# The variance-gamma ratio distribution

## TL;DR

This paper derives the exact probability density function and distributional properties of the ratio of two independent variance-gamma random variables, including tail behavior and moments, with special cases for symmetric variables.

## Contribution

It provides the first exact formulas for the ratio distribution of variance-gamma variables, including density, tail probabilities, and moments, expanding understanding of this ratio's behavior.

## Key findings

- Exact density function of the ratio derived
- Parameter regimes for bounded density identified
- Asymptotic tail probability approximations provided

## Abstract

Let $X$ and $Y$ be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the ratio $X/Y$ is derived. Some basic distributional properties are also derived, including identification of parameter regimes under which the density is bounded, asymptotic approximations of tail probabilities, and fractional moments; in particular, we see that the mean is undefined. In the case that $X$ and $Y$ are independent symmetric variance-gamma random variables, an exact formula is also given for the cumulative distribution function of the ratio $X/Y$.