text:
\section{Introduction}

This paper is motivated by the opportunity to produce explicit cumulative
distribution functions (cdf) and probability distribution functions (pdf) of
multivariate gamma distributions and multi-factor gamma distributions. In this
way, we present two new classes of multivariate copulas generalizing the
Farlie-Gumbel-Morgenstern Copulas. From Sklar (Sklar, 1959) who states that
the cdf $F$ of a random vector $\mathbf{X}=\left(  X_{1},\ldots,X_{n}\right)
$ with continuous marginal cdfs can be uniquely written in the form%
\begin{equation}
F\left(  x_{1},\ldots,x_{n}\right)  =C\left[  F_{1}\left(  x_{1}\right)
,\ldots,F_{n}\left(  x_{n}\right)  \right]  ,\text{ }\mathbf{x}=\left(
x_{1},\ldots,x_{n}\right)  \in\mathbb{R}^{n}, \label{Sklar1}%
\end{equation}
where $n$ is the dimension of the random vector $\mathbf{X}$, where $C:\left[
0,1\right]  ^{n}\rightarrow\left[  0,1\right]  $ is a \emph{copula} and where
$F_{1},\ldots,F_{n}$ are the marginal cdfs of $\mathbf{X}$.

If $f$ is the pdf of the random vector $\mathbf{X}$ and $f_{1},\ldots,f_{n}$
the marginal pdfs of $\mathbf{X,}$ then%
\[
c\left(  u_{1},\ldots,u_{n}\right)  =\frac{\partial^{n}}{\partial u_{1}%
\ldots\partial u_{n}}C\left(  u_{1},\ldots,u_{n}\right)  ,
\]
and we have the following equality%
\begin{equation}
f\left(  x_{1},\ldots,x_{n}\right)  =c\left[  F_{1}\left(  x_{1}\right)
,\ldots,F_{n}\left(  x_{n}\right)  \right]  f_{1}\left(  x_{1}\right)  \ldots
f_{n}\left(  x_{n}\right)  . \label{Sklar2}%
\end{equation}

From Equality (\ref{Sklar1}) it is possible to express the cdf $F$ of
$\mathbf{X}$. This expression cannot simply explicitly give the copula $C$ for
the \textit{multivariate gamma distribution associated with} $\left(
P,\lambda\right)  $ and the \textit{multi-factor gamma distribution associated
with} $\left(  P,\Lambda\right)  .$ But with Joe (Joe, 1997), we can give
copulas deduced of the Laplace transform of the \textit{multivariate gamma
distribution associated with} $\left(  P,\lambda\right)  $ and the Laplace
transform of the \textit{multi-factor gamma distribution associated with}
$\left(  P,\Lambda\right)  .$ By applying the formulas (\ref{Sklar1}) and
(\ref{Sklar2}) we can give a new \textit{multivariate gamma distribution
associated with} $\left(  P,\lambda\right)  $ and a new\textit{\ multi-factor
gamma distribution associated with} $\left(  P,\Lambda\right)  $ for which we
have an explicit formula for its cdf and its pdf.

The paper is organized as follows. Section 2 gives definitions of multivariate
gamma distributions and multi-factor gamma distributions for which Laplace
transform is given, and considers the bidimentional and tridimentional case.
Section 3 defines the Laplace copula. Section 4 states the two main results.
Section 5 applies the main cases to bidimensional and tridimensional cases. In
particular, the Joe's family BB10 is generalized. For ease of the fluent
exposition of the paper, proofs are collected in the Appendix.

\section{Multivariate gamma distributions and multi-factor\textit{\ }gamma
distributions}

In the literature, the multivariate gamma distributions on $\mathbb{R}^{n}$
have several non-equivalent definitions. Many authors require only that the
marginal distributions are ordinary gamma distributions (Balakrishnan
\textit{et al}., 1997). In the present paper, we use the extension of the
classical one-dimensional definition to $\mathbb{R}^{n}$ obtained as follows:
we consider an affine polynomial $P\left(  \boldsymbol{\theta}\right)  $ in
$\boldsymbol{\theta}=\left(  \theta_{1},\ldots,\theta_{n}\right)  $ where
`affine' means that, for $j=1,\ldots,n,$ $\partial^{2}P/\partial\theta_{j}%
^{2}=0$. We also assume that $P\left(  \mathbf{0}\right)  =1.$ For instance,
for $n=2,$ we have $P\left(  \theta_{1},\theta_{2}\right)  =1+p_{1}\theta
_{1}+p_{2}\theta_{2}+p_{\left\{  1,2\right\}  }\theta_{1}\theta_{2}$.

We denote by $\mathfrak{P}_{n}=\mathfrak{P}\left(  \left[  n\right]  \right)
$ the family of all subsets of $\left[  n\right]  $ and $\mathfrak{P}%
_{n}^{\ast} $ the family of non-empty subsets of $\left[  n\right]  =\left\{
1,\ldots,n\right\}  .$ For simplicity, if $n$ is fixed and if there is no
ambiguity, we denote these families by $\mathfrak{P}$ and $\mathfrak{P}^{\ast
}$, respectively.

We denote by $\mathbb{N}$ the set of non-negative integers. If $\mathbf{z}%
=\left(  z_{1},\ldots,z_{n}\right)  \in\mathbb{R}^{n}$ and $\mathbf{\alpha
}=\left(  \alpha_{1},\ldots,\alpha_{n}\right)  \in\mathbb{N}^{n},$ then
$\mathbf{\alpha}!=\alpha_{1}!\ldots\alpha_{n}!$, $\left\vert \mathbf{\alpha
}\right\vert =\alpha_{1}+\ldots+\alpha_{n}$, $a_{\mathbf{\alpha}}%
=a_{\alpha_{1},\ldots,\alpha_{n}}$ and
\[
\mathbf{z}^{\mathbf{\alpha}}=\prod\limits_{i=1}^{n}z_{i}^{\alpha_{i}}%
=z_{1}^{\alpha_{1}}\ldots z_{n}^{\alpha_{n}}.
\]
For $T\ $in $\mathfrak{P}_{n},$ we simplify the above notation by writing
$\mathbf{z}^{T}=\prod\nolimits_{t\in T}z_{t}$ instead of $\mathbf{z}%
^{\mathbf{1}_{T}}$ where
\[
\mathbf{1}_{T}=\left(  \alpha_{1},\ldots,\alpha_{n}\right)  \text{ with
}\alpha_{i}=1\text{ if }i\in T\text{ and }\alpha_{i}=0\text{ if }i\notin T.
\]
We also write $\mathbf{z}^{-T}$ for $\prod\nolimits_{t\in T}1/z_{t}.$ For a
mapping $a:\mathfrak{P}\rightarrow\mathbb{R}$, we shall use the notation
$a:\mathfrak{P}\rightarrow\mathbb{R}$, $T\mapsto a_{T}.$ In this notation, an
affine polynomial with constant term equal to $1$ is $P\left(
\boldsymbol{\theta}\right)  =\sum_{T\in\mathfrak{P}}p_{T}\boldsymbol{\theta
}^{T},$ with $p_{\varnothing}=1.$ For simplicity, if $T=\left\{  t_{1}%
,\ldots,t_{k}\right\}  ,$ we denote $a_{\left\{  t_{1},\ldots,t_{k}\right\}
}=a_{t_{1}\ldots t_{k}}.$ The indicator function of a set $S$ is denoted by
$\mathbbm{1}_{S},$ that is, $\mathbbm{1}_{S}\left(  x\right)  =1$ for $x\in S$
and $0$ for $x\notin S$.

We fix $\lambda>0.$ If a random vector $\mathbf{X=}\left(  X_{1},\ldots
,X_{n}\right)  $ on $\mathbb{R}^{n}$ with probability distribution (pd)
$\mu_{\mathbf{X}}$ is such that its Laplace transform is%
\begin{equation}
\mathbb{E}\left\{  \exp\left[  -\left(  \theta_{1}X_{1}+\cdots+\theta_{n}%
X_{n}\right)  \right]  \right\}  =\left[  P\left(  \boldsymbol{\theta}\right)
\right]  ^{-\lambda}, \label{TLMGD}%
\end{equation}
where $\mathbb{E}$ denotes the expectation, for a set of $\boldsymbol{\theta}$
with non-empty interior, then we denote $\mu_{\mathbf{X}}=\gamma_{\left(
P,\lambda\right)  },$ and $\gamma_{\left(  P,\lambda\right)  }$ will be called
the \textit{multivariate gamma distribution associated with} $\left(
P,\lambda\right)  $. These multivariate gamma distributions occur naturally in
the classification of natural exponential families in $\mathbb{R}^{n}$
(Bar-Lev \textit{et al.,} 1994).

The marginal distributions of the \textit{multivariate gamma distribution
associated with} $\left(  P,\lambda\right)  $ are ordinary gamma distributions
of parameters $\left(  p_{i},\lambda\right)  $ for $i=1,\ldots,n,$ with
Laplace transform $\left(  1+p_{i}\theta_{i}\right)  ^{-\lambda},$ and pd
$\gamma_{\left(  p_{i},\lambda\right)  }\left(  \text{\emph{d}}x\right)
=x^{\lambda-1}p_{i}^{-\lambda}/\Gamma\left(  \lambda\right)  \exp\left(
-x/p_{i}\right)  \mathbbm{1}_{\left(  0,\infty\right)  }\left(  x\right)
$\emph{d}$x.$ We extend the first definition to the\textit{\ multi-factor
gamma distribution associated with} $\left(  P,\Lambda\right)  $ where
$\Lambda=\left(  \lambda,\lambda_{1},\ldots,\lambda_{n}\right)  $ and
$\lambda_{i}\geqslant\lambda>0$ for all $i=1,\ldots,n$ by its Laplace
transform%
\begin{equation}
\mathbb{E}\left\{  \exp\left[  -\left(  \theta_{1}X_{1}+\cdots+\theta_{n}%
X_{n}\right)  \right]  \right\}  =\left[  P\left(  \boldsymbol{\theta}\right)
\right]  ^{-\lambda}\prod_{i=1}^{n}\left(  1+p_{i}\theta_{i}\right)
^{-\left(  \lambda_{i}-\lambda\right)  }. \label{TLMMGD}%
\end{equation}
The marginal distributions of the \textit{multi-factor gamma distribution
associated with} $\left(  P,\Lambda\right)  $ are ordinary gamma distributions
of parameters $\left(  p_{i},\lambda_{i}\right)  $ for $i=1,\ldots,n.$

We state first a proposition, whose proof is obvious.

\begin{proposition}
\label{buildMFGD}A random vector $\mathbf{X}$ with distribution
$\mathbf{\gamma}_{P,\Lambda}$ can be obtained in the following way: \newline
Let $\mathbf{Y}$ be a vector with distribution $\mathbf{\gamma}_{P,\lambda}.$
Let $\mathbf{Z}=\left(  Z_{1},\dots,Z_{n}\right)  $ be a random vector
constituted of independent margins for which its pds are $\gamma_{\left(
p_{i},\lambda_{i}\right)  },$ and such that $Z$ and $Y$ are independent. Then
the vector $\mathbf{X}=\mathbf{Y}+\mathbf{Z}$ has Laplace transform
(\ref{TLMMGD}), and consequently is a\textit{\ multi-factor gamma distribution
}associated to $\left(  P,\Lambda\right)  .$
\end{proposition}

For the bidimensional case, Dussauchoy and Berland, (Dussauchoy and Berland,
1972) consider the random vector $\mathbf{X}=(X_{1},X_{2})$ with Laplace
transform
\begin{align}
L_{\mathbf{X}}\left(  \theta_{1},\theta_{2}\right)   &  =\left(  1+p_{1}%
\theta_{1}\right)  ^{-\lambda_{1}}\left(  1+p_{2}\theta_{2}\right)
^{-\lambda_{2}}\left[  1-\frac{r_{12}p_{1}p_{2}\theta_{1}}{\left(
1+p_{1}\theta_{1}\right)  \left(  1+p_{2}\theta_{2}\right)  }\right]
^{-\lambda}\nonumber\\
&  =\left(  1+p_{1}\theta_{1}\right)  ^{-\left(  \lambda_{1}-\lambda\right)
}\left(  1+p_{2}\theta_{2}\right)  ^{-\left(  \lambda_{2}-\lambda\right)
}\left(  1+p_{1}\theta_{1}+p_{2}\theta_{2}+p_{12}\theta_{1}\right)
^{-\lambda}, \label{DuchaussoyBerland}%
\end{align}
where $r_{12}=1-p_{12}/\left(  p_{1}p_{2}\right)  >0$ and $p_{1},p_{2}%
,p_{12}>0 $.

Unfortunately the Laplace transform of these pds is simple, but its pdfs and
cdfs are unknown, except for the case $n=2$ for the \textit{multivariate gamma
distribution associated with} $\left(  P,\lambda\right)  $. Let $F_{m}^{p}$ be
the generalized hypergeometric function (Slater, 1966) defined by%
\begin{equation}
F_{m}^{p}\left(  \alpha_{1},\ldots,\alpha_{p};\beta_{1},\ldots,\beta
_{m};z\right)  =\sum_{k=0}^{\infty}\frac{\left(  \alpha_{1}\right)  _{k}%
\cdots\left(  \alpha_{p}\right)  _{k}}{\left(  \beta_{1}\right)  _{k}%
\cdots\left(  \beta_{m}\right)  _{k}}\frac{z^{k}}{k!}, \label{Hypergeomconf}%
\end{equation}
where $\left(  a\right)  _{k}=\Gamma\left(  a+k\right)  /\Gamma\left(
a\right)  $ for $a>0$ and $k\in\mathbb{N}$, or more generally by $(a)_{0}=1$,
$(a)_{n+1}=(a+n)(a),$ $\forall n\in\mathbb{N}$, $\forall a\in\mathbb{R}$, is
the Pochhammer's symbol. For simplification, we denote $F_{m}^{0}$ by $F_{m}$.
Bernardoff (Bernardoff, 2006) gives the

\begin{proposition}
Let $P\left(  \theta_{1},\theta_{2}\right)  =1+p_{1}\theta_{1}+p_{2}\theta
_{2}+p_{12}\theta_{1}\theta_{2}$ be an affine polynomial where $p_{1},p_{2}>0$
and $p_{1,2}>0.$ Let $\mu=\boldsymbol{\gamma}_{\left(  P,\lambda\right)  }$ be
the gamma distribution associated to $\left(  P,\lambda\right)  .$ The measure
$\mu$ exists if and only if $c=\left(  p_{1}p_{2}-p_{12}\right)  /p_{12}%
^{2}=p_{1}p_{2}/p_{12}^{2}r_{12}>0$. Then we have%
\begin{equation}
\boldsymbol{\gamma}_{\left(  P,\lambda\right)  }\left(  \text{\emph{d}}%
x_{1},\text{\emph{d}}x_{2}\right)  =\frac{p_{12}^{-\lambda}}{\Gamma\left(
\lambda\right)  ^{2}}\mathbf{e}^{-\frac{p_{2}}{p_{12}}x_{1}-\frac{p_{1}%
}{p_{12}}x_{2}}\left(  x_{1}x_{2}\right)  ^{\lambda-1}F_{1}\left(
\lambda;cx_{1}x_{2}\right)  \mathbbm{1}_{\left(  0,\infty\right)  ^{2}}\left(
\mathbf{x}\right)  \,\text{\emph{d}}\left(  \mathbf{x}\right)  . \label{mud2}%
\end{equation}

\end{proposition}

For the case $\Lambda=\left(  \lambda,\lambda,\lambda_{2}\right)  $, the
\textit{multi-factor gamma distribution associated with} $\left(
P,\Lambda\right)  $ is named by Chatelain \emph{et al.} (Chatelain \emph{et
al., }2008) the\textit{\ multisensor gamma distribution associated with}
$\left(  P,\lambda,\lambda_{2}\right)  $ and they have proved that its pd is
given by the equality%
\begin{align}
\boldsymbol{\gamma}_{\left(  P,\Lambda\right)  }\left(  \text{d}x_{1}%
,\text{d}x_{2}\right)   &  =\frac{p_{12}^{-\lambda}p_{2}^{-\left(  \lambda
_{2}-\lambda\right)  }1}{\Gamma\left(  \lambda\right)  \Gamma\left(
\lambda_{2}\right)  }x_{1}^{\lambda-1}x_{2}^{\lambda_{2}-1}\mathbf{e}%
^{-\frac{p_{2}}{p_{12}}x_{1}-\frac{p_{1}}{p_{12}}x_{2}}\Phi_{3}(\lambda
_{2}-\lambda;\lambda_{2};c\frac{p_{12}}{p_{2}}x_{2};cx_{1}x_{2})\nonumber\\
&  \times\mathbbm{1}_{\left(  0,\infty\right)  ^{2}}\left(  x_{1}%
,x_{2}\right)  \,\text{d}x_{1}\text{d}x_{2}\text{,} \label{BMGD}%
\end{align}
where%
\begin{equation}
\Phi_{3}\left(  a;b;x,y\right)  =\sum_{m,n\geqslant0}\frac{\left(  a\right)
_{m}}{\left(  b\right)  _{m+n}}\frac{x^{m}}{m!}\frac{y^{n}}{n!}
\label{HornFunction}%
\end{equation}
is the Horn function.

For the bidimensional general case, we have the following Theorem. Let $F_{I}$
be the function defined by%
\begin{equation}
F_{I}\left(  a,b,c,\mathbf{z}\right)  =\sum_{m_{1},m_{2},m_{3}=0}^{\infty
}\frac{\left(  a\right)  _{m_{1}}\left(  b\right)  _{m_{2}}\left(  c\right)
_{m_{3}}}{\left(  a+c\right)  _{m_{1}+m_{3}}\left(  b+c\right)  _{m_{2}+m_{3}%
}}\frac{z_{1}^{m_{1}}}{m_{1}!}\frac{z_{2}^{m_{2}}}{m_{2}!}\frac{z_{3}^{m_{3}}%
}{m_{3}!}; \label{genLauricella1}%
\end{equation}
it is a particular generalized Lauricella function defined, by example, in
(Panda, 1973).

\begin{theorem}
\label{Proposition_multifactorbig2}The pd of $\boldsymbol{\gamma}_{\left(
P,\left(  \lambda,\lambda_{1},\lambda_{2}\right)  \right)  }$ is given by the
equality%
\begin{align}
\boldsymbol{\gamma}_{\left(  P,\left(  \lambda,\lambda_{1},\lambda_{2}\right)
\right)  }\left(  \text{\emph{d}}x_{1},\text{\emph{d}}x_{2}\right)   &
=\frac{p_{12}^{-\lambda}p_{1}^{-\left(  \lambda_{1}-\lambda\right)  }%
p_{2}^{-\left(  \lambda_{2}-\lambda\right)  }}{\Gamma\left(  \lambda
_{1}\right)  \Gamma\left(  \lambda_{2}\right)  }x_{1}^{\lambda_{1}-1}%
x_{2}^{\lambda_{2}-1}\mathbf{e}^{-\left(  \frac{p_{2}}{p_{12}}x_{1}%
+\frac{p_{1}}{p_{12}}x_{2}\right)  }\nonumber\\
&  \hspace*{-3cm}\times F_{I}\left(  \lambda_{1}-\lambda,\lambda_{2}%
-\lambda,\lambda,\frac{p_{12}}{p_{1}}x_{1},\frac{p_{12}}{p_{2}}x_{2}%
,cx_{1}x_{2}\right)  \mathbbm{1}_{\left(  0,\infty\right)  ^{2}}\left(
x_{1},x_{2}\right)  \,\text{\emph{d}}x_{1}\text{\emph{d}}x_{2}\text{,}
\label{mulbigamma2}%
\end{align}

\end{theorem}

If we get $\lambda_{1}=\lambda$ in the equality (\ref{mulbigamma2}), we obtain
Chatelain and Tourneret's result (\ref{BMGD}) because%
\begin{align*}
F_{I}\left(  0,\lambda_{2}-\lambda,\lambda,z_{1},z_{2},z_{3}\right)   &  =\\
\hspace*{-3cm}\sum_{m_{2},m_{3}=0}^{\infty}\frac{\left(  b\right)  _{m_{2}}%
}{\left(  b+c\right)  _{m_{2}+m_{3}}}\frac{z_{2}^{m_{2}}}{m_{2}!}\frac
{z_{3}^{m_{3}}}{m_{3}!}  &  =\Phi_{3}\left(  b;b+c;z_{2},z_{3}\right)  .
\end{align*}

Bernardoff (Bernardoff, 2006) gives the following Proposition:

\begin{proposition}
Let $\mu$ be a \textit{multivariate gamma distribution }on $\mathbb{R}^{n}$
associated to $\left(  P,\lambda\right)  .$ Assume that $\mu$ is not
concentrated on a linear subspace of $\mathbb{R}^{n}$ of the form
$\{x\in\mathbb{R}^{n};\ x_{k}=0\}$ for some $k$ in $\left[  n\right]
=\{1,\ldots,n\}.$ Then:

\begin{enumerate}
\item[\emph{(i)}] For all $i\in\left[  n\right]  ,$ $p_{i}\neq0$.

\item[\emph{(ii)}] If $p_{1},\ldots,p_{k}<0$ and $p_{k+1},\ldots,p_{n}>0,$
then $\text{Supp}\left(  \mu\right)  \subset\left(  -\infty,0\right]
^{k}\times\left[  0,\infty\right)  ^{n-k}$.

\item[\emph{(iii)}] If $p_{1},\ldots,p_{n}>0$ then $p_{\left[  n\right]
}\geqslant0.$
\end{enumerate}
\end{proposition}

Bernardoff (Bernardoff, 2006) gives a necessary and sufficient condition for
infinite divisibility of the \textit{multivariate gamma distribution
associated with} $\left(  P,\lambda\right)  $, in the sense that the Laplace
transform of $\gamma_{\left(  P,\lambda\right)  }$ power $t$ for all positive
$t$ is still the Laplace transform of a positive measure, by the following theorem:

\begin{theorem}
\label{mainresult} Let $\mu=\gamma_{P,\lambda}$ be a gamma distribution
associated with $\left(  P,\lambda\right)  ,$ where $\lambda>0$ and $P\left(
\boldsymbol{\theta}\right)  =\sum\nolimits_{T\in\mathfrak{P}_{n}}%
p_{T}\boldsymbol{\theta}^{T}$ is such that $p_{i}>0$, for all $i\in\left[
n\right]  ,$ and $p_{\left[  n\right]  }>0$. Let $\widetilde{P}\left(
\boldsymbol{\theta}\right)  =\sum\nolimits_{T\in\mathfrak{P}_{n}}%
\widetilde{p}_{T}\boldsymbol{\theta}^{T}$ be the affine polynomial such that
$\widetilde{p}_{T}=-p_{\overline{T}}/p_{\left[  n\right]  }$ for all
$T\in\mathfrak{P}_{n},$ where $\overline{T}=\left[  n\right]  \smallsetminus
T$. Let
\[
\widetilde{b}_{S}=b_{S}(\widetilde{P})=\sum_{k=1}^{\left\vert S\right\vert
}\left(  k-1\right)  !\sum_{\mathcal{T\in}\Pi_{S}^{k}}\prod_{T\in\mathcal{T}%
}\widetilde{p}_{T},
\]
where $\left\vert S\right\vert $ denotes the cardinality of the set $S$, and
$\Pi_{S}^{k}$ denotes the set of all partitions of $S$ into k \emph{non-empty}
subsets of $S$. Then the measure $\mu$ is infinitely divisible if and only if%
\begin{equation}
\widetilde{\text{$p$}}_{i}<0\text{ for all }i\in\left[  n\right]  ,
\label{cond1}%
\end{equation}
and%
\begin{equation}
\widetilde{b}_{S}\geqslant0\text{ for all }S\in\mathfrak{P}_{n}^{\ast}\text{
such that }\left\vert S\right\vert \geqslant2. \label{cond2}%
\end{equation}

\end{theorem}

\begin{corollary}
By the properties of infinite divisible distributions we conclude that the
necessary and sufficient conditions for infinite divisibility of a
\textit{multivariate gamma distribution }associated to $\left(  P,\lambda
\right)  $ of theorem (\ref{mainresult}), are also necessary and sufficient
conditions for infinite divisibility of \textit{multivariate multi-factor
gamma distribution }associated to $\left(  P,\Lambda\right)  .$
\end{corollary}

To illustrate the difficulty to calculate the \textit{multivariate gamma
distribution associated with} $\left(  P,\lambda\right)  $ we give, for the
tridimensional case, the following theorem. Let $F_{II}$ be the function
defined by%
\begin{equation}
F_{II}\left(  \lambda_{1},\lambda_{2},z_{1},z_{2},z_{3},z_{4}\right)
=\sum_{m_{1},\ldots,m_{4}=0}^{\infty}\frac{1}{\left(  \lambda_{1}\right)
_{m_{1}+m_{2}+m_{3}}\left(  \lambda_{2}\right)  _{2m_{1}+m_{2}+m_{4}}}%
\frac{z_{1}^{m_{1}}}{m_{1}!}\frac{z_{2}^{m_{2}}}{m_{2}!}\frac{z_{3}^{m_{3}}%
}{m_{3}!}\frac{z_{4}^{m_{4}}}{m_{4}!}; \label{genLauricella2}%
\end{equation}
it is still a particular generalized Lauricella function.

\begin{theorem}
\label{gamma3}In the case $n=3$, $p_{i}>0$ for $i\in\left[  3\right]  ,$
$p_{ij}>0$ for $\left(  i,j\right)  \in\left[  3\right]  ^{2}$, $\widetilde{b}%
_{ij}=-\frac{b_{k}}{p_{123}}+\frac{p_{jk}p_{ik}}{p_{123}^{2}}\geqslant0$ for
$i\neq j$ and $\left\{  i,j,k\right\}  =\left[  3\right]  ,$ $p_{123}>0,$ and
\[
\widetilde{b}_{123}=-\frac{1}{p_{123}}+\frac{p_{12}p_{1}}{p_{123}^{2}}%
+\frac{p_{13}p_{2}}{p_{123}^{2}}+\frac{p_{23}p_{1}}{p_{123}^{2}}+2\frac
{p_{12}p_{13}p_{23}}{p_{123}^{3}}\geqslant0,
\]
the infinitely divisible \textit{multivariate gamma distribution }%
$\gamma_{\left(  \lambda,P\right)  }$ \textit{associated with} $\left(
P,\lambda\right)  ,$ is given by the formula%
\begin{align}
\gamma_{\left(  \lambda,P\right)  }\left(  \text{\emph{d}}\mathbf{x}\right)
&  =\frac{p_{123}^{-\lambda}}{\left[  \Gamma\left(  \lambda\right)  \right]
^{3}}\exp(\widetilde{p}_{1}x_{1}+\widetilde{p}_{2}x_{2}+\widetilde{p}_{3}%
x_{3})\left(  x_{1}x_{2}x_{3}\right)  ^{\lambda-1}\nonumber\\
&  \times F_{II}(\lambda,\lambda,\widetilde{b}_{13}x_{1}x_{3}\widetilde{b}%
_{23}x_{2}x_{3},\widetilde{b}_{123}x_{1}x_{2}x_{3},\widetilde{b}_{12}%
x_{1}x_{2},\widetilde{b}_{13}x_{1}x_{3}+\widetilde{b}_{23}x_{2}x_{3}%
)\mathbbm{1}_{\left(  0,\infty\right)  ^{3}}\left(  \mathbf{x}\right)
\,\text{\emph{d}}\mathbf{x}. \label{MGD3}%
\end{align}

\end{theorem}

\begin{remark}
The case $p_{123}=0$ is solved by Letac and Weso\l owski (Letac and
Weso\l owski, 2008).
\end{remark}

\begin{remark}
If $\widetilde{b}_{12}=\widetilde{b}_{13}=\widetilde{b}_{23}=0,$ Theorem
\ref{gamma3} gives
\[
\gamma_{\left(  \lambda,P\right)  }\left(  \text{\emph{d}}\mathbf{x}\right)
=\frac{p_{123}^{-\lambda}}{\left[  \Gamma\left(  \lambda\right)  \right]
^{3}}\exp(\widetilde{p}_{1}x_{1}+\widetilde{p}_{2}x_{2}+\widetilde{p}_{3}%
x_{3})\left(  x_{1}x_{2}x_{3}\right)  ^{\lambda-1}F_{2}\left(  \lambda
,\lambda;\widetilde{b}_{123}x_{1}x_{2}x_{3}\right)  \mathbbm{1}_{\left(
0,\infty\right)  ^{3}}\left(  \mathbf{x}\right)  \,\text{\emph{d}}\mathbf{x},
\]
and if we put $\lambda=1$ in this last equality, we obtain the Kibble and
Moran distribution given in (Balakrishnan \emph{et al.}, 2000).
\end{remark}

\section{Laplace copula}

We recall the following theorem (Marshall and Olkin, 1988), Theorem 2.1, p.
835, and his Corollary

\begin{theorem}
\label{MarshallO}Let $H_{1},\ldots,H_{n},$ be univariate cdfs, and let $G$ be
an $n$-variate cdf such that $\overline{G}\left(  0,\ldots,0\right)  =1,$ with
univariate margins $G_{i}$ $(i=1,\ldots,n)$. Denote the Laplace transforms of
$G$ and $G_{i}$, respectively, by $\phi$ and $\phi_{i}$ $(i=1,\ldots,n)$. Let
$K$ be a copula. If $F_{i}\left(  x\right)  =\exp\left\{  -\phi_{i}%
^{-1}\left[  H_{i}\left(  x\right)  \right]  \right\}  $ $\left(
i=1,\ldots,n\right)  ,$ then
\begin{equation}
H\left(  x_{1},\ldots,x_{n}\right)  =\int\cdots\int K\left\{  \left[
F_{1}\left(  x_{1}\right)  \right]  ^{\theta_{1}},\ldots,\left[  F_{n}\left(
x_{n}\right)  \right]  ^{\theta_{n}}\right\}  \emph{d}G\left(  \theta
_{1},\ldots,\theta_{n}\right)  \label{Formula 1 Marshall and Olkin}%
\end{equation}
is an $n$-variate cdf with marginals $H_{1},\ldots,H_{n}.$
\end{theorem}

\begin{corollary}
\label{corollaireMO1}Under the conditions of the Theorem \ref{MarshallO},
with\newline$K\left(  x_{1},\ldots,x_{n}\right)  =\prod_{i=1}^{n}x_{i},\left(
0\leqslant x_{i}\leqslant1;\text{ }i=1,\ldots,n\right)  ,$%
\begin{equation}
H\left(  x_{1},\ldots,x_{n}\right)  =\phi\left\{  \phi_{1}^{-1}\left[
H_{1}\left(  x_{1}\right)  \right]  ,\ldots,\phi_{n}^{-1}\left[  H_{n}\left(
x_{n}\right)  \right]  \right\}  \label{Formula 2 Marshall and Olkin}%
\end{equation}
defines an $n$-variate cdf with marginals $H_{1},\ldots,H_{n}.$
\end{corollary}

Therefore copula for $H$ is associated to $G$ and is given by the formula%
\begin{equation}
\mathbf{C}\left(  u_{1},\ldots,u_{n}\right)  =\phi\left[  \phi_{1}^{-1}\left(
u_{1}\right)  ,\ldots\phi_{n}^{-1}\left(  u_{n}\right)  \right]  .
\label{Copula Marshall and Olkin}%
\end{equation}

\begin{remark}
If we know $\phi$ and $\phi_{i}$, for $i=1,\ldots,n,$ we can find the copula
$\mathbf{C,}$ that is the case for the multivariate gamma distributions and
for the multivariate multi-factor gamma distributions. For the case
$H_{i}=G_{i},$ we can write :
\begin{equation}
H\left(  x_{1},\ldots,x_{n}\right)  =\phi\left\{  \phi_{1}^{-1}\left[
G_{1}\left(  x_{1}\right)  \right]  ,\ldots\phi_{n}^{-1}\left[  G_{n}\left(
x_{n}\right)  \right]  \right\}  , \label{other gamma}%
\end{equation}
and this last formula defines $n$-variate cdf gamma $H$ with margin
distributions $G_{1},\ldots,G_{n}.$\newline Then the copula associated to $H$
is
\[
\mathbf{C}\left(  u_{1},\ldots,u_{n}\right)  =\phi\left[  \phi_{1}^{-1}\left(
u_{1}\right)  ,\ldots,\phi_{n}^{-1}\left(  u_{n}\right)  \right]  .
\]

\end{remark}

Now, from the Formula \ref{Copula Marshall and Olkin}, we can give the
following Definition

\begin{definition}
Let $\mathbf{X}=\left(  X_{1},X_{2},\ldots,X_{n}\right)  $ be a random vector
in $\left]  0,+\infty\right[  ^{n}$ with Laplace transform $\varphi
_{\mathbf{X}}$ defined by $L_{\mathbf{X}}\left(  \boldsymbol{\theta}\right)
=L_{\mathbf{X}}\left(  \theta_{1},\ldots,\theta_{n}\right)  =\varphi
_{\mathbf{X}}\left(  \theta_{1},\ldots,\theta_{n}\right)  =\mathbb{E}\left\{
\exp\left[  -\left(  \theta_{1}X_{1}+\cdots+\theta_{n}X_{n}\right)  \right]
\right\}  $, and let $\varphi_{X_{i}}$ be the Laplace transform of the random
variable $X_{i},$ defined by $L_{X_{i}}\left(  \theta_{i}\right)
=\varphi_{X_{i}}\left(  \theta_{i}\right)  =\mathbb{E}\left[  \exp\left(
-\theta_{i}X_{i}\right)  \right]  .$ The function $\varphi_{X_{i}}$ is a one
and onto, and is decreasing of $\left[  0,+\infty\right[  $ onto $\left]
0,1\right]  $. Its inverse function is denoted by $\varphi_{X_{i}}^{-1}$. Then
the function $C_{L_{_{\mathbf{X}}}}$ defined by%
\[
C_{L_{_{\mathbf{X}}}}\left(  u_{1},\ldots,u_{n}\right)  =\varphi_{\mathbf{X}%
}\left[  \varphi_{\mathbf{X}_{1}}^{-1}\left(  u_{1}\right)  ,\ldots
,\varphi_{\mathbf{X}_{n}}^{-1}\left(  u_{n}\right)  \right]
\]
is a copula. We call this copula $C_{L_{_{\mathbf{X}}}}$ the Laplace copula
associated to the random vector $\mathbf{X}$. If $\mathbf{X}$ has pd
$\mu_{\mathbf{X}}\left(  \text{\emph{d}}\mathbf{x}\right)  ,$ then we denote
still $C_{L_{_{\mathbf{X}}}}\left(  u_{1},\ldots,u_{n}\right)
=C_{L_{_{\mathbf{\mu}_{\mathbf{X}}}}}\left(  u_{1},\ldots,u_{n}\right)  .$
\end{definition}

\begin{proposition}
Let $\mathbf{F}_{1},\ldots,\mathbf{F}_{n},$ be $n$ univariate cdfs on $\left]
0,+\infty\right[  .$ The relation%
\begin{align*}
\mathbf{F}\left(  x_{1},\ldots,x_{n}\right)   &  =\varphi_{\mathbf{X}}\left\{
\varphi_{\mathbf{X}_{1}}^{-1}\left[  \mathbf{F}_{1}\left(  x_{1}\right)
\right]  ,\ldots,\varphi_{\mathbf{X}_{n}}^{-1}\left[  \mathbf{F}_{n}\left(
x_{n}\right)  \right]  \right\} \\
&  =C_{L_{_{\mathbf{X}}}}\left[  \mathbf{F}_{1}\left(  x_{1}\right)
,\ldots,\mathbf{F}_{n}\left(  x_{n}\right)  \right]
\end{align*}
defines a cdf $\mathbf{F}$ with marginal cdfs $\mathbf{F}_{1},\ldots
,\mathbf{F}_{n}.$ Then the copula associated to $\mathbf{F}$ is
$C_{L_{_{\mathbf{X}}}}.$\newline If we choose, for $i=1,2,\ldots,n$,
$\mathbf{F}_{i}=\mathbf{F}_{X_{i}},$ the cdf of $X_{i},$ then $\mathbf{F}$ is
the cdf of a random vector $\mathbf{X}$ for which the marginal cdfs are the
cdfs of $X_{i}$, and we have $\mathbf{F}\neq\mathbf{F}_{\mathbf{X}}$ where
$\mathbf{F}_{\mathbf{X}}$ calls the cdf of $\mathbf{X}$. In the case of
multivariate gamma distributions and multi-factor gamma distributions, we get
other multivariate gamma distributions and multi-factor gamma distributions
with given copula.
\end{proposition}

Laplace copula can be seen as a generalization of Archimedean copula (Joe, 1997,2014).

\section{Main results}

Now, we are in the capacity to give the two main results of this paper

\begin{theorem}
\label{copulagammaM}Let $P$ an affine polynomial in the $n$ variables
$\theta_{i},$ $i=1,\ldots,n$, with $P\left(  \mathbf{0}\right)  =1,$ then the
Laplace copula of the multivariate gamma distribution $\gamma_{\left(
P,\lambda\right)  }$ associated to $\left(  P,\lambda\right)  $ is%
\begin{equation}
C_{L_{\gamma_{\left(  P,\lambda\right)  }}}\left(  \mathbf{v}\right)
=\mathbf{v}^{\left[  n\right]  }[1+\sum_{T\subset\left[  n\right]  ,\left\vert
T\right\vert >1}\left(  -1\right)  ^{\left\vert T\right\vert }P(-\frac
{1}{\mathbf{p}}\mathbf{1}_{T})\prod_{t\in T}(1-v_{t}^{\frac{1}{\lambda}%
})]^{-\lambda}, \label{LC_MGD}%
\end{equation}
where $\mathbf{v}=\left(  v_{1},\ldots,v_{n}\right)  ,$ $\left\vert
T\right\vert $ is the cardinality of $T$ and the vector $\frac{1}{\mathbf{p}%
}\mathbf{1}_{T}$ is defined by $\left(  \frac{1}{\mathbf{p}}\mathbf{1}%
_{T}\right)  _{i}=\frac{1}{p_{i}}$ if $i\in T$, $\left(  \frac{1}{\mathbf{p}%
}\mathbf{1}_{T}\right)  _{i}=0$ if $i\notin T$, for $i\in\left\{
1,2,\ldots,n\right\}  $.
\end{theorem}

This family is given, by example, for the simpler case $P\left(
\boldsymbol{\theta}\right)  =\prod_{i=1}^{n}(1+p_{i}\theta_{i})]-\beta
\mathbf{p}^{\left[  n\right]  }\boldsymbol{\theta}^{\left[  n\right]  },$ with
$0\leqslant\beta<1$, corresponding to the family gived by (Fang \textit{et
al.,} 2000), namely $C\left(  \mathbf{v}\right)  =\mathbf{v}^{\left[
n\right]  }[1-\beta\prod_{i=1}^{n}(1-v_{i}^{\frac{1}{\lambda}})]^{-\lambda}.$

\begin{theorem}
\label{copulagammaMM}Let $P$ an affine polynomial in the $n$ variables
$\theta_{i},$ $i=1,\ldots,n$, with $P\left(  \mathbf{0}\right)  =1,$ let
$\Lambda=\left(  \lambda,\lambda_{1},\ldots,\lambda_{n}\right)  $ with
$\lambda_{i}>\lambda>0,$ for $i\in\left\{  1,2,\ldots,n\right\}  ,$ then the
Laplace copula of the multivariate multi-factor gamma distribution
$\gamma_{\left(  P,\Lambda\right)  }$ associated to $\left(  P,\Lambda\right)
$ is%
\begin{equation}
C_{L_{\gamma_{\left(  P,\Lambda\right)  }}}\left(  \mathbf{v}\right)
=\mathbf{v}^{\left[  n\right]  }[1+\sum_{T\subset\left[  n\right]  ,\left\vert
T\right\vert >1}\left(  -1\right)  ^{\left\vert T\right\vert }P(-\frac
{1}{\mathbf{p}}\mathbf{1}_{T})\prod_{t\in T}(1-v_{t}^{\frac{1}{\lambda_{t}}%
})]^{-\lambda}. \label{LC_MMGD}%
\end{equation}

\end{theorem}

This copula family is a new generalization of Farlie-Gumbel-Morgenstern Copulas.

Bekrizadeh \textit{et al.,} 2012 propose a similar formula only for $\lambda$
being a negative integer.

We note that, if the conditions of Theorem \ref{mainresult} are checked,
formulas (\ref{LC_MGD}) and (\ref{LC_MMGD}) define copulas.

For these copulas, if we inject univariate gamma distributions $\gamma
_{\left(  p_{i},\lambda\right)  }$ in the first case or $\gamma_{\left(
p_{i},\lambda_{i}\right)  }$ in the second case, then we obtain other
multivariate gamma distributions and other multivariate multi-factor gamma
distributions in the sense that their marginal distributions are respectively
$\gamma_{\left(  p_{1},\lambda\right)  }$ and $\gamma_{\left(  p_{1}%
,\lambda_{i}\right)  }$. Their cdf and pdf are given by (\ref{Sklar1}) and
(\ref{Sklar2}) respectively, and have a link with \textit{multivariate gamma
distribution associated with} $\left(  P,\lambda\right)  $ and
\textit{multi-factor gamma distribution associated with} $\left(
P,\Lambda\right)  $ respectively.

\section{The bidimensional and tridimensional cases}

For the bidimensional case, Theorem (\ref{copulagammaM}) and Theorem
(\ref{copulagammaMM}) give the following corollary

\begin{corollary}
For the bivariate gamma distribution such that $L_{\gamma_{\left(
P,\lambda\right)  }}\left(  \boldsymbol{\theta}\right)  =\left(  P\left(
\boldsymbol{\theta}\right)  \right)  ^{-\lambda}$, we have
\[
C_{L_{\gamma_{\left(  P,\lambda\right)  }}}\left(  v_{1},v_{2}\right)
=v_{1}v_{2}[1-r_{12}(1-v_{1}^{\frac{1}{\lambda}})(1-v_{2}^{\frac{1}{\lambda}%
})]^{-\lambda},
\]
where, for $\mathbf{X}=\left(  X_{1},X_{2}\right)  $ with pd $\gamma_{\left(
P,\lambda\right)  },$ $r_{12}=-P(-p_{1}^{-1},-p_{2}^{-1})=1-p_{12}/\left(
p_{1}p_{2}\right)  $ is the linear correlation coefficient of the random
variables $X_{1},$ $X_{2}$ ; it checks $0\leqslant r_{12}\leqslant1$. It is
the copula of the BB10 family p. 154 in the Joe's book (Joe 1997). This result
is due to the following equalities : $E\left(  X_{i}\right)  =\lambda p_{i,}$
$Var\left(  X_{i}\right)  =\lambda p_{i}^{2},$ $Cov(X_{1},X_{2})=\lambda
\left(  -p_{12}+p_{1}p_{2}\right)  $ obtained by derivating $L_{\gamma
_{\left(  P,\lambda\right)  }}\left(  \boldsymbol{\theta}\right)  $ at
$\mathbf{0}.$\newline For the bivariate multi-factor gamma distribution such
that $L_{\gamma_{\left(  P,\Lambda\right)  }}\left(  \theta\right)  =\left(
P\left(  \theta\right)  \right)  ^{-\lambda}\prod_{i=1}^{2}\left(
1+p_{i}\theta_{i}\right)  ^{-\left(  \lambda_{i}-\lambda\right)  }$ , we have
a more general family
\begin{equation}
C_{L_{\gamma_{\left(  P,\Lambda\right)  }}}\left(  v_{1},v_{2}\right)
=v_{1}v_{2}[1-r_{12}(1-v_{1}^{\frac{1}{\lambda_{1}}})(1-v_{2}^{\frac
{1}{\lambda_{2}}})]^{-\lambda}, \label{bmfgLc}%
\end{equation}
where, for $\mathbf{X}=\left(  X_{1},X_{2}\right)  =\left(  Y_{1}%
,Y_{2}\right)  +\left(  Z_{1},Z_{2}\right)  =\mathbf{Y}+\mathbf{Z}$ as defined
in Proposition \ref{buildMFGD} with pd $\gamma_{\left(  P,\Lambda\right)  },$
and $0\leqslant r_{12}=-P(-p_{1}^{-1},-p_{2}^{-1})=1-p_{12}/\left(  p_{1}%
p_{2}\right)  \leqslant1,$ is the linear correlation coefficient of the random
variables $Y_{1},$ $Y_{2}$ in the bivariate gamma distribution such that
$L_{_{\mathbf{Y}}}\left(  \theta\right)  =\left[  P\left(  \theta\right)
\right]  ^{-\lambda}$.
\end{corollary}

This bivariate family is not given in the Joe's books (Joe, 1997, 2014).
Bekrizadeh \textit{et al.,} 2012 give a similar formula for $\lambda$ being a
negative integer.

We recall the formulas (Joe, 2014) for the computation of $\tau,$ the
Kendall's tau, and $\rho_{S},$ the Spearmann's rho%
\begin{align}
\tau &  =1-4%
{\displaystyle\iint_{\left[  0,1\right]  ^{2}}}
\frac{\partial C}{\partial u}\left(  u,v\right)  \frac{\partial C}{\partial
v}\left(  u,v\right)  \,\text{d}u\text{d}v,\label{tau formula}\\
\rho_{S}  &  =12%
{\displaystyle\iint_{\left[  0,1\right]  ^{2}}}
C\left(  u,v\right)  \,\text{d}u\text{d}v-3. \label{rho formula}%
\end{align}
Then, we can give the following result

\begin{proposition}
\label{Kendall's tau & Spearman's rho}The Kendall's tau and the Spearman's rho
of the Laplace copula of the bivariate multi-factor gamma distribution are
given by the following formulas%
\begin{align}
\tau &  =1-F_{2}^{3}\left(  2\lambda,1,1;2\lambda_{1}+1,2\lambda_{2}%
+1;r_{12}\right) \nonumber\\
&  +\frac{4\lambda}{\left(  2\lambda_{1}+1\right)  \left(  2\lambda
_{2}+1\right)  }r_{12}F_{2}^{3}\left(  2\lambda+1,1,2;2\lambda_{1}%
+2,2\lambda_{2}+2;r_{12}\right) \nonumber\\
&  -\frac{\lambda^{2}}{\left(  2\lambda_{1}+1\right)  \left(  2\lambda
_{2}+1\right)  \left(  \lambda_{1}+1\right)  \left(  \lambda_{2}+1\right)
}r_{12}^{2}F_{2}^{3}\left(  2\lambda+2,2,2;2\lambda_{1}+3,2\lambda
_{2}+3;r_{12}\right)  , \label{KtauHyper}%
\end{align}
and%
\begin{align}
\rho_{S}  &  =3\left[  F_{2}^{3}(1,1,\lambda;2\lambda_{1}+1,2\lambda
_{2}+1;r_{12})-1\right] \label{SrhoHyper1}\\
&  =\frac{3\lambda}{\left(  2\lambda_{1}+1\right)  \left(  2\lambda
_{2}+1\right)  }r_{12}F_{2}^{3}(1,2,\lambda+1;2\lambda_{1}+2,2\lambda
_{2}+2;r_{12}). \label{SrhoHyper2}%
\end{align}

\end{proposition}

For the tridimensional case, Theorem (\ref{copulagammaM}) and Theorem
(\ref{copulagammaMM}) give the following corollary

\begin{corollary}
For the trivariate gamma distribution $\gamma_{\left(  P,\lambda\right)  }$
such that $L_{\gamma_{\left(  P,\lambda\right)  }}\left(  \boldsymbol{\theta
}\right)  =\left(  P\left(  \boldsymbol{\theta}\right)  \right)  ^{-\lambda}$,
we have
\begin{align*}
C_{L_{\gamma_{\left(  P,\lambda\right)  }}}\left(  v_{1},v_{2},v_{3}\right)
&  =v_{1}v_{2}v_{3}[1-r_{12}(1-v_{1}^{\frac{1}{\lambda}})(1-v_{2}^{\frac
{1}{\lambda}})-r_{13}(1-v_{1}^{\frac{1}{\lambda}})(1-v_{3}^{\frac{1}{\lambda}%
})\\
&  -r_{23}(1-v_{2}^{\frac{1}{\lambda}})(1-v_{3}^{\frac{1}{\lambda}}%
)+2r_{123}(1-v_{1}^{\frac{1}{\lambda}})(1-v_{2}^{\frac{1}{\lambda}}%
)(1-v_{3}^{\frac{1}{\lambda}})]^{-\lambda},
\end{align*}
\newline where, for $\mathbf{X}=\left(  X_{1},X_{2},X_{3}\right)  $ with pd
$\gamma_{\left(  P,\lambda\right)  },$ we denote by $r_{ij}=-P(-\frac{1}%
{p}\mathbf{1}_{\left\{  i,j\right\}  })=1-p_{ij}/(p_{i}p_{j}),$ $1\leqslant
i\neq j\leqslant3,$ the linear correlation coefficient of the random variables
$X_{i},X_{j},$ they check $0\leqslant r_{ij}\leqslant1,$ and we denote by
$r_{123}$ the number defined by
\begin{align}
r_{123}  &  =\mathbb{E}\{\prod_{i=1}^{3}\left[  X_{i}-\mathbb{E}\left(
X_{i}\right)  \right]  \}/\prod_{i=1}^{3}(\mathbb{E}\{\left[  X_{i}%
-\mathbb{E}\left(  X_{i}\right)  \right]  ^{3}\})^{1/3}\label{r123}\\
&  =-\tfrac{1}{2}P(-p_{1}^{-1},-p_{2}^{-1},-p_{3}^{-1})\nonumber
\end{align}
(to compare with $r_{12}=\mathbb{E}\{\prod_{i=1}^{2}\left[  X_{i}%
-\mathbb{E}\left(  X_{i}\right)  \right]  \}/\prod_{i=1}^{2}(\mathbb{E}%
\{\left[  X_{i}-\mathbb{E}\left(  X_{i}\right)  \right]  ^{2}\})^{1/2}%
$).\newline This last result is due to the following equalities :
$\mathbb{E}\left(  X_{i}\right)  =\lambda p_{i,}$ $\mathbb{E}\{\left[
X_{i}-\mathbb{E}\left(  X_{i}\right)  \right]  ^{3}\}=2\lambda p_{i}^{3},$
\newline$\mathbb{E}\{\prod_{i=1}^{3}\left[  X_{i}-\mathbb{E}\left(
X_{i}\right)  \right]  \}=-\lambda p_{1}p_{2}p_{3}P(-p_{1}^{-1},-p_{2}%
^{-1},-p_{3}^{-1})$ obtained by derivating $L_{\gamma_{\left(  P,\lambda
\right)  }}\left(  \boldsymbol{\theta}\right)  $ at $\mathbf{0}.$\newline For
the trivariate multi-factor gamma distribution such that $L_{\gamma_{\left(
P,\Lambda\right)  }}\left(  \boldsymbol{\theta}\right)  =\left[  P\left(
\boldsymbol{\theta}\right)  \right]  ^{-\lambda}\prod_{i=1}^{3}\left(
1+p_{i}\theta_{i}\right)  ^{-\left(  \lambda_{i}-\lambda\right)  }$
($\lambda_{i}\geqslant\lambda,$ $i=1,2,3$ ), we have%
\begin{align*}
C_{L_{\gamma_{\left(  P,\Lambda\right)  }}}\left(  v_{1},v_{2},v_{3}\right)
&  =v_{1}v_{2}v_{3}[1-r_{12}(1-v_{1}^{\frac{1}{\lambda_{1}}})(1-v_{2}%
^{\frac{1}{\lambda_{2}}})-r_{13}(1-v_{1}^{\frac{1}{\lambda_{1}}}%
)(1-v_{3}^{\frac{1}{\lambda_{3}}})\\
&  -r_{23}(1-v_{2}^{\frac{1}{\lambda_{2}}})(1-v_{3}^{\frac{1}{\lambda_{3}}%
})+2r_{123}(1-v_{1}^{\frac{1}{\lambda_{1}}})(1-v_{2}^{\frac{1}{\lambda_{2}}%
})(1-v_{3}^{\frac{1}{\lambda_{3}}})]^{-\lambda},
\end{align*}
where, for $\mathbf{X}=\left(  X_{1},X_{2},X_{3}\right)  =\left(  Y_{1}%
,Y_{2},Y_{3}\right)  +\left(  Z_{1},Z_{2},Z_{3}\right)  =\mathbf{Y}%
+\mathbf{Z}$ as defined in Proposition \ref{buildMFGD} with pd $\gamma
_{\left(  P,\Lambda\right)  },$ we denote by $r_{ij}=-P(-\frac{1}{\mathbf{p}%
}\mathbf{1}_{\left\{  i,j\right\}  })=1-\frac{p_{ij}}{p_{i}p_{j}},$
$1\leqslant i\neq j\leqslant3,$ the linear correlation coefficient of the
random variables $Y_{i},$ $Y_{j}$ in the trivariate gamma distribution such
that $L_{_{Y}}\left(  \boldsymbol{\theta}\right)  =\left(  P\left(
\boldsymbol{\theta}\right)  \right)  ^{-\lambda},$ they check $0\leqslant
r_{ij}\leqslant1,$ and $r_{123}$ is defined by equality (\ref{r123}).
\end{corollary}

\section{Appendix}

\subsection{Proof of Theorem \ref{Proposition_multifactorbig2}}

Let $\mathbf{X}=(X_{1},X_{2})$ and $\mathbf{Y}=(Y_{1},Y_{2})$ be independent
random variables. The pdf of $\mathbf{Z}=(X_{1}+Y_{1},X_{2}+Y_{2})$ where
$X_{i}$ has pd $\gamma\left(  p_{i},\lambda_{i}-\lambda\right)  ,$ $i=1,2$ and
$\mathbf{Y}$ has pd $\gamma\left(  P,\lambda\right)  $, is obtained by
convolution. By changing variables $1-v_{i}/z_{i}=u_{i}$, $i=1,2,$ and with
the notation $c=b_{12}(\widetilde{P})=-p_{12}^{-1}+p_{1}p_{2}p_{12}^{-2},$
$d_{i}=cp_{12}p_{i}^{-1},$ $i=1,2$ we obtain the pdf $f_{\mathbf{Z}}$ of
$\mathbf{Z}$ by $f_{\boldsymbol{Z}}\left(  z_{1},z_{2}\right)  =0$ if $\left(
z_{1},z_{2}\right)  \notin\left(  0,\infty\right)  ^{2}$ and for $\left(
z_{1},z_{2}\right)  \in\left(  0,\infty\right)  ^{2}$%
\begin{align}
f_{\mathbf{Z}}\left(  z_{1},z_{2}\right)   &  =\int_{0}^{z_{1}}\int_{0}%
^{z_{2}}\frac{1}{\left[  \Gamma\left(  \lambda\right)  \right]  ^{2}%
p_{12}^{\lambda}}\mathbf{e}^{-\left(  \frac{p_{2}}{p_{12}}v_{1}+\frac{p_{1}%
}{p_{12}}v_{2}\right)  }\left(  v_{1}v_{2}\right)  ^{\lambda-1}F_{1}\left(
\lambda;cv_{1}v_{2}\right) \nonumber\\
&  \times\frac{\mathbf{e}^{-\frac{z_{1}-v_{1}}{p_{1}}}}{\Gamma\left(
\lambda_{1}-\lambda\right)  }\left(  \frac{z_{1}-v_{1}}{p_{1}}\right)
^{\lambda_{1}-\lambda}\frac{\mathbf{e}^{-\frac{z_{2}-v_{2}}{p_{2}}}}%
{\Gamma\left(  \lambda_{2}-\lambda\right)  }\left(  \frac{z_{2}-v_{2}}{p_{2}%
}\right)  ^{\lambda_{2}-\lambda}\frac{\text{d}v_{1}}{z_{1}-v_{1}}%
\frac{\text{d}v_{2}}{z_{2}-v_{2}}\nonumber\\
&  =\frac{z_{1}^{\lambda_{1}-1}z_{2}^{\lambda_{2}-1}\mathbf{e}^{-\left(
\frac{p_{2}}{p_{12}}z_{1}+\frac{p_{1}}{p_{12}}z_{2}\right)  }}{\left(
\Gamma\left(  \lambda\right)  \right)  \Gamma\left(  \lambda_{1}%
-\lambda\right)  \Gamma\left(  \lambda_{2}-\lambda\right)  p_{12}^{\lambda
}p_{1}^{\lambda_{1}-\lambda}p_{2}^{\lambda_{2}-\lambda}}\sum_{k\geqslant
0}\frac{\left(  cz_{1}z_{2}\right)  ^{k}}{\Gamma\left(  \lambda+k\right)
k!}\nonumber\\
&  \times\int_{0}^{1}\mathbf{e}^{d_{1}z_{1}u_{1}}u_{1}^{\lambda_{1}-\lambda
-1}\left(  1-u_{1}\right)  ^{\lambda+k-1}\,\text{d}u_{1}\int_{0}^{1}%
\mathbf{e}^{d_{2}z_{2}u_{2}}u_{2}^{\lambda_{2}-\lambda-1}\left(
1-u_{2}\right)  ^{\lambda+k-1}\,\text{d}u_{2}. \label{convolution_gamma_b}%
\end{align}
As we have, with the notation $B\left(  \alpha,\beta\right)  =\int_{0}%
^{1}u^{\alpha-1}\left(  1-u\right)  ^{\beta-1}\,$d$u=\Gamma\left(
\alpha\right)  \Gamma\left(  \beta\right)  /\Gamma\left(  \alpha+\beta\right)
$, $\alpha,\beta>0$ for the Euler's Beta function,%
\[
\int_{0}^{1}\mathbf{e}^{\delta u}u^{\alpha-1}\left(  1-u\right)  ^{\beta
-1}\,\text{d}u=B\left(  \alpha,\beta\right)  \sum_{n=0}^{\infty}\frac{\left(
\alpha\right)  _{n}}{\left(  \alpha+\beta\right)  _{n}}\frac{\delta^{n}}{n!},
\]
we obtain%
\begin{align*}
f_{\mathbf{Z}}\left(  \boldsymbol{z}\right)   &  =\frac{z_{1}^{\lambda_{1}%
-1}z_{2}^{\lambda_{2}-1}\mathbf{e}^{-\left(  \frac{p_{2}}{p_{12}}z_{1}%
+\frac{p_{1}}{p_{12}}z_{2}\right)  }}{\Gamma\left(  \lambda\right)
\Gamma\left(  \lambda_{1}-\lambda\right)  \Gamma\left(  \lambda_{2}%
-\lambda\right)  p_{12}^{\lambda}p_{1}^{\lambda_{1}-\lambda}p_{2}^{\lambda
_{2}-\lambda}}\sum_{k\geqslant0}\frac{\left(  cz_{1}z_{2}\right)  ^{k}}%
{\Gamma\left(  \lambda+k\right)  k!}\frac{\Gamma\left(  \lambda_{1}%
-\lambda\right)  \Gamma\left(  \lambda+k\right)  }{\Gamma\left(  \lambda
_{1}+k\right)  }\\
&  \times\sum_{n=0}^{\infty}\frac{\left(  \lambda_{1}-\lambda\right)  _{n}%
}{\left(  \lambda_{1}+k\right)  _{n}}\frac{\left(  d_{1}z_{1}\right)  ^{n}%
}{n!}\frac{\Gamma\left(  \lambda_{2}-\lambda\right)  \Gamma\left(
\lambda+k\right)  }{\Gamma\left(  \lambda_{2}+k\right)  }\sum_{m=0}^{\infty
}\frac{\left(  \lambda_{2}-\lambda\right)  _{m}}{\left(  \lambda_{2}+k\right)
_{m}}\frac{\left(  d_{2}z_{2}\right)  ^{m}}{m!}\\
&  =\frac{z_{1}^{\lambda_{1}-1}z_{2}^{\lambda_{2}-1}\mathbf{e}^{-\left(
\frac{p_{2}}{p_{12}}z_{1}+\frac{p_{1}}{p_{12}}z_{2}\right)  }}{\Gamma\left(
\lambda_{1}\right)  \Gamma\left(  \lambda_{2}\right)  p_{12}^{\lambda}%
p_{1}^{\lambda_{1}-\lambda}p_{2}^{\lambda_{2}-\lambda}}\sum_{k=0}^{\infty}%
\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{\left(  \lambda_{1}-\lambda
\right)  _{n}\left(  \lambda_{2}-\lambda\right)  _{m}\left(  \lambda\right)
_{k}}{\left(  \lambda_{1}\right)  _{k+n}\left(  \lambda_{2}\right)  _{k+m}%
}\frac{\left(  d_{1}z_{1}\right)  ^{n}}{n!}\frac{\left(  d_{2}z_{2}\right)
^{m}}{m!}\frac{\left(  cz_{1}z_{2}\right)  ^{k}}{k!}.
\end{align*}
Hence, we have proved the formula (\ref{mulbigamma2}).

\subsection{Proof of Theorem \ref{gamma3}}

We start from the Laplace transform of the trivariate gamma distribution
$\gamma_{\left(  \lambda,P\right)  }$ associated to $\left(  \lambda,P\right)
$. First, we write%

\begin{align*}
L_{\gamma_{\left(  \lambda,P\right)  }}\left(  \boldsymbol{\theta}\right)   &
=\left(  1+p_{1}\theta_{1}+p_{2}\theta_{2}+p_{3}\theta_{3}+p_{12}\theta
_{1}\theta_{2}+p_{13}\theta_{1}\theta_{3}+p_{23}\theta_{2}\theta_{3}%
+p_{123}\theta_{1}\theta_{2}\theta_{3}\right)  ^{-\lambda}\\
&  =\left(  1+p_{3}\theta_{3}\right)  ^{-\lambda}\left(  1+\frac{p_{1}%
+p_{13}\theta_{3}}{1+p_{3}\theta_{3}}\theta_{1}+\frac{p_{2}+p_{23}\theta_{3}%
}{1+p_{3}\theta_{3}}\theta_{2}+\frac{p_{12}+p_{123}\theta_{3}}{1+p_{3}%
\theta_{3}}\theta_{1}\theta_{2}\right)  ^{-\lambda}.
\end{align*}
Let $Q\left(  \theta_{1},\theta_{2}\right)  =1+q_{1}\theta_{1}+q_{2}\theta
_{2}+q_{12}\theta_{1}\theta_{2}$ where $q_{1}=\left(  p_{1}+p_{13}\theta
_{3}\right)  /\left(  1+p_{3}\theta_{3}\right)  ,$ $q_{2}=\left(  p_{2}%
+p_{23}\theta_{3}\right)  /\left(  1+p_{3}\theta_{3}\right)  ,$ and
$q_{12}=\left(  p_{12}+p_{123}\theta_{3}\right)  /\left(  1+p_{3}\theta
_{3}\right)  $. With these notations, we have
\[
L_{\gamma_{\left(  \lambda,P\right)  }}\left(  \boldsymbol{\theta}\right)
=\left(  1+p_{3}\theta_{3}\right)  ^{-\lambda}\left(  1+q_{1}\theta_{1}%
+q_{2}\theta_{2}+q_{12}\theta_{1}\theta_{2}\right)  ^{-\lambda}.
\]
\newline Let $\widetilde{p}_{T}=-p_{\overline{T}}/p_{123}$ ,$\widetilde{b}%
_{ij}=\widetilde{p}_{ij}+\widetilde{p}_{i}\widetilde{p}_{j},$ and
$\widetilde{b}_{123}=\widetilde{p}_{123}+\widetilde{p}_{1}\widetilde{p}%
_{23}+\widetilde{p}_{2}\widetilde{p}_{13}+\widetilde{p}_{3}\widetilde{p}%
_{12}+2\widetilde{p}_{1}\widetilde{p}_{2}\widetilde{p}_{3},$ then we have%
\begin{align*}
b_{1,2}(\widetilde{Q})  &  =\frac{-q_{12}+q_{1}q_{2}}{q_{12}^{2}}\\
&  =\widetilde{b}_{12}+\frac{\widetilde{b}_{123}}{\left(  \theta
_{3}-\widetilde{p}_{3}\right)  }+\frac{\widetilde{b}_{13}\widetilde{b}_{23}%
}{\left(  \theta_{3}-\widetilde{p}_{3}\right)  ^{2}}.
\end{align*}

Let $\gamma_{\left(  \lambda,Q\right)  }$ be the bivariate gamma distribution
associated to $\left(  \lambda,Q\right)  $, its Laplace transform
is\newline$\left(  1+q_{1}\theta_{1}+q_{2}\theta_{2}+q_{12}\theta_{2}%
\theta_{3}\right)  ^{-\lambda}$. We have%
\[
\gamma_{\left(  \lambda,Q\right)  }\left(  dx\right)  =\frac{q_{12}^{-\lambda
}}{\left[  \Gamma\left(  \lambda\right)  \right]  ^{2}}e^{-\frac{q_{2}}%
{q_{12}}x_{1}-\frac{q_{1}}{q_{12}}x_{2}}\left(  x_{1}x_{2}\right)
^{\lambda-1}F_{1}[\lambda,b_{1,2}(\widetilde{Q})x_{1}x_{2}%
]\mathbbm{1}_{\left(  0,\infty\right)  ^{2}}\left(  x_{1},x_{2}\right)
\,\text{d}x_{1}\text{d}x_{2}.
\]
Second, we are looking for the unidimensional distribution for which its
Laplace transform with respect to the variable $\theta_{3}$ is equal to%
\begin{align}
&  \left(  1+p_{3}\theta_{3}\right)  ^{-\lambda}\frac{q_{12}^{-\lambda}%
}{\left(  \Gamma\left(  \lambda\right)  \right)  ^{2}}e^{-\frac{q_{2}}{q_{12}%
}x_{1}-\frac{q_{1}}{q_{12}}x_{2}}\left(  x_{1}x_{2}\right)  ^{\lambda-1}%
F_{1}[\lambda,b_{1,2}(\widetilde{Q})x_{1}x_{2}]\mathbbm{1}_{\left(
0,\infty\right)  ^{2}}\left(  x_{1},x_{2}\right)  \,\text{d}x_{1}\text{d}%
x_{2}\nonumber\\
&  =\frac{p_{123}^{-\lambda}}{\left[  \Gamma\left(  \lambda\right)  \right]
^{2}}\exp\left(  \widetilde{p}_{1}x_{1}+\widetilde{p}_{2}x_{2}\right)  \left(
x_{1}x_{2}\right)  ^{\lambda-1}\left(  \theta_{3}-\widetilde{p}_{3}\right)
^{-\lambda}\exp(\frac{\widetilde{b}_{13}x_{1}+\widetilde{b}_{23}x_{2}}%
{\theta_{3}-\widetilde{p}_{3}})\nonumber\\
&  \times F_{1}\{\lambda,[\widetilde{b}_{12}+\frac{\widetilde{b}_{123}}%
{\theta_{3}-\widetilde{p}_{3}}+\frac{\widetilde{b}_{13}\widetilde{b}_{23}%
}{\left(  \theta_{3}-\widetilde{p}_{3}\right)  ^{2}}]x_{1}x_{2}%
\}\mathbbm{1}_{\left(  0,\infty\right)  ^{2}}\left(  x_{1},x_{2}\right)
\,\text{d}x_{1}\text{d}x_{2}. \label{MGD3STEP1}%
\end{align}

From equality (\ref{MGD3STEP1}) and (Hladik, 1969), we obtain the
unidimensional distribution for which its Laplace transform is%
\begin{align}
&  \frac{p_{123}^{-\lambda}}{\left[  \Gamma\left(  \lambda\right)  \right]
^{2}}\exp\left(  \widetilde{p}_{1}x_{1}+\widetilde{p}_{2}x_{2}+\widetilde{p}%
_{3}x_{3}\right)  \left(  x_{1}x_{2}\right)  ^{\lambda-1}\theta_{3}^{-\lambda
}\exp(\frac{\widetilde{b}_{13}x_{1}+\widetilde{b}_{23}x_{2}}{\theta_{3}%
})\nonumber\\
&  \times F_{1}[\lambda,(\frac{\widetilde{b}_{13}\widetilde{b}_{23}}%
{\theta_{3}^{2}}+\frac{\widetilde{b}_{123}}{\theta_{3}}+\widetilde{b}%
_{12})x_{1}x_{2}]\mathbbm{1}_{\left(  0,\infty\right)  ^{2}}\left(
x_{1},x_{2}\right)  \,\text{d}x_{1}\text{d}x_{2}. \label{MGDSTEP1B}%
\end{align}

Third, we are looking for the unidimensional distribution for which its
Laplace transform with respect to the variable $\theta_{3}$ is equal to%
\begin{align}
&  \theta_{3}^{-\lambda}\exp(\frac{\widetilde{b}_{13}x_{1}+\widetilde{b}%
_{23}x_{2}}{\theta_{3}})F_{1}[\lambda,(\frac{\widetilde{b}_{13}\widetilde{b}%
_{23}}{\theta_{3}^{2}}+\frac{\widetilde{b}_{123}}{\theta_{3}}+\widetilde{b}%
_{12})x_{1}x_{2}]\nonumber\\
&  =\sum_{k=0}^{\infty}\frac{1}{\left(  \lambda\right)  _{k}}\left(
x_{1}x_{2}\right)  ^{k}\sum_{\ell+m+n=k}\frac{1}{\ell!m!n!}\widetilde{b}%
_{13}^{\ell}\widetilde{b}_{23}^{\ell}\widetilde{b}_{123}^{m}\widetilde{b}%
_{12}^{n}\theta_{3}^{-\left(  \lambda+2\ell+m\right)  }\exp(\frac
{\widetilde{b}_{13}x_{1}+\widetilde{b}_{23}x_{2}}{\theta_{3}})
\label{MGD3STEP2}%
\end{align}
From (Hadlik,1986), we have the following equality%
\begin{equation}
L_{\left[  \Gamma\left(  \lambda\right)  \right]  ^{-1}t^{\lambda-1}%
F_{1}\left(  \lambda,at\right)  \mathbbm{1}_{\left(  0,\infty\right)  }\left(
t\right)  \,\text{d}t}\left(  s\right)  =s^{-\lambda}\exp\left(  \frac{a}%
{s}\right)  \label{hladik}%
\end{equation}
We utilize the equality (\ref{hladik}) in (\ref{MGD3STEP2}) and we obtain the
following unidimensional distribution%

\begin{align}
&  \sum_{k=0}^{\infty}\frac{1}{\left(  \lambda\right)  _{k}}\left(  x_{1}%
x_{2}\right)  ^{k}\sum_{\ell+m+n=k}\frac{1}{\ell!m!n!}\widetilde{b}_{13}%
^{\ell}\widetilde{b}_{23}^{\ell}\widetilde{b}_{123}^{m}\widetilde{b}_{12}%
^{n}\left[  \Gamma\left(  \lambda+2\ell+m\right)  \right]  ^{-1}x_{3}%
^{\lambda+2\ell+m-1}\nonumber\\
&  \times F_{1}\left[  \lambda+2\ell+m,(\widetilde{b}_{13}x_{1}+\widetilde{b}%
_{23}x_{2})x_{3}\right]  \mathbbm{1}_{\left(  0,\infty\right)  }\left(
x_{3}\right)  \,\text{d}x_{3}\nonumber\\
&  =\left[  \Gamma\left(  \lambda\right)  \right]  ^{-1}x_{3}^{\lambda
-1}\mathbbm{1}_{\left(  0,\infty\right)  }\left(  x_{3}\right) \nonumber\\
&  \times\sum_{\left(  \ell,m,n\right)  \in\mathbb{N}^{3}}\frac{1}{\left(
\lambda\right)  _{\left(  \ell+m+n\right)  }\left(  \lambda\right)  _{\left(
2\ell+m\right)  }\ell!m!n!}(\widetilde{b}_{13}x_{1}x_{3}\widetilde{b}%
_{23}x_{2}x_{3})^{\ell}(\widetilde{b}_{123}x_{1}x_{2}x_{3})^{m}(\widetilde{b}%
_{12}x_{1}x_{2})^{n}\nonumber\\
&  \times F_{1}(\lambda+2\ell+m,\widetilde{b}_{13}x_{1}x_{3}+\widetilde{b}%
_{23}x_{2}x_{3})\,\text{d}x_{3}\nonumber\\
&  =\left[  \Gamma\left(  \lambda\right)  \right]  ^{-1}x_{3}^{\lambda
-1}\mathbbm{1}_{\left(  0,\infty\right)  }\left(  x_{3}\right) \nonumber\\
&  \times\sum_{\left(  \ell,m,n,p\right)  \in\mathbb{N}^{4}}\frac{1}{\left(
\lambda\right)  _{\left(  \ell+m+n\right)  }\left(  \lambda\right)  _{\left(
2\ell+m+p\right)  }\ell!m!n!p!}\nonumber\\
&  \times(\widetilde{b}_{13}x_{1}x_{3}\widetilde{b}_{23}x_{2}x_{3})^{\ell
}(\widetilde{b}_{123}x_{1}x_{2}x_{3})^{m}(\widetilde{b}_{12}x_{1}x_{2}%
)^{n}(\widetilde{b}_{13}x_{1}x_{3}+\widetilde{b}_{23}x_{2}x_{3})^{p}%
\,\text{d}x_{3}\nonumber\\
&  =\left[  \Gamma\left(  \lambda\right)  \right]  ^{-1}x_{3}^{\lambda
-1}F_{II}(\lambda,\lambda,\widetilde{b}_{13}x_{1}x_{3}\widetilde{b}_{23}%
x_{2}x_{3},\widetilde{b}_{123}x_{1}x_{2}x_{3},\widetilde{b}_{12}x_{1}%
x_{2},\widetilde{b}_{13}x_{1}x_{3}+\widetilde{b}_{23}x_{2}x_{3}%
)\mathbbm{1}_{\left(  0,\infty\right)  }\left(  x_{3}\right)  \,\text{d}x_{3}
\label{MGD3STEP3}%
\end{align}
We carry this last equality (\ref{MGD3STEP3}) in (\ref{MGDSTEP1B}) and we
obtain the pd (\ref{MGD3}).

\subsection{Proof of Theorem \ref{copulagammaM}}

We utilize the following notations, for $i\in\left[  n\right]  ,$
$L_{\gamma_{\left(  p_{i},\lambda\right)  }}\left(  \theta_{i}\right)
=\left(  1+p_{i}\theta_{i}\right)  ^{-\lambda}=v_{i},0<\lambda_{i}<\lambda$
and $1+p_{i}\theta_{i}=u_{i}$. In particular $\theta_{i}=\left(
u_{i}-1\right)  /p_{i}=\left(  v_{i}^{-1/\lambda}-1\right)  /p_{i}$ and
$u_{i}=v_{i}^{-1/\lambda}.$ We remark that $u_{i}=0\Leftrightarrow\theta
_{i}=-1/p_{i}.$

We start from the Laplace transform of the \textit{multivariate gamma
distribution }associated to $\left(  P,\lambda\right)  $. From the definition
of $\gamma_{\left(  P,\lambda\right)  }$, we obtain%
\[
\lbrack L_{\gamma_{\left(  P,\lambda\right)  }}\left(  \boldsymbol{\theta
}\right)  ]^{-\frac{1}{\lambda}}=\sum_{T\subset\left[  n\right]  }%
p_{T}\boldsymbol{\theta}^{T}.
\]
Hence, we have%
\begin{align*}
\lbrack L_{\gamma_{\left(  P,\lambda\right)  }}\left(  \boldsymbol{\theta
}\right)  ]^{-\frac{1}{\lambda}}  &  =\sum_{T\subset\left[  n\right]  }%
p_{T}\left(  \mathbf{u}-\mathbf{1}_{\left[  n\right]  }\right)  ^{T}\left(
\frac{1}{\mathbf{p}}\right)  ^{T}\\
&  =\sum_{T\subset\left[  n\right]  }q_{T}\mathbf{u}^{T}=Q\left(
\mathbf{u}\right)
\end{align*}
where $\mathbf{u}=\left(  u_{1},\ldots,u_{n}\right)  ,$ and
\[
Q\left(  \mathbf{u}\right)  =\sum_{T\subset\left[  n\right]  }\alpha
_{T}\left(  \mathbf{u}-\mathbf{1}_{\left[  n\right]  }\right)  ^{T}%
\mathbf{u}^{\overline{T}},
\]
indeed, the polynomials $\left(  \mathbf{u}-\mathbf{1}_{\left[  n\right]
}\right)  ^{T}\mathbf{u}^{\overline{T}},$ $T\subset\left[  n\right]  $ form a
basis of the vector space of affine polynomials of degree less or equal to
$n.$ Because $u_{i}=0\Leftrightarrow\theta_{i}=-\frac{1}{p_{i}},$ for
$i\in\left[  n\right]  ,$ we have%
\[
Q\left(  0\right)  =P(-\frac{1}{\mathbf{p}}\mathbf{1}_{\left[  n\right]
})=\left(  -1\right)  ^{n}\alpha_{\left[  n\right]  },
\]
consequently
\[
\alpha_{\left[  n\right]  }=\left(  -1\right)  ^{n}P(-\frac{1}{\mathbf{p}%
}\mathbf{1}_{\left[  n\right]  }).
\]
Since $P$ is an affine polynomial, we have
\[
\alpha_{T}=\left(  -1\right)  ^{\left\vert T\right\vert }P(-\frac
{1}{\mathbf{p}}\mathbf{1}_{T}),
\]
in particular $\alpha_{T}=0,$ if $\left\vert T\right\vert =1$ and
$\alpha_{\varnothing}=1.$ Finally we obtain%
\begin{align}
P\left(  \boldsymbol{\theta}\right)   &  =\sum_{T\subset\left[  n\right]
}\left(  -1\right)  ^{\left\vert T\right\vert }P(-\frac{1}{\mathbf{p}%
}\mathbf{1}_{T})\left(  \mathbf{u}-\mathbf{1}_{\left[  n\right]  }\right)
^{T}\mathbf{u}^{\overline{T}}\label{d_p_affine}\\
&  =\mathbf{u}^{\left[  n\right]  }+\sum_{T\subset\left[  n\right]
,\left\vert T\right\vert >1}\left(  -1\right)  ^{\left\vert T\right\vert
}P(-\frac{1}{\mathbf{p}}\mathbf{1}_{T})\left(  \mathbf{u}-\mathbf{1}_{\left[
n\right]  }\right)  ^{T}\mathbf{u}^{\overline{T}}\nonumber
\end{align}
We deduce from equality (\ref{d_p_affine})%
\begin{align*}
L_{\mathbf{\gamma}_{\left(  P,\lambda\right)  }}\left(  \boldsymbol{\theta
}\right)   &  =[\sum_{T\subset\left[  n\right]  }\left(  -1\right)
^{\left\vert T\right\vert }P(-\frac{1}{\mathbf{p}}\mathbf{1}_{T})\left(
\mathbf{u}-\mathbf{1}_{\left[  n\right]  }\right)  ^{T}\mathbf{u}%
^{\overline{T}}]^{-\lambda}\\
&  =\left(  \mathbf{u}^{\left[  n\right]  }\right)  ^{-\lambda}[\sum
_{T\subset\left[  n\right]  }\left(  -1\right)  ^{\left\vert T\right\vert
}P(-\frac{1}{\mathbf{p}}\mathbf{1}_{T})(\mathbf{1}_{\left[  n\right]  }%
-\frac{1}{\mathbf{u}})^{T}]^{-\lambda}\\
&  =\mathbf{v}^{\left[  n\right]  }[1+\sum_{T\subset\left[  n\right]
,\left\vert T\right\vert >1}\left(  -1\right)  ^{\left\vert T\right\vert
}P(-\frac{1}{\mathbf{p}}\mathbf{1}_{T})\prod_{t\in T}(1-v_{t}^{\frac
{1}{\lambda}})]^{-\lambda}\\
&  =C_{L_{\gamma_{\left(  P,\lambda\right)  }}}\left(  \mathbf{v}\right)  .
\end{align*}

\subsection{Proof of Theorem \ref{copulagammaMM}}

We start from the Laplace transform of the \textit{multivariate multi-factor
gamma distribution }associated to $\left(  P,\Lambda\right)  ,$ $\Lambda
=\left(  \lambda,\lambda_{1},\ldots\lambda_{n}\right)  $, where $P$ is an
affine polynomial with respect to $\boldsymbol{\theta}=\left(  \theta
_{1},\ldots,\theta_{n}\right)  $ and $\lambda_{i}\geqslant\lambda,$ for
$i\in\left[  n\right]  .$ We use the following notations $u_{i}=\left(
1+p_{i}\theta_{i}\right)  ^{-\lambda}$ and $v_{i}=\left(  1+p_{i}\theta
_{i}\right)  ^{-\lambda_{i}}$ so that $\left(  1+p_{i}\theta_{i}\right)
^{-\left(  \lambda_{i}-\lambda\right)  }=v_{i}/u_{i}$ and $\left(
1+p_{i}\theta_{i}\right)  ^{-1}=u_{i}^{1/\lambda}=v_{i}^{1/\lambda_{i}}.$
According to equality \ref{LC_MGD}, we have%
\begin{align*}
L_{\mathbf{\gamma}_{\left(  P,\Lambda\right)  }}\left(  \boldsymbol{\theta
}\right)   &  =\left[  P\left(  \boldsymbol{\theta}\right)  \right]
^{-\lambda}\prod_{i\in\left[  n\right]  }\left(  1+p_{i}\theta_{i}\right)
^{-\left(  \lambda_{i}-\lambda\right)  }\\
&  =\mathbf{u}^{\left[  n\right]  }[1+\sum_{T\subset\left[  n\right]
,\left\vert T\right\vert >1}\left(  -1\right)  ^{\left\vert T\right\vert
}P(-\frac{1}{\mathbf{p}}\mathbf{1}_{T})\prod_{t\in T}(1-u_{t}^{\frac
{1}{\lambda}})]^{-\lambda}\prod_{i\in\left[  n\right]  }\frac{v_{i}}{u_{i}}\\
&  =\mathbf{v}^{\left[  n\right]  }[1+\sum_{T\subset\left[  n\right]
,\left\vert T\right\vert >1}\left(  -1\right)  ^{\left\vert T\right\vert
}P(-\frac{1}{\mathbf{p}}\mathbf{1}_{T})\prod_{t\in T}(1-u_{t}^{\frac
{1}{\lambda}})]^{-\lambda}\\
&  =\mathbf{v}^{\left[  n\right]  }[1+\sum_{T\subset\left[  n\right]
,\left\vert T\right\vert >1}\left(  -1\right)  ^{\left\vert T\right\vert
}P(-\frac{1}{\mathbf{p}}\mathbf{1}_{T})\prod_{t\in T}(1-v_{t}^{\frac
{1}{\lambda_{t}}})]^{-\lambda}\\
&  =C_{L_{\gamma_{\left(  P,\Lambda\right)  }}}\left(  \mathbf{v}\right)  .
\end{align*}

\subsection{Proof of Proposition \ref{Kendall's tau & Spearman's rho}}

\subsubsection{Kendall's tau}

By injecting the given copula in formula (\ref{bmfgLc}) in formula
(\ref{tau formula}), we obtain%

\begin{align*}
\frac{1-\tau}{4}  &  =%
{\displaystyle\iint_{\left[  0,1\right]  ^{2}}}
u_{1}u_{2}[1-r_{12}(1-u_{1}^{\frac{1}{\lambda_{1}}})(1-u_{2}^{\frac{1}%
{\lambda_{2}}})]^{-2\lambda-2}\\
&  \{1-r_{12}[1-(1-\frac{\lambda}{\lambda_{1}})u_{1}^{\frac{1}{\lambda_{1}}%
}](1-u_{2}^{\frac{1}{\lambda_{2}}})\}\{1-r_{12}[1-(1-\frac{\lambda}%
{\lambda_{2}})u_{2}^{\frac{1}{\lambda_{2}}}](1-u_{1}^{\frac{1}{\lambda_{1}}%
})\}\,\text{d}u_{1}\text{d}u_{2}.
\end{align*}
\newline By changing the variables $t_{i}=1-u_{i}^{1/\lambda_{i}},$ $i=1,2$,
we obtain%
\begin{align*}
\frac{1-\tau}{4\lambda_{1}\lambda_{2}}  &  =%
{\displaystyle\iint_{\left[  0,1\right]  ^{2}}}
\left(  1-r_{12}t_{1}t_{2}\right)  ^{-2\lambda}\left(  1-t_{1}\right)
^{2\lambda_{1}-1}\left(  1-t_{2}\right)  ^{2\lambda_{2}-1}\\
&  -\frac{\lambda}{\lambda_{1}}r_{12}\left(  1-r_{12}t_{1}t_{2}\right)
^{-2\lambda-1}\left(  1-t_{1}\right)  ^{2\lambda_{1}}t_{2}\left(
1-t_{2}\right)  ^{2\lambda_{2}-1}\\
&  -\frac{\lambda}{\lambda_{2}}r_{12}\left(  1-r_{12}t_{1}t_{2}\right)
^{-2\lambda-1}t_{1}\left(  1-t_{1}\right)  ^{2\lambda_{1}-1}\left(
1-t_{2}\right)  ^{2\lambda_{2}}\\
&  +\frac{\lambda^{2}}{\lambda_{1}\lambda_{2}}r_{12}^{2}\left(  1-r_{12}%
t_{1}t_{2}\right)  ^{-2\lambda-2}t_{1}\left(  1-t_{1}\right)  ^{2\lambda_{1}%
}t_{2}\left(  1-t_{2}\right)  ^{2\lambda_{2}}\,\text{d}t_{1}\text{d}t_{2}.
\end{align*}
By using equality $\left(  1-r_{12}t_{1}t_{2}\right)  ^{-2\lambda}=\sum
_{k=0}^{\infty}\left(  2\lambda\right)  _{k}\frac{\left(  r_{12}^{k}t_{1}%
^{k}t_{2}^{k}\right)  }{k!}$ in the last result, we obtain,%
\begin{align*}
\frac{1-\tau}{4\lambda_{1}\lambda_{2}}  &  =\sum_{k=0}^{\infty}\left(
2\lambda\right)  _{k}\frac{r_{12}^{k}}{k!}B\left(  k+1,2\lambda_{1}\right)
B\left(  k+1,2\lambda_{2}\right) \\
&  -\frac{\lambda}{\lambda_{1}}r_{12}\sum_{k=0}^{\infty}\left(  2\lambda
+1\right)  _{k}\frac{r_{12}^{k}}{k!}B\left(  k+1,2\lambda_{1}+1\right)
B\left(  k+2,2\lambda_{2}\right) \\
&  -\frac{\lambda}{\lambda_{2}}r_{12}\sum_{k=0}^{\infty}\left(  2\lambda
+1\right)  _{k}\frac{r_{12}^{k}}{k!}B\left(  k+2,2\lambda_{1}\right)  B\left(
k+1,2\lambda_{2}+1\right) \\
&  +\frac{\lambda^{2}}{\lambda_{1}\lambda_{2}}r_{12}^{2}\sum_{k=0}^{\infty
}\left(  2\lambda+2\right)  _{k}\frac{r_{12}^{k}}{k!}B\left(  k+2,2\lambda
_{1}+1\right)  B\left(  k+2,2\lambda_{2}+1\right)  .
\end{align*}
With the equality $B\left(  \alpha,\beta\right)  =\Gamma\left(  \alpha\right)
\Gamma\left(  \beta\right)  /\Gamma\left(  \alpha+\beta\right)  $ for
$\alpha,\beta>0,$ we obtain%
\begin{align*}
1-\tau &  =\sum_{k=0}^{\infty}\frac{\left(  1\right)  _{k}\left(  1\right)
_{k}\left(  2\lambda\right)  _{k}}{\left(  2\lambda_{1}+1\right)  _{k}\left(
2\lambda_{2}+1\right)  _{k}}\frac{r_{12}^{k}}{k!}\\
&  -\frac{4\lambda}{\left(  2\lambda_{1}+1\right)  \left(  2\lambda
_{2}+1\right)  }r_{12}\sum_{k=0}^{\infty}\frac{\left(  1\right)  _{k}\left(
2\right)  _{k}\left(  2\lambda+1\right)  _{k}}{\left(  2\lambda_{1}+2\right)
_{k}\left(  2\lambda_{2}+2\right)  _{k}}\frac{r_{12}^{k}}{k!}\\
&  +\frac{\lambda^{2}}{\left(  2\lambda_{1}+1\right)  \left(  2\lambda
_{2}+1\right)  \left(  \lambda_{1}+1\right)  \left(  \lambda_{2}+1\right)
}r_{12}^{2}\sum_{k=0}^{\infty}\frac{\left(  2\right)  _{k}\left(  2\right)
_{k}\left(  2\lambda+2\right)  _{k}}{\left(  2\lambda_{1}+3\right)
_{k}\left(  2\lambda_{2}+3\right)  _{k}}\frac{r_{12}^{k}}{k!}.
\end{align*}
Finally, by using equality (\ref{Hypergeomconf}), we have proved equality
(\ref{KtauHyper}).

\subsubsection{Spearman's rho}

By injecting the given copula in formula (\ref{bmfgLc}) in formula
((\ref{rho formula}), we obtain%

\begin{equation}
\varrho_{S}=12\int_{0}^{1}\int_{0}^{1}u_{1}u_{2}[1-r_{12}(1-u_{1}^{\frac
{1}{\lambda_{1}}})(1-u_{2}^{\frac{1}{\lambda_{2}}})]^{-\lambda}\,\text{d}%
u_{1}\text{d}u_{2}-3. \label{rho inter}%
\end{equation}
By changing the variables $v_{i}=u_{i}^{\frac{1}{\lambda_{i}}},$ $i=1,2$, we
obtain%
\begin{align}
&  \hspace{-3cm}\int_{0}^{1}\int_{0}^{1}u_{1}u_{2}[1-r_{12}\left(
(1-u_{1}^{\frac{1}{\lambda_{1}}})\right)  (1-u_{2}^{\frac{1}{\lambda_{2}}%
})]^{-\lambda}\,\text{d}u_{1}\text{d}u_{2}\nonumber\\
&  =\int_{0}^{1}\int_{0}^{1}\lambda_{1}\lambda_{2}v_{1}^{2\lambda_{1}-1}%
v_{2}^{2\lambda_{2}-1}\left[  1-r_{12}\left(  1-v_{1}\right)  \left(
1-v_{2}\right)  \right]  ^{-\lambda}\,\text{d}v_{1}\text{d}v_{2}\nonumber\\
&  =\lambda_{1}\lambda_{2}\sum_{k=0}^{\infty}\frac{\left(  \lambda\right)
_{k}}{k!}r_{12}^{k}\int_{0}^{1}v_{1}^{2\lambda_{1}-1}\left(  1-v_{1}\right)
^{k}\text{d}v_{1}\int_{0}^{1}v_{2}^{2\lambda_{2}-1}\left(  1-v_{2}\right)
^{k}\,\text{d}v_{2}\nonumber\\
&  =\lambda_{1}\lambda_{2}\sum_{k=0}^{\infty}\frac{\left(  \lambda\right)
_{k}}{k!}r_{12}^{k}B\left(  2\lambda_{1},k+1\right)  B\left(  2\lambda
_{2},k+1\right) \nonumber\\
&  =\frac{1}{4}\sum_{k=0}^{\infty}\frac{\left(  \lambda\right)  _{k}\left(
1\right)  _{k}^{2}}{\left(  2\lambda_{1}+1\right)  _{k}\left(  2\lambda
_{2}+1\right)  _{k}}\frac{r_{12}^{k}}{k!}\nonumber\\
&  =\frac{1}{4}F_{2}^{3}\left(  1,1,\lambda;2\lambda_{1}+1,2\lambda
_{2}+1;r_{12}\right)  \label{integral rho}%
\end{align}
By injecting equality (\ref{integral rho}) in equality (\ref{rho inter}), we
obtain equality (\ref{SrhoHyper1}). We remark that we have%
\[
F_{2}^{3}\left(  \lambda,1,1;2\lambda_{1}+1,2\lambda_{2}+1;r_{12}\right)
-1=\frac{\lambda}{\left(  2\lambda_{1}+1\right)  \left(  2\lambda
_{2}+1\right)  }r_{12}\left[  \sum_{k=0}^{\infty}\left(  k+1\right)
!\frac{\left(  \lambda+1\right)  _{k}}{\left(  2\lambda_{1}+2\right)
_{k}\left(  2\lambda_{2}+2\right)  _{k}}r_{12}^{k}\right]  .
\]
This gives equality (\ref{SrhoHyper2}).

\section{Acknowledgement}

I thank G\'{e}rard Letac for many helpful conversations.

redpajama_set_name: RedPajamaArXiv


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text:
Vredeveld Wins Top Swimmer as Baylor Girls Take 8th Straight Tennessee Title
Kristen Vredeveld and Dan Flack were named swimmer and coach of the meet at the Tennessee High School State Championships.
More High School
There were 3 Pool Records set at the 2020 Girls Inter-AC Invite. A couple of those marks were formerly held by some noteworthy national team swimmers.
Watch Bacon's 51.3 100 Back/52.8 100 Fly at 2020 ISL HS Champs (Full Recap)
Bacon set 2 individual League Records, while teammate Erin Gemmell broke 50 seconds to win the 100 free as a freshman. Both contributed to a pair of relays.
Two Chicago-Area HS Swimmers Robbed, One Shot While Returning From Meet
Two swimmers from Lindblom Math & Science Academy High were robbed at gunpoint this week as they were returning from a swim meet. One was shot in the chest.
Aaron Sequeira Breaks 3 Meet Records at Southwest Ohio HS Classic
The Southwest Ohio Classic, which held its 35th edition in 2020, is one of the fastest pre-state level high school meets in the country.
McCorvy & Lawson Removed From SafeSport Database, 3 Others Briefly Appear
Michael McCorvy and Joshua Lawson have been removed from the U.S. Center for SafeSport's database of banned individuals, two of several changes this week.
PA Swim Coach Under Investigation Over Inappropriate Posts, Video
Swim coach David Schultz is under investigation by his school district over a pornographic video of Schultz under a different name, local media report.
Indiana HS Diving Coach Accused of Inappropriate Touching, Charged With Battery
Indiana high school diving coach Jordan P. Hatch has been charged with misdemeanor battery after witnesses say he inappropriately touched a diver.
2003 Worlds Silver Medalist Kevin Clements Diagnosed with Rare Form of Leukemia
Kevin Clements, pictured above with his family, is a former assistant coach at USC and currently is the head coach at JSerra High School.
by Braden Keith 5
February 13th, 2013 High School, News
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The Baylor School winning 200 and 400 medley relays (L-R) Hanna Peiffer, Kristen Vredeveld, Kimberlee John-Williams, Breann Roman.
The Tennessee High School State Championship meet is a very unique one. It is truly a meet "of the people, for the people, by the people," as the TSSAA, the official governing body of secondary sports, doesn't recognize swimming. So, instead, high school swimming is run by the Tennessee Interscholastic Swim Coaches Association. That's not the only reason this league is unique. Many states split up their state championship into two days, but this meet, hosted in Knoxville by the Pilot Aquatic Club, makes their split a bit differently.
Friday's events ran prelims and finals through the 100 fly, and then Saturday started anew with prelims and finals of the 100 freestyle onward. That left an opportunity for a double that many swimmers would probably like to attempt, but isn't feasible in a normal high school meet: the 100 free and 100 fly; at this meet, Mt. Juliet sophomore Carrie Johnson pulled that off most successfully with a victory in the 100 fly on Friday (54.62) and a third-place finish in the 100 free to begin Saturday (51.88).
The winner of that 100 free was the star of this women's meet, Cal-bound senior Kristen Vredeveld. She has been a stalwort of the Baylor School for the last four years of State and National Record breaking, state title winning streak of triumph in Chattanooga, Tennessee. What better way to end her high school career with two individual victories, two relay wins, and a 7th-straight team State Championship.
As mentioned, Vredeveld's first victory came in the 100 freestyle, where she swam a 49.57. That made her the only swimmer to dip under 50 seconds in the race. The runner-up, Sophie Pilkinton, looks to be the face of Tennessee sprinting for the future, as the Harpeth Hall sophomore knocked a second off of her best time with a 50.88 for 2nd-place.
The Harpeth Hall program also has a great sprint group, for Pilkinton to train with. She and her Harpeth Hall teammates won the 400 free relay in 3:30.45, besting the runners-up from Baylor in 3:31.40. Last season, Baylor swept the relays, but that was with Vredeveld swimming all three of them and just one individual. This year, they tweaked their strategy and she went with a more traditional two individuals and two relays.
Her other event this year, and victory, came in the 50 free where she was a 22.91. Pilkinton took a silver-medal in this race for the second-straight year (both times behind a senior) in 23.53.
Among other non-Baylor winners was another Harpeth Hall swimmer: Elizabeth Stinson. She won the 500 free in 4:50.03. Christ Presbyterian freshman Tatum Wade had a very good swim for 2nd in 4:52.55; she and Stinson paced their race almost identically, but Stinson was just a smidge faster on each 50.
While many women's teams this year around the country, it seems, have won state titles on the back of young freshmen and sophomores, the striking thing about the program at Baylor is that they always have experience. That veteran presence is how they stay so consistently good for so long. They kicked off the meet with a win in the 200 medley relay in 1:42.16, including a 25.08 butterfly from Kimberlee John-Williams and a 22.11 anchor from Vredeveld. They were joined by another senior, Hannah Peiffer, and a lone junior in Bree Roman on the breaststroke leg. This relay beat last year's winning time by nearly two seconds.
That group would be the dominant names in this meet. Peiffer took the 200 IM in 2:05.10, beating Michelle Chintanapohl (2:05.48) with a great freestyle split of 29.0. Peiffer would win the 100 back as well in 54.18.
John-Williams, who is bound for Georgia, was the runner-up in the 100 fly in 55.10 and the 100 back in 54.80. Those times are both slower than she was at last year's meet, where she took both titles. Roman added a 5th individual title for the team by taking the 100 breaststroke in 1:03.44. This is her first year in yards swimming, as the junior transferred in to the country from Jamaica, giving a big boost to this team that didn't have a ton of depth in their breaststroke group. Maddie Hoff (no relationship) and Elena Escalas tied for 2nd in 1:05.10, just beating-out John Paul II's Paige Scheriger (1:05.13).
The Baylor School also took a win in the 200 free relay in 1:33.05. Vredeveld led that race off in 22.76, and was joined by John-Williams, Hannah Peiffer, and Bree Roman. In this race, they missed their State and National Record time from last year.
Kristen Vredeveld was named the meet's top swimmer, and her coach Dan Flack was named the Women's Coach of the Year.
Full meet results.
Full team standings.
Place School Points
1 Baylor School 388
2 Harpeth Hall 316
3 Brentwood High School Bruins 165. 50
4 Science Hill High School 127. 50
5 Ravenwood High School 118
6 Girls Preparatory School 108
6 Hardin Valley Academy 108
8 Mhea 98. 50
9 Bearden High School 74
10 Oakland High School 70
« 2013 SEC Women's Championship Fan Guide: Olympians, Olympians, Everywhere
Open Turn: What's hotter, High School Swimming or The Harlem Shake? »
Kimberlee John-Williams
Kristen Vredeveld
Who's this Chatham Dobbs character on the boy's side? 16 years old according to USA Swimming, 19.9 relay split, 45.2 100 free, 48.4 100 back, 1:48.1 200 IM.
Chatham Dobbs
me! haha
completely conquered
Interesting meet format. I am making my poor swimmer do the 100 fly and 100 free this year and she won't get but about 9 minutes in between each.
About Braden Keith
Braden Keith is the Editor-in-Chief and a co-founder of SwimSwam.com. He first got his feet wet by building The Swimmers' Circle beginning in January 2010, and now comes to SwimSwam to use that experience and help build a new leader in the sport of swimming. Aside from his life on the InterWet, …
More from Braden Keith
Noosa Club Dominates as Lee and Sloman Sweep 5K Time Trials in Australia
5-Time Georgia High School State Finalist Madigan Starr Commits to Air Force

redpajama_set_name: RedPajamaCommonCrawl


text:
Cyberpunk 2077 Reveals Cool New Details About Keanu Reeves' Johnny Silverhand
There are several good reasons why Cyberpunk 2077 is one of the most highly-anticipated video ga...
There are several good reasons why Cyberpunk 2077 is one of the most highly-anticipated video games of the year. In addition to some awesome RPG features, the title will also feature Keanu Reeves as one of the cool characters who help you along the way. But just how amazing is Johnny Silverhand? New details about the character have just been revealed and just like Reeves, he's a rockstar.
The new details were revealed in Cyberpunk 2077's Night City Wire: Episode 2 which featured a spotlight on Johnny. Interestingly, it was confirmed that the character is actually the lead singer of a fictional punk group Samurai and although Reeves is Johnny's speaking voice, his songs are performed by Dennis Lyxzén of the band Refused.
Lyxzén also shared some cool details about how Johnny will be an important part of Cyberpunk 2077's storyline. The frontman stated that Johnny is one of the characters that the game player will meet in Night City. The meeting will lead to the player's character being inspired by Johnny's anti-establishment views and propel them further in the gameplay.
First of all, it's already fantastic that Reeves is playing Johnny but we think he's even cooler now that he's a confirmed rockstar who will be performing punk songs in the game. Needless to say, we can't wait to start playing and meeting Johnny this November.
Cyberpunk 2077 will be available on PlayStation 4, Windows, and Xbox One in November 2020 before being released on Stadia by the end of the year. The game will be available on PlayStation 5 and Xbox Series X in 2021.
Related: Keanu Reeves Releasing Comic Book Series About An Immortal Superhero Who Looks Just Like Him


A public action was brought before the Constitutional Court of Colombia in which the petitioners challenged the constitutionality of article 30 of the Labour Code and article 238 (partial) of Decree-Law 2737 of 1989 (Juvenile Code) because they were in violation of articles 44 and 67 of the Constitution.
The Court confined its decision to the constitutionality of article 238 of the Juvenile Code, which authorizes young persons under the age of 18 years to enter into employment contracts, subject to the fulfilment of certain requirements.
1 ILO Convention on Minimum Age, 1973 (No. 138); ILO Convention on the Worst Forms of Child Labour, 1999 (No. 182); UN Convention on the Rights of the Child, 1989.
2 Paragraph 41 of the decision.
3 Paragraph 19 of the decision.


So what's the life of a cat like? Pretty much like you'd imagine. You have a handful of areas to explore, with territory ranging from an alley to the "mush room" and even a tree top. You'll come across little tip boxes from time to time, as well as plaques that broadcast your position when you step on them. There are even a few commands you can unleash from the chat box, including napping, meowing, screeching, and purring. Cat stuff, basically!
There's sort of a goal in ChatChat, which is to capture mice and bring them back to your doorstep, but it doesn't give you anything beyond points, which are about as useful as mittens for lobsters. You can also get turned into a dog, which functions sort of like a game of tag, adding a bit of craziness to the otherwise laid back social atmosphere. If you're looking for some time to kill and would like to pretend you're a cat, well, now you know where to go!
Just to let you guys know; the rating on this one applies to the fact that it's an online game, so other players may change the experience for you.
Your experience greatly depends upon the people you meet and how cool they are. For example: I once played an epic game of dog tag with another kitty, only to block him out of the tree by getting a bunch of other cats to sleep near the base. Priceless! If you don't find cool people at first, keep trying, they're out there!
Surprisingly fun, especially if you find a good group.
I've got a problem. I'm trying to play this game because it sounds really cool, but it won't load for me. It just shows a black screen where the game should be. I'm playing on a Windows 7 computer, on Firefox, if that helps.
Takes a while to find awesome people, but totally worth it when you do.
By the way, has anyone else come across a certain spammer by the name of Jerkcat? Ruined our room.
I love this game. Btw if you turn into a dog but want to be changed back, either tag another player or catch a mouse and bring it to the doorstep.
It is a fun game! I found a group of bronies with names of the characters and we played together. I just wish you could decide your cat's color.
It won't let me go in anywhere?
It keeps saying "Enter Room Name to Create" but when I try, it just says that that room isn't available or something...I tried KittyRpg and KittyRpg2 but they didn't work either. What's going on?
I keep trying to play this game but I keep getting what the game is calling a "connection error". I've played this game before and this has'nt happend. Is it because I'm using internet explorer 7? Do I need to update my internet browser?
Sorry to hear someone's being a jerk on the game. Just meow and walk away! :-) Games with lots of online interactions always carry the risk of someone pulling a stunt like that. At least with this game they're easier to ignore.
I found a hidden music room!
It's in the southeastern part of the twisty cavern.
Someone says they're stuck following me. How do they undo that?
Hi guys i am new here!
No you dont. Sometimes when I play it says that. It's only because the game is gonna crash. That's all.
That's right. despite that I was once a Thunderclan, I have decided to give up my cozy life to come and risk my hononorable clan member. I want to help and be friends with all of you. FEWW!!
aww, is the game broke?


text:
About Alaska's Volcanoes | Hazards from Alaska's Volcanoes | Map & Alphabetical List | Interactive Map | Eruption Search | Volcano Search | Buldir
You are here: Home > Volcano Information > Regional Map & Alphabetical List > Buldir
Buldir bibliography: all known references that deal with Buldir.
BULDIR LINKS
Kiska
38 REFERENCES FOUND
Sort results by: Publication YearAuthorTitle
Generation of calc-alkaline magmas during crystallization at high oxygen fugacity: an experimental and petrologic study of tephras from Buldir Volcano, western Aleutian arc, Alaska, USA, 2020
Waters, L.E., Cottrell, Elizabeth, Coombs, M.L., and Kelley, K.A., 2020, Generation of calc-alkaline magmas during crystallization at high oxygen fugacity: an experimental and petrologic study of tephras from Buldir Volcano, western Aleutian arc, Alaska, USA: Journal of Petrology, p. 1-36, doi:10.1093/petrology/egaa104.
2018 update to the U.S. Geological Survey national volcanic threat assessment, 2018
Ewert, J.W., Diefenbach, A.K., and Ramsey, D.W., 2018, 2018 update to the U.S. Geological Survey national volcanic threat assessment: U.S. Geological Survey Scientific Investigations Report 2018-5140, 40 p., https://doi.org/10.3133/ sir20185140.
download from USGS
Quantifying arc migration and the role of forearc subduction erosion in the central Aleutians, 2018
Jicha, B.R., and Kay, S.M., 2018, Quantifying arc migration and the role of forearc subduction erosion in the central Aleutians: Journal of Volcanology and Geothermal Research, v, 360, p. 84-99, doi: http://dx.doi.org/10.1016/j.jvolgeores.2018.06.016.
The role of subducted basalt in the source of island arc magmas: evidence from seafloor lavas of the Western Aleutians, 2015
Yogodzinski, G.M., Brown, S.T., Kelemen, P.B., Vervoort, J.D., Portnyagin, Maxim, Sims, K.W.W., Hoernle, Kaj, Jicha, B.R., and Werner, Reinhard, 2015, The role of subducted basalt in the source of island arc magmas: evidence from seafloor lavas of the Western Aleutians: Journal of Petrology, v. 56, n. 3, p. 441-492, doi:10.1093/petrology/egv006
Preliminary database of Quaternary vents in Alaska, 2014
Cameron, C.E., and Nye, C.J., 2014, Preliminary database of Quaternary vents in Alaska: Alaska Division of Geological & Geophysical Surveys Miscellaneous Publication 153, 11 p., doi:10.14509/27357 .
Controls of melt genesis and evolution at Buldir volcano in the western Aleutian island arc, 2011
Arndt, Shawn, 2011, Controls of melt genesis and evolution at Buldir volcano in the western Aleutian island arc: University of South Carolina unpublished Masters thesis. Https://scholarcommons.sc.edu/etd/1318/.
Subduction controls of Hf and Nd isotopes in lavas of the Aleutian island arc, 2010
Yogodzinski, G.M., Vervoort, J.D., Brown, S.T., and Gerseny, M., 2010. Subduction controls of Hf and Nd isotopes in lavas of the Aleutian island arc: Earth and Planetary Science Letters, v. 300, p. 226-238, doi: 10.1016/j.epsl.2010.09.035 .
Preliminary spreadsheet of eruption source parameters for volcanoes of the world, 2009
Mastin, L.G., Guffanti, Marianne, Ewert, J.E., and Spiegel, Jessica, 2009, Preliminary spreadsheet of eruption source parameters for volcanoes of the world: U.S. Geological Survey Open-File Report 2009-1133, v. 1.2, 25 p., available at http://pubs.usgs.gov/of/2009/1133/ .
USGS website with links to PDF and spreadsheets
Western Aleutian and Kamchatka high-Mg# andesite and dacite, 2008
Yogodzinski, G.M., and Bryant, J.A., 2008, Western Aleutian and Kamchatka high-Mg# andesite and dacite [abs.]: in Abstracts of the 18th annual V.M. Goldschmidt conference, Geochimica et Cosmochimica Acta, v. 72, n. 12S, p. A1062.
System for ranking relative threats of U.S. volcanoes, 2007
Ewert, John, 2007, System for ranking relative threats of U.S. volcanoes: Natural Hazards Review, v. 8, n. 4, p. 112-124.
Along-strike trace element and isotopic variation in Aleutian Island arc basalt: subduction melts sediments and dehydrates serpentine, 2007
Singer, B.S., Jicha, B.R., Leeman, W.P., Rogers, N.W., Thirlwall, M.F., Ryan, Jeff, and Nicolaysen, K.E., 2007, Along-strike trace element and isotopic variation in Aleutian Island arc basalt: subduction melts sediments and dehydrates serpentine: Journal of Geophysical Research, v. 112, n. B6, 26 p., doi: 10.1029/2006JB004897 .
Digital data for the reconnaissance geologic map of the western Aleutian islands, Alaska, 2006
Wilson, F.H., Mohadjer, Solmaz, Labay, K.A., Shew, Nora, and Grey, D.M., 2006, Digital data for the reconnaissance geologic map of the western Aleutian islands, Alaska: U.S. Geological Survey Open-File Report 2006-1302, 1 p., available online at http://pubs.usgs.gov/of/2006/1302/
link to publication on USGS website
Volcanoes of the world: an illustrated catalog of Holocene volcanoes and their eruptions, 2003
Siebert, L., and Simkin, T., 2002-, Volcanoes of the world: an illustrated catalog of Holocene volcanoes and their eruptions: Smithsonian Institution, Global Volcanism Program Digital Information Series GVP-3, http://volcano.si.edu/search_volcano.cfm, unpaged internet resource.
Bibliography of information on Alaska volcanoes, 2003
Cameron, C. E., Triplehorn, J. H., and Robar, C. L., 2003, Bibliography of information on Alaska volcanoes: Alaska Division of Geological & Geophysical Surveys Miscellaneous Publication MP 131, 1 CD-ROM.
Along-strike variation in the Aleutian Island Arc: Genesis of high Mg# andesite and implications for continental crust, 2003
Kelemen, P.B., Yogodzinski, G.M., and Scholl, D.M., 2003, Along-strike variation in the Aleutian Island Arc: Genesis of high Mg# andesite and implications for continental crust, in Eiler, J., ed., Inside the Subduction Factory: AGU Monograph 138, p. 223-276.
Volcanoes of Alaska, 1998
Nye, C. J., Queen, Katherine, and McCarthy, A. M., 1998, Volcanoes of Alaska: Alaska Division of Geological & Geophysical Surveys Information Circular IC 0038, unpaged, 1 sheet, scale 1:4,000,000, available at http://www.dggs.dnr.state.ak.us/pubs/pubs?reqtype=citation&ID=7043 .
website with links to sheets in PDF format
Alaska Division of Geological & Geophysical Surveys, 1995, Volcanoes of Alaska: Alaska Division of Geological & Geophysical Surveys Information Circular IC 0038, unpaged, 1 sheet, scale 1:4,000,000.
Aleutian magmas in space and time, 1994
Kay, S. M., and Kay, R. W., 1994, Aleutian magmas in space and time: in Plafker, George and Berg, H. C., (eds.), The Geology of Alaska, Geological Society of America The Geology of North America series v. G-1, p. 687-722.
Volcanoes of the world [2nd edition], 1994
Simkin, Tom, and Siebert, Lee, 1994, Volcanoes of the world [2nd edition]: Tucson, Arizona, Geoscience Press, 349 p.
The geology, geochemistry and petrology of the recent magmatic phase of the central and western Aleutian Arc, 1994
Myers, J. D., 1994, The geology, geochemistry and petrology of the recent magmatic phase of the central and western Aleutian Arc: unpublished manuscript unpaged.
Age, character, and significance of Aleutian arc volcanism, 1994
Fournelle, J. H., Marsh, B. D., and Myers, J. D., 1994, Age, character, and significance of Aleutian arc volcanism: in Plafker, George and Berg, H. C., (eds.), The Geology of Alaska, Geological Society of America The Geology of North America Series v. G-1, p. 723-758.
Geothermal resources of the Aleutian Arc, 1993
Motyka, R. J., Liss, S. A., Nye, C. J., and Moorman, M. A., 1993, Geothermal resources of the Aleutian Arc: Alaska Division of Geological & Geophysical Surveys Professional Report PR 0114, 17 p., 4 sheets, scale 1:1,000,000.
website with links to PDF files
Volcanoes of North America: United States and Canada, 1990
Wood, C. A., and Kienle, Juergen, (eds.), 1990, Volcanoes of North America: United States and Canada: New York, Cambridge University Press, 354 p.
Aleutian magmatic systems: an integrated view, 1989
Kay, R. W., and Kay, S. M., 1989, Aleutian magmatic systems: an integrated view [abs.]: in Continental magmatism: abstracts, New Mexico Bureau of Mines and Mineral Resources Bulletin 0131, p. 148.
An evaluation of the global variations in the major element chemistry of arc basalts, 1988
Plank, Terry, and Langmuir, C.H., 1988, An evaluation of the global variations in the major element chemistry of arc basalts: Earth and Planetary Science Letters, v. 90, p. 349-370.
Spacial variations in trace element ratios in the Aleutian Arc, 1986
Kay, S. M., Kay, R. W., Romick, J. D., and Yogodzinski, G. M., 1986, Spacial variations in trace element ratios in the Aleutian Arc [abs.]: Abstracts with Programs - Geological Society of America, v. 18, n. 6, p. 651.
Map showing distribution, composition, and age of Late Cenozoic volcanic centers in Alaska, 1986
Luedke, R. G., and Smith, R. L., 1986, Map showing distribution, composition, and age of Late Cenozoic volcanic centers in Alaska: U.S. Geological Survey Miscellaneous Investigations Series Map I 1091-F, unpaged, 3 sheets, scale 1:1,000,000.
ADGGS website with links to map PDFs
Mafic mineralogy in calc-alkaline Aleutian volcanic rocks: evidence for primitive magmas and magma mixing, 1982
Kay, S. M., and Kay, R. W., 1982, Mafic mineralogy in calc-alkaline Aleutian volcanic rocks: evidence for primitive magmas and magma mixing [abs.]: Abstracts with Programs - Geological Society of America, v. 14, n. 7, p. 526.
The Aleutians, 1982
Marsh, B. D., 1982, The Aleutians: in Thorpe, R. S., (ed.), Andesites: orogenic andesites and related rocks, Chichester, United Kingdom, John Wiley & Sons, p. 99-114.
Volcanoes of the world, 1981
Simkin, Tom, Siebert, Lee, McClelland, Lindsay, Bridge, David, Newhall, Christopher, and Latter, J. H., 1981, Volcanoes of the world: Stroudsburg, PA, Hutchinson Publishing Company, 233 p.
Comprehensive tables giving physical data and thermal energy estimates for young igneous systems of the United States, 1978
Smith, R. L., Shaw, H. R., Luedke, R. G., and Russell, S. L., 1978, Comprehensive tables giving physical data and thermal energy estimates for young igneous systems of the United States: U.S. Geological Survey Open-File Report 78-0925, p. 1-25.
Alaska Peninsula-Aleutian Islands, 1958
Powers, H. A., 1958, Alaska Peninsula-Aleutian Islands: in Williams, H., (ed.), Landscapes of Alaska, Los Angeles, CA, University of California Press, p. 61-75.
Reconnaissance geology of some western Aleutian Islands, Alaska, 1956
Coats, R. R., 1956, Reconnaissance geology of some western Aleutian Islands, Alaska: in Investigations of Alaskan volcanoes, U.S. Geological Survey Bulletin 1028-E, p. 83-100, 1 sheet, scale unknown.
full-text PDF : 4.4 MB
plate 17 PDF : 1.6 MB
Geology of Buldir Island, Aleutian Islands, Alaska, 1953
Coats, R. R., 1953, Geology of Buldir Island, Aleutian Islands, Alaska: U.S. Geological Survey Bulletin 0989-A, 26 p.
full-text PDF : 4 MB
Configuration of the Aleutian Ridge: Rat Islands-Semisopochnoi I. to west of Buldir I., 1953
Gibson, William, and Nichols, H., 1953, Configuration of the Aleutian Ridge: Rat Islands-Semisopochnoi I. to west of Buldir I.: Abstracts with Programs - Geological Society of America, v. 64, n. 10, p. 1173-1188, 2 sheets, scale unknown.
Reconnaissance geology of some western Aleutian Islands, 1947
Coats, R. R., 1947, Reconnaissance geology of some western Aleutian Islands: U.S. Geological Survey Alaskan Volcano Investigations Report 0002, p. 95-105.
United States coast pilot, Alaska Part 2, Yakutat Bay to Arctic Ocean, 1947
U.S. Department of Commerce, and Coast and Geodetic Survey, 1947, United States coast pilot, Alaska Part 2, Yakutat Bay to Arctic Ocean: Washington DC, United States Government Printing Office, 659 p.
Geological notes on some of the coasts and islands of the Bering Sea and vicinity, 1894
Dawson, G.M., 1894, Geological notes on some of the coasts and islands of the Bering Sea and vicinity: Geological Society of America Bulletin, v. 5, p. 117-146.
URL: avo.alaska.edu/volcanoes/volcbib.php


If Skepta Didn't Break Through in the US, Does Stormzy Have a Chance?
Grime continues to fight for a legitimate moment in the spotlight across the pond.
From Texas to Tottenham, RiFF RaFF and Skepta team up for a terrifying trunk rattler.


The Fleurieu Peninsula council requested a new website that would bring their website up to date with the latest in website technologies and user experiences.
JABA built a regional tourism website using Australian Tourism Data Warehouse (ATDW) information with V3 technology for online bookings. JABA also designed the website with style changes to colours and textures, as well as using capitals in headline fonts with key feature words highlighted by a set theme colour. The concept behind this design was proven through extensive eye tracking surveys to be the most effective for user retention online. The Fleurieu Peninsula council are incredibly pleased with the current website design as well as the simple to learn Content Managment System (CMS).
"The Fleurieu Peninsula Tourism website was recently rebuilt and looks great. The after service JABA provided was amazing and the staff are great to deal with. The design is fantastic and JABA are very helpful in this aspect. Managing the CMS has been very easy and the training session was clear and valueable"
Fleurieu Peninsula is situated just 45 minutes south of Adelaide and offers some of the best food, wine, wildlife and events in the State.
As a peninsula, water plays a major part in the recreational activities with long sandy beaches, rugged cliff-scapes, and sheltered coves introduce a range of year round holiday activities that include fishing, surfing, scuba and snorkelling, sailing and swimming. There is plenty of things to do, places to eat and a wide range of accommodation to rest at.


This intensive 1-day comprehensive course provides participants with a solid understanding of the Medical Device Regulations in Canada and the USA as well as the procedures for Investigational Testing involving Human Subjects. It will also give insight to the requirements for advertisement, labelling, study conduct and monitoring, reporting Adverse Events, complaint handling and how all of these apply to a quality management system.
Peivand holds a Ph.D. in Physiology and Biophysics from School of Medicine, University of Sherbrook, and MBA from American Business Administration Institute, and he is accredited by the Canadian Council of Pharmaceutical Education with certifications in Evidence Based Medicine, Psychiatry, and Management.
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He has authored more than 65 scientific realizations, including 12 publications in the National Library of Canada, articles in peer-reviewed journals, oral and poster presentations, and more than 1000 technical and scientific reports.


text:
We need a better understanding of 'good' research impacts
Paul Benneworth, Senior Researcher, Center for Higher Education Policy Studies, University of Twente, the Netherlands
My starting point is to welcome the recently published Federation for the Humanities and Social Sciences report, Approaches to assessing impact in the Humanities and Social Sciences as a valuable addition to a growing policy understanding of the diversity of ways in which humanities and social sciences research (HSS) creates societal impact. It matches what has been found elsewhere by the British Academy in the UK, the AWTI in the Netherlands and the Norwegian Research Council, and reflects a nuanced understanding among policy-makers that there are no simple metric-based ways to measure HSS impact.
Those countries that have introduced assessment methods for impact have had to use relatively qualitative, descriptive approaches, such as the UK's Impact Case Studies or the Netherlands Standard Evaluation Protocol. Both these approaches rely on stylisation and peer review to turn particular exemplary activities into scores allowing comparison between research groups. And research from within our own ENRESSH network shows that these two approaches are being adopted more widely across most research councils within Europe seeking to assess HSS impact.
But despite the widespread understanding of the diversity of ways in which HSS creates societal impact, there remains a persistent concern that HSS researchers have made little, or at least fragmented, progress in achieving societal impacts. One explanation for this phenomenon is organizational: Researchers have neither the time, the training nor the incentives to make achieving impacts a primary goal. Our own ENRESSH research (reported recently at the Eu-SPRI conference) started identifying a second general explanation: the difficulty of determining what constitutes "good" impact in research.
In particular, we focus on the question of what constitutes a benefit, who benefits from the research impact, and how closely that fits with the researchers' own ethical frameworks. In much of the work undertaken on HSS research impact, there is a tendency to highlight case studies that are unambiguously "positive" for society. And herein lies the problem: what is positive for "society" is not fixed, unlike economic impact defined in terms of GDP, but is instead a politically defined characteristic.
At ENRESSH, for example, we found an example of a historical research group studying a country whose popular self-image as an independent nation-state was bound up with a particular conservative-nationalist political current. Challenging that historical narrative was bound up with challenging that conservative political capital, the impacts of which could be regarded as negative, unpatriotic or worse.
Likewise, urban socologists working in the field of social exclusion find their research increasingly co-opted by "resilience studies." Resilience approaches involve equipping weaker communities to deal with difficult circumstances and turbulence and therefore carry the imprimatur of being socially positive. But from certain progressive political perspectives, the concept of resilience has become embroiled in a wider political climate in which state responsibilities for welfare are passed back to smaller groups, and therefore abandoning those more vulnerable groups to the market.
And just as no ethical medical researcher is going to develop a drug that significantly harms their patients, it is unsurprising that HSS researchers are concerned that promoting impact may lead to their research harming their research subjects or others. From my perspective, the elephant in the corner of the room for the HSS debate is this wider question of "which publics benefit in what ways from research impact?" If we do not seriously consider such questions about the assessments of public value, such assessments risk looking anecdotal, subjective, and ultimately as less convincing than more economic impact assessments that hide behind the faux objectivity of pricing.
This question clearly needs more serious consideration and resolution of the tensions arising in using research before we will really experience a qualitative improvement in the wider societal benefits created by HSS.
About this blog series: Following the publication of the Federation's new report, "Approaches to Assessing Impacts in the Humanities and Social Sciences," we have reached out to other members of the research community to share their thoughts on the challenges and opportunities associated with assessing scholarly impacts. It is our hope that this series of blogs and our new report will help support a productive conversation in the HSS community about the important topic of scholarly impact assessment.


text:
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New posters for 9, Transformers: Revenge of the Fallen, Whatever Works, I Come With The Rain, Shrink, Homecoming, The Goods.
A script review is out for Robert Rodriguez's "Machete" spin-off feature film.
Jaron Pitts, the man behind that fan made "Green Lantern" movie trailer that got the Net all buzzing last week, talks about the work to Popdose.