Inversion of analytic characteristic functions and infinite convolutions of exponential and Laplace densities

TL;DR

This paper explores conditions under which certain entire function quotients serve as characteristic functions, revealing that their associated probability densities can be expressed as infinite series of exponential or Laplace densities, with applications to various examples.

Contribution

It establishes new criteria for characteristic functions derived from entire functions and expresses their densities as generalized Dirichlet series involving exponential or Laplace densities.

Findings

Certain quotients of entire functions are characteristic functions.

Probability densities can be represented as infinite linear combinations of exponential or Laplace densities.

Applications demonstrate the practical relevance of the theoretical results.

Abstract

We prove that certain quotients of entire functions are characteristic functions. Under some conditions, the probability measure corresponding to a characteristic function of that type has a density which can be expressed as a generalized Dirichlet series, which in turn is an infinite linear combination of exponential or Laplace densities. These results are applied to several examples.