Characterization of strong limits of randomly indexed sequences (Q1600590)

Let \((X_n: n\geq 1)\) be a sequence of random variables with values in a Polish space, and \(X\) a given random variable. The author gives a necessary and sufficient condition under which there exists a sequence \((N_n: n\geq 1)\) of positive integer-valued random variables satisfying \(X_{N_n}\to X\) almost surely as \(n\to\infty\). A similar result is proved for convergence in probability and in \(L_p\). In each case, the proof gives explicitly the required sequence \((N_n: n\geq 1)\).