The characteristic function of the discrete Cauchy distribution (In Memory of T. Cacoullos)

TL;DR

This paper introduces the Cauchy-Cacoullos family of discrete distributions, derives their characteristic functions, and explores their properties, extending to discrete Student distributions with odd degrees of freedom.

Contribution

It presents a new family of integer-valued Cauchy-type distributions and analyzes their characteristic functions and properties, extending to discrete Student distributions.

Findings

Derived the characteristic function of the Cauchy-Cacoullos family.

Identified distributional properties similar to continuous Cauchy distributions.

Extended results to discrete Student distributions with odd degrees of freedom.

Abstract

A new family of integer-valued Cauchy-type distributions is introduced, the {\it Cauchy-Cacoullos family}. The characteristic function is evaluated, showing some interesting distributional properties, similar to the ordinary (continuous) Cauchy scale family. The results are extendable to discrete Student-type distributions with odd degrees of freedom. Keywords: Fourier series; discrete Student distribution; Cauchy-Cacoullos family.