On a Problem of M. Kac on Laplace Distributions

We give counterexamples to a problem of M. Kac in the Scottish Book, which asks whether a certain nonlinear operation on two characteristic functions characterizes Laplace distributions, in analogy with the Cram\'er--L\'evy theorem for Gaussian distributions. We then give an affirmative answer to a refined version of the problem. Finally, we develop a general framework for such characterization problems, construct generalized counterexamples, and pose some open questions.