On the large O−rates of convergence in limit theorems for compound random sums of arrays of row-wise independent random variables

Abstract

Compound random sums are extensions of random sums when the random number of summands is a partial sum of independent and identically distributed positive integer-valued random variables, which assumed independent of summands. In the paper, upper bounds for the large O?rates of convergence in weak limit theorems for compound random sums of arrays of row-wise independent random variables, in term of Trotter distance are studied. The main results are approximation theorems which give the Trotter distance between normalized compound random sums of the given independent random variables and the compound ??decomposable random variables. By these results the converging rates in central limit theorem, weak law of large numbers and stable limit theorem for compound random sums are then established. The obtained results in this paper are closely related to the classical ones.