On a Problem of M. Kac on Laplace Distributions

TL;DR

This paper addresses a problem posed by M. Kac regarding the characterization of Laplace distributions through nonlinear operations on characteristic functions, providing counterexamples, an affirmative refined solution, and a general framework.

Contribution

It offers the first counterexamples to Kac's problem, a positive solution to a refined version, and a new framework for distribution characterization problems.

Findings

Counterexamples to Kac's original problem.

An affirmative answer to a refined problem version.

Development of a general framework for characterization problems.

Abstract

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.