On the Weak Laws of Large Numbers for Compound Random Sums of Independent Random Variables with Convergence Rates

The weak laws of large numbers are extremely intuitive and applicable results in various fields of probability theory and mathematical statistics. Compound random sums are extensions of classical random sums where the random number of summands is a partial sum of independent and identically distributed positive integer-valued random variables, assuming independence of summands. In this paper, a weak laws of large numbers for normalized compound random sums of independent (not necessarily identically distributed) random variables is studied and the convergence rates in types of “Small-o” and “Large-O” error estimates are established, using Trotter’s distance approach. The obtained results in this paper are extensions and generalizations of the known classical ones.