An extension of the subsequence principle (Q2754977)

Assume that \((X_{n})\) is a sequence of random variables such that there exists a subsequence \((X_{n_{k}})\) satisfying: NEWLINE\[NEWLINE\forall A \hbox{with} P(A)>0, \exists F_{A} \hbox{distribution function s.t.} P(X_{n_{k}}<t \mid A)=F_{A}(t).NEWLINE\]NEWLINE Then by \textit{D. J. Aldous} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 40, 59-82 (1977; Zbl 0571.60027)] or \textit{I. Berkes} and the author [Probab. Theory Relat. Fields 73, 395-413 (1986; Zbl 0572.60027)] there exists a random measure \({\tilde\mu}\) satisfying NEWLINE\[NEWLINE\forall A \hbox{with} P(A)>0, F_{A}(t)=E({\tilde\mu}(-\infty,t) \mid A), \forall t \hbox{point of continuity of} F_{A}.NEWLINE\]NEWLINE Denote by \({\mathcal S}\) a Borel set in the space of probability measures endowed with the Prokhorov metric. For \(\mu\in{\mathcal S}\) one denotes by \(G_{\mu}\) a probability measure such that \(\mu\mapsto G_{\mu}\) is measurable, and for \(k\geq 1\), we consider measures \(f_{k}:R^{\infty}\times{\mathcal S}\rightarrow R\) such that \(f_{k}(\xi_{1},\xi_{2},\ldots,\mu)\) converges in law to \(G_{\mu}\). Here \((\xi_{n})\) is a sequence of i.i.d. random variables of distribution \(\mu\). The author proves the following result: Assume some regularity conditions on functions \(f_{k}\) and assume \(P({\tilde\mu}\in{\mathcal S})=1\). Then there exists a subsequence \((X_{n_{k}})=(Y_{k})\) such that for any permutation \((Y_k')\) of \((Y_{k})\) we have \(f_{k}(Y_1',Y_2',\ldots,{\tilde\mu})\) converges in law to \(\int G_{{\tilde\mu}}dP\).