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We show, how the classical Berry-Esseen theorem for normal approximation maybe used to derive rates of convergence for random sums of centerd, real-valuedrandom variables with respect to a certain class of probability metrics,including the Kolmogorov and the Wasserstein distances. This technique isapplied to several examples, including the approximation by a Laplacedistribution of a geometric sum of centered random variables with finite thirdmoment, where a concrete Berry-Esseen bound is derived. This bound reduces to abound of the supposedly optimal order $\sqrt{p}$ in the i.i.d. case.

On rates of convergence and Berry-Esseen bounds for random sums of centered random variables with finite third moments