On a class of distributions stable under random summation

TL;DR

This paper explores a class of distributions called ta-stable, which remain stable under random summation where the number of summands is also random, revealing connections to Chebyshev polynomials and hyperbolic secant distributions.

Contribution

It introduces the concept of ta-stability, analyzes the semigroup structure, and characterizes the characteristic functions using analytic function iteration, linking to Chebyshev polynomials and hyperbolic secant distributions.

Findings

Characteristic function expressed via Chebyshev polynomials

ta-normal distribution corresponds to hyperbolic secant distribution

Identifies properties and examples of ta-stable distributions

Abstract

We investigate a family of distributions having a property of stability-under-addition, provided that the number $\nu$ of added-up random variables in the random sum is also a random variable. We call the corresponding property a \,$\nu$-stability and investigate the situation with the semigroup generated by the generating function of $\nu$ is commutative. Using results from the theory of iterations of analytic functions, we show that the characteristic function of such a $\nu$-stable distribution can be represented in terms of Chebyshev polynomials, and for the case of $\nu$-normal distribution, the resulting characteristic function corresponds to the hyperbolic secant distribution. We discuss some specific properties of the class and present particular examples.