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On a Problem of M. Kac on Laplace Distributions

Robert Koirala

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 Cramér--Lévy 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.

On a Problem of M. Kac on Laplace Distributions

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 Cramér--Lévy 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.

Paper Structure

This paper contains 7 sections, 14 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Counterexamples
  3. A positive result under geometric infinite divisibility
  4. Some generalities
  5. Generalized counterexamples
  6. The Gaussian limit of counterexamples and some open questions
  7. A discrete analog

Key Result

Proposition 2

The functions are characteristic functions on $\mathbb{R}$ satisfying eq:defining-equation, but neither is of the form eq:laplace. In particular, Problem problem-kac is false as stated. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (35)

  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Definition 4: MR773445
  • Example 5
  • Theorem 6: MR773445; steutel1990set
  • Theorem 7
  • proof
  • Remark 8
  • ...and 25 more