Distributions that are both log-symmetric and R-symmetric

TL;DR

This paper characterizes distributions that are both log-symmetric and R-symmetric, revealing they form a subset of those moment-equivalent to the lognormal, including some known and new distributions with specific properties.

Contribution

It provides a complete characterization of doubly symmetric distributions, expanding understanding of their structure and relation to the lognormal distribution.

Findings

Doubly symmetric distributions are a subset of those moment-equivalent to the lognormal.

Includes the lognormal and some Berg/Askey class distributions.

Stieltjes classes are not doubly symmetric.

Abstract

Two concepts of symmetry for the distributions of positive random variables $Y$ are log-symmetry (symmetry of the distribution of $\log Y$) and R-symmetry [7]. In this paper, we characterise the distributions that have both properties, which we call doubly symmetric. It turns out that doubly symmetric distributions constitute a subset of those distributions that are moment-equivalent to the lognormal distribution. They include the lognormal, some members of the Berg/Askey class of distributions, and a number of others for which we give an explicit construction (based on work of A.J. Pakes) and note some properties; Stieltjes classes, however, are not doubly symmetric.