The Limiting Behaviour of Randomly Indexed Sequences

Abstract

Let {Nn, n≥1} be an arbitrary sequence of positive integer-valued random variables and let F and G be given distribution functions. We present necessary and sufficient conditions under which there exists an array {Xn, k, 1≤k≤kn, n≥1} of random variables such that \(X_{n,k} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\mathcal{D}} F,{\text{ }}1 \leqslant k \leqslant k_n ,{\text{ }}n \geqslant 1\), and \(X_{n,N_n } \xrightarrow{\mathcal{D}}G\), as n→∞. Furthermore, we consider the speed of weak convergence of \(X_{n,N_n }\) to G, as n→∞.