Weak convergence of random sums and maximum random sums under nonrandom norming

Necessary and sufficient conditions are obtained for the weak convergence of random sums and maximum random sums of independent identically distributed random variables. Limit distributions for these sums are described. The indices are not assumed to be independent of the summands. Supported in part by Grants NFW000 and NFW300 from the International Science Foundation and the Russian Government and by Grant No. 93-011-1446 from the Russian Foundation for Fundamental Research. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.