Elect. Comm. in Probab. 10 (2005), 70{81 ELECTRONIC COMMUNICATIONS in PROBABILITY WHEN DOES A RANDOMLY WEIGHTED SELF{NORMALIZED SUM CONVERGE IN DISTRIBUTION?

We determine exactly when a certain randomly weighted self{normalized sum converges in distribution, partially verifying a 1965 conjecture of Leo Breiman, and then apply our results to characterize the asymptotic distribution of relative sums and to provide a short proof of a 1973 conjecture of Logan, Mallows, Rice and Shepp on the asymptotic distribution of self{ normalized sums in the case of symmetry. 0.1 A conjecture of Breiman Throughout this paper fYigi¸1 will denote a sequence of i.i.d. Y random variables, where Y is non{negative with distribution function G. Let Y 2 D (®), with 0 < ® · 2, denote that Y is in the domain of attraction of a stable law of index ®. We shall use the notation Y 2 D (0) to mean that 1¡G is a slowly varying function at in¯nity. Now let fXigi¸1 be a sequence of i.i.d. X random variables independent of fYigi¸1; where X satis¯es EjXj <1 and EX = 0: (1) Consider the randomly weighted self{normalized su