Stable convergence of random sequences with random indices (Q1585966)

Let \(\{X_n\}\) be a sequence of random elements which are defined on a probability space \((\Omega,{\mathcal F},P)\) and take values in a metric space \(E\). Let \({\mathcal G}\) be a sub-\(\sigma\)-algebra of \({\mathcal F}\). We say that \(\{X_n\}\) converges \({\mathcal G}\)-stably to an \(E\)-valued random element \(X\), defined on \((\Omega, {\mathcal F},P)\), if for any bounded continuous function \(f\) on \(E\) and any event \(A\in{\mathcal G}\), \(EI(A)f (X_n)\to EI(A) f(X)\). This definition is equivalent to the original definition of \textit{A. Rényi} [Sankhyā, Ser. A 25, 293-302 (1963; Zbl 0141.16401)]. The author proves, among others, the Anscombe-type theorems which say that under some conditions if \(X_n\) converges \({\mathcal G}\)-stably to \(X\), then \(X_{N_n}\) converges \({\mathcal G}\)-stably to \(X\), where \(\{k_n\}\) is a sequence of positive numbers tending to \(\infty\) and \(\{N_n\}\) is a sequence of random indices, defined on the same probability space, such that \(N_n/k_n\) converges in probability to some positive random variable.