On the L_p-convergence for compound random sums of pairwise independent random variables

A random sum is the sum of a random number of random variables, and it has attracted much attention from many researchers. Compound random sums are extensions of classical random sums, in which the random index is determined by the partial sum of independent and identically distributed, non-negative integer-valued random variables. This paper establishes mean convergence theorems for compound random sums of pairwise independent (not necessarily identically distributed) random variables under the Cesàro uniform integrability condition. Our main results extend Theorem 3.1 of Giang and Hung [1] and Theorem 3.1 of Hung [2]. As immediate corollaries, we obtain weak laws of large numbers for negative-binomial sums of pairwise independent random variables.