Estimates of the shift parameter of the Student's distribution with a small number of degrees of freedom

In this paper we discuss a possible explanation of the emergence of heavy-tailed distributions observed in practice instead of the expected normal laws. Limit theorems for statistics constructed from samples with random sizes are the base for this explanation. As examples of the application of general theorems, conditions are presented for the convergence of the distributions of asymptotically normal statistics to Student distributions. New statistical estimations of the location parameter of this distribution are presented.