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

Kalyan Barman, Palaniappan Vellaisamy

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 Lévy processes are finite-variation processes with BG distributed increments having random parameters. Finally, we apply our results in exponential stock models.

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

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 Lévy processes are finite-variation processes with BG distributed increments having random parameters. Finally, we apply our results in exponential stock models.
Paper Structure (19 sections, 12 theorems, 113 equations)

This paper contains 19 sections, 12 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Bilateral-gamma distributions
  4. Function spaces and probability metrics
  5. Essence of Stein's Method
  6. Linear combinations of BG random variables
  7. Notations
  8. Probability density function
  9. Moment generating function
  10. Moments
  11. Characterizations
  12. Stein equation
  13. Approximation of linear combinations of BG random variables
  14. Approximation for sums
  15. Limit of compound Poisson distributions
  16. ...and 4 more sections

Key Result

Theorem 3.1

Let $T_n$ be defined as in defTn. Then $T_n\sim BG(\frac{\alpha}{1-\alpha}, L_n+p,\frac{\beta}{1-\beta},M_n+q)$, where $\alpha, \beta$ are defined in ref1 and the r.v.s $L_n$ and $M_n$ have distributions defined in ref2.

Theorems & Definitions (30)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • Theorem 3.7
  • ...and 20 more