Math. Proc. Camb. Phil. Soc. (1980), 88, 171 Printed in Great Britain Weak convergence of sequences of random elements with random indices BY M. CSORGO AND Z. RYCHLIK Carleton University and Maria Curie-Sklodowska University, Lublin (Received 15 June 1979, revised 18 December 1979) 1. Introduction. Let (S, d) be a separable metric space equipped with its Borel <r- field 3$. Let {Y n , n ^ 1} be a sequence of S-valued random elements defined on a prob- ability space (Q, s&, P). Assume Y n => Y converges weakly to an S-valued random element Y. Let {N n , n ^ 1} be a sequence of positive integer-valued random variables defined on the same probability space (Q, s/, P). Following the work of AnscombeO), many authors (see, for example, (4), (5), (6), (9) and (1)) have investigated the limit distribution of Y N as n->co. The results obtained P have the following form. Suppose Y n => Y. If N n /k n > A (in probability), where here and in what follows A is a positive random variable and k n are constants going to infinity, P then under some additional assumptions Y N => Y. The condition N n /k n > A in the case A = 1 was first discussed by Anscombe(3), who also introduced the following uniform continuity condition on {Y n , n ^ 1}. For each e > 0 there exists 8 > 0 such that ^) ^ e) < e, where the maximum is taken over all i that |i — n| ^ Sn. This condition, known as Anscombe's condition, plays a very important role in the proofs of the results given in the above mentioned papers. Recently Aldous(l) has proved that Anscombe's condition P is exactly the right one when N n /k n > 1. He also has given some necessary and P sufficient conditions for Y N => Y in the case when N n /k n > A. The necessary con- ditions are new, however many of the sufficient ones are essentially known. Thus, even in the spacial case, when Y n = S n /B n ,S n = X x +...+ X n , JB| = DS n , and {X k , k > 1} are independent random variables with means zero and finite variances, from Aldous' P results (l) we can only anticipate that if N n /k n *• A, DX k = c 2 > 0, k ^ 1, and SJeni=>N'(0,1), then S N /cN\ => N (0,1), where N(0,1) denotes a random variable with the standard normal distribution function. On the other hand, {S N JB Nn , n ^ 1}, in general, may P not be asymptotically normal even if N n /n > 1 and {X k , k > 1} satisfy Lindeberg's condition (for an appropriate example, cf. (4)). But taking into account the results given in (4), or (n) it is easy to see that if {X k , k ^ 1} satisfy Lindeberg's condition and B N JB n —^-> A, thenS y JBlr n => N{0,1). This remark proves that when {Y n ,n > l}does P not satisfy Anscombe's condition, then the assumption N n /k n > A is not applicable 0305-0041/8n/OOnn.74.')O $O3.. r >0 © I080 Oamhriclgo Philosophical Socioty