The Limiting Behaviour of Randomly Indexed Sequences

Let {N n , 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 {X n, k , 1<= k<= k n , n => 1} of random variables such that X(n,K) ~ F , 1 <= k <= kn, n=> 1 , and X(n,Nn) - G, as n- (infinity). Furthermore, we consider the speed of weak convergence of X(n,Nn) to G, as n-(infinity).