each \(f\in C_b(S)\). This result, proved in Pratelli and Rigo (J Theoret Probab 36:372-389, 2023) , is the starting point of this paper. Three types of contributions are provided. First, \(\sigma (g)\) is replaced by an arbitrary sub-\(\sigma \)-field \(\mathcal {G}\subset \mathcal {B}(S)\). Second, the result is applied to some specific frameworks, including equivalence couplings, total variation distances, and the decomposition of cadlag processes with finhite activity. Third, following Hansen et al. (Tempered Bayesian analysis, Unpublished manuscript, 2024), the result is extended to models and kernels. This extension has a fairly natural interpretation in terms of decision theory, mass transportation and statistics.