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Simulations of multivariate gamma distributions and multifactor gamma distributions

Philippe Bernardoff, Bénédicte Puig

TL;DR

This work extends simulations from bivariate to multivariate and multifactor gamma distributions by developing a general mgd framework defined via $\mathbb{E}\{\exp(-\boldsymbol{\theta}^{\top}\mathbf{X})\}=[P(\boldsymbol{\theta})]^{-\lambda}$ with an affine $P_n$, and by constructing $\mathbf{X}=\mathbf{Y}+\mathbf{Z}$ to enable practical simulation. It derives a general joint distribution form $\boldsymbol{\gamma}_{(P_n,\lambda)}$, marginal gamma laws, and multifactor extensions with conditional Laplace transforms, including explicit infinite-divisibility criteria and special functions (e.g., Lauricella/Horn) for densities in low dimensions. The paper then provides concrete conditional-Laplace-transform expressions and detailed simulation algorithms for key cases ($n=2,3,4$) in the $k=1$ setting, and discusses a Markovian simplification that yields an efficient sequential simulation scheme for general $n$. It culminates with extensive simulations in dimensions 2–5 illustrating bgd, tgd, quadrivariate gamma, and Mmgd constructions, validating the methods and clarifying parameter choices and correlation structures. The results significantly extend the toolbox for simulating high-dimensional infinitely divisible gamma-type models, with potential applications in multivariate risk, classification within exponential families, and stochastic modeling of dependent gamma processes.

Abstract

This article provides a general expression for infinitely divisible multivariate gamma distributions defined by their Laplace transforms, as well as the conditional Laplace transform of infinitely divisible multivariate gamma distributions.We give algorithms for simulating infinitely divisible gamma distributions and infinitely divisible multifactor gamma distributions in dimension 2,3,4 and for all dimensions greater than 2 in the Markovian case. We give examples of simulations in dimension 2,3,4 and in dimension 5 in the Markovian case.

Simulations of multivariate gamma distributions and multifactor gamma distributions

TL;DR

This work extends simulations from bivariate to multivariate and multifactor gamma distributions by developing a general mgd framework defined via with an affine , and by constructing to enable practical simulation. It derives a general joint distribution form , marginal gamma laws, and multifactor extensions with conditional Laplace transforms, including explicit infinite-divisibility criteria and special functions (e.g., Lauricella/Horn) for densities in low dimensions. The paper then provides concrete conditional-Laplace-transform expressions and detailed simulation algorithms for key cases () in the setting, and discusses a Markovian simplification that yields an efficient sequential simulation scheme for general . It culminates with extensive simulations in dimensions 2–5 illustrating bgd, tgd, quadrivariate gamma, and Mmgd constructions, validating the methods and clarifying parameter choices and correlation structures. The results significantly extend the toolbox for simulating high-dimensional infinitely divisible gamma-type models, with potential applications in multivariate risk, classification within exponential families, and stochastic modeling of dependent gamma processes.

Abstract

This article provides a general expression for infinitely divisible multivariate gamma distributions defined by their Laplace transforms, as well as the conditional Laplace transform of infinitely divisible multivariate gamma distributions.We give algorithms for simulating infinitely divisible gamma distributions and infinitely divisible multifactor gamma distributions in dimension 2,3,4 and for all dimensions greater than 2 in the Markovian case. We give examples of simulations in dimension 2,3,4 and in dimension 5 in the Markovian case.
Paper Structure (16 sections, 26 theorems, 115 equations, 13 figures, 6 algorithms)

This paper contains 16 sections, 26 theorems, 115 equations, 13 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Probability distributions of multivariate gamma distributions
  3. Conditional Laplace transform
  4. Conditional Laplace transform in the particular case $k=1$
  5. Applications of the main results
  6. A particular case in the case $k=1$
  7. The case $n=2$ and $k=1$
  8. The case $n=3$ and $k=1$
  9. The case $n=4$ and $k=1$
  10. Markovian multivariate gamma distributions
  11. Simulations
  12. Simulations in dimension 2
  13. Simulations in dimension 3
  14. Simulations in dimension 4
  15. Simulations of Mmgd in dimension 5
  16. ...and 1 more sections

Key Result

Proposition 1

A random vector $\boldsymbol{X}$ with distribution $\boldsymbol{\gamma}$$_{P,\boldsymbol{\Lambda}}$ can be obtained in the following way: Let $\boldsymbol{Y}$ be a random vector with distribution $\boldsymbol{\gamma}$$_{\left( P,\lambda\right) }.$ Let $\boldsymbol{Z}=\left( Z_{1},\dots,Z_{n}\righ

Figures (13)

  • Figure 1: Distributions and simulations of $\mathbf{X}_{\left[2\right]}$ and $\mathbf{Y}_{\left[2\right]}$
  • Figure 2: Distributions and simulations of $\mathbf{X}^{\prime}_{\left[2\right]}$ and $\mathbf{Y}^{\prime}_{\left[2\right]}$
  • Figure 3: Distributions of $X_{1},X_{2},X_{3}$
  • Figure 4: Distribution and simulation of $\mathbf{X}_{\left[3\right]}$
  • Figure 5: Distributions of $Y_{1},Y_{2},Y_{3}$
  • ...and 8 more figures

Theorems & Definitions (37)

  • Proposition 1
  • Remark 2
  • Proposition 3
  • Theorem 4
  • Proposition 5
  • Theorem 6
  • Corollary 7
  • Theorem 8
  • Remark 9
  • Remark 10
  • ...and 27 more