  Using Stein's method techniques, we develop a framework which allows one to
bound the error terms arising from approximation by the Laplace distribution
and apply it to the study of random sums of mean zero random variables. As a
corollary, we deduce a Berry-Esseen type theorem for the convergence of certain
geometric sums. Our results make use of a second order characterizing equation
and a distributional transformation which is related to zero-biasing.
