On the asymptotic distribution of the sum of a random number of independent random variables

Abstract

Let ~1, ~2, . . . , ~ , . . . denote a sequence of independent random variables and put (1) ~,~ = ~l-k ~2+ "" + ~ ( n ~ l , 2 , . . . ) . Several authors (see e. g. [1] and [2]) investigated the asymptotic distribution of ~,(,) for t ~ + o~ where r(t) is a positive integer-valued random variable, for t>0 , which converges in probability to + ~ for t + + ~ . The most general results in this direction have been obtained by DOBRUgIN [3]. In all these investigations it has been supposed that r(t) is for any t > 0 independent of the random variables ~,, (n ~ 1 , 2 , . . . ) . A general and very useful theorem without this supposition has been proved by F. J. ANSCOMBE [4]. In a recent paper [5] TAKAC$ has proved a theorem, which can be considered also as a result on the asymptotic distribution of the sum of a random number of independent random variables, i. e. using the above notations on the asymptotic distribution of g~(,) where ~,~ is defined by (1). In this case r(t) depends .essentially on the variables ~ (n ~ I , 2 , . . . ) . The aim of the present paper is to show that the mentioned result of TAKACS can b e easily deduced from a special case of the theorem of ANSCOMBE mentioned above. To make the paper self-contained, we give in w 1 a short proof of the special case of ANSCOMBE'S theorem which is needed for our purpose (Theorem 1). Using this theorem, in w 2 a new and simple proof of the result of TAKACS mentioned above is given.