# A Note on the Existence of the Multivariate Gamma Distribution

@article{Royen2016ANO, title={A Note on the Existence of the Multivariate Gamma Distribution}, author={Thomas Royen}, journal={arXiv: Probability}, year={2016} }

The p-variate gamma distribution in the sense of Krishnamoorthy and Parthasarathy exists for all positive integer degrees of freedom d and at least for all real values d > p-2, p > 1. For special structures of the "associated" covariance matrix it also exists for all positive d. In this paper a relation between central and non-central multivariate gamma distributions is shown, which implies the existence of the p-variate gamma distribution at least for all non-integer d greater than the integer…

## 6 Citations

Non-Central Multivariate Chi-Square and Gamma Distributions

- Mathematics
- 2016

A (p-1)-variate integral representation is given for the cumulative distribution function of the general p-variate non-central gamma distribution with a non-centrality matrix of any admissible rank.…

A combinatorial proof of the Gaussian product inequality in the MTP 2 case

- Mathematics, Computer Science
- 2022

: A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector X = ( X 1 ,...,X d ) of arbitrary length can be…

Product Inequalities for Multivariate Gaussian, Gamma, and Positively Upper Orthant Dependent Distributions

- Mathematics, Computer Science
- 2022

A strong version of the Gaussian product inequality for multivariate gamma distributions in the case of nonnegative correlations is established, thereby extending a result recently derived by Genest and Ouimet.

Stein characterizations for linear combinations of gamma random variables

- Mathematics
- 2017

In this paper we propose a new, simple and explicit mechanism allowing to derive Stein operators for random variables whose characteristic function satisfies a simple ODE. We apply this to study…

A combinatorial proof of the Gaussian product inequality beyond the MTP2 case

- Mathematics, Computer Science
- 2021

: A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector X = ( X 1 ,...,X d ) of arbitrary length can be…

Some improved Gaussian correlation inequalities for symmetrical n-rectangles extended to some multivariate gamma distributions and some further probability inequalities

- Mathematics
- 2020

The Gaussian correlation inequality (GCI) for symmetrical n-rectangles is improved if the absolute components have a joint MTP2-distribution (multivariate totally positive of order 2). Inequalities…

## References

SHOWING 1-10 OF 11 REFERENCES

Non-Central Multivariate Chi-Square and Gamma Distributions

- Mathematics
- 2016

A (p-1)-variate integral representation is given for the cumulative distribution function of the general p-variate non-central gamma distribution with a non-centrality matrix of any admissible rank.…

Some probability inequalities for multivariate gamma and normal distributions

- Mathematics
- 2015

The Gaussian correlation inequality for multivariate zero-mean normal probabilities of symmetrical n-rectangles can be considered as an inequality for multivariate gamma distributions (in the sense…

Which multivariate gamma distributions are infinitely divisible

- Mathematics
- 2006

In the literature, the multivariate gamma distributions on Rn have several non-equivalent definitions. Many authors require only that the marginal distributions are ordinary gamma distributions…

Existence and non-existence of the non-central Wishart distributions.

- Mathematics
- 2011

The problem considered in this paper is to find when the non-central Wishart distribution, defined on the cone Pd of semi positive definite matrices of order d and with a real valued shape parameter,…

Integral Representations and Approximations for Multivariate Gamma Distributions

- Mathematics
- 2007

AbstractLet R be a p×p-correlation matrix with an “m-factorial” inverse R−1 = D − BB′ with diagonal D minimizing the rank m of B. A new
$$\left(m+1 \atop 2\right)$$-variate integral representation is…

A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions

- Mathematics
- 2014

An extension of the Gaussian correlation conjecture (GCC) is proved for multivariate gamma distributions (in the sense of Krishnamoorthy and Parthasarathy). The classical GCC for Gaussian probability…

Infinite divisibility of multivariate gamma distributions and M - matrices

- Mathematics
- 1989

Suppose X=(X1,..,Xp)', has the Laplace transform ψ (t) = ∣Ι + VT∣-½, where V is a positive definite matrix and T= diag(t1,..,tp). It is shown that ψ(t) is infinitely divisible if and only if DV-1 D…

Relationships Between Biased and Unbiased Rayleigh Distributions

- Mathematics
- 1967

A conjecture made in [8] regarding the relationship between the p-dimensional biased Rayleigh distribution and the $( {p + 1} )$-dimensional unbiased Rayleigh distribution is proved. Other results…

Modern Multivariate Statistical Analysis: A Graduate Course and Handbook

- Psychology, Mathematics
- 1985