**Abstract**

This chapter contains a possible explanation of the emergence of heavy-tailed distributions observed in practice instead of the expected normal laws. The bases for this explanation are limit theorems for random sums and statistics constructed from samples with random sizes. As examples of the application of general theorems, conditions are presented for the convergence of the distributions of random sums of independent random vectors *with finite covariance matrices* to multivariate elliptically contoured stable and Linnik distributions. Also, conditions are presented for the convergence of the distributions of asymptotically normal (in the traditional sense) statistics to multivariate Student distributions. The joint asymptotic behavior of sample quantiles is also considered.

**Keywords:** random sum, random sample size, multivariate normal mixtures, heavy-tailed distributions, multivariate stable distribution, multivariate Linnik distribution, Mittag-Leffler distribution, multivariate Student distribution, sample quantiles

**AMS 2000 Subject Classification**: 60F05, 60G50, 60G55, 62E20, 62G30
