Linear combination of bilateral gamma random variables: distributional theory and approximations

TL;DR

This paper derives the exact distribution and properties of linear combinations of bilateral gamma variables, develops Stein characterizations for approximations, and applies findings to exponential stock models.

Contribution

It provides the first exact distributional results for linear combinations of bilateral gamma variables and develops Stein-based approximation methods.

Findings

Exact distribution formulas for linear combinations of bilateral gamma variables.

Explicit error bounds for distributional approximations in Kolmogorov and Wasserstein distances.

Application of theoretical results to exponential stock models.

Abstract

In this article, we obtain the exact distribution of a linear combination of bilateral gamma (BG) random variables (r.v.s). Next, we discuss the distributional properties of the linear combination of BG r.v.s, including probability density function, cumulant generating function and characteristic function. A Stein characterization is developed, which leads us to several distributional approximation results with explicit error bounds in both Kolmogorov and Wasserstein distances. Related limit theorems are also discussed. Furthermore, we show that the associated L\'evy processes are finite-variation processes with BG distributed increments having random parameters. Finally, we apply our results in exponential stock models.