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

Dominic Edelmann ,¶, Donald Richards†, and Thomas Royen‡ Division of
Biostatistics, German Cancer Research Center, Im Neuenheimer Feld 280, 69120
Heidelberg, Germany. E-mail<EMAIL_ADDRESS>¶Corresponding Author, E-mail<EMAIL_ADDRESS>†Department of Statistics, Pennsylvania State University, University Park, PA
16802, U.S.A. E-mail<EMAIL_ADDRESS>‡University of Applied Sciences,
D-55411 Bingen, Germany. E-mail<EMAIL_ADDRESS>MSC 2010 subject
classifications: Primary 60E15; Secondary 62E15, 62H10. Keywords and phrases:
Moment inequalities, multivariate gamma distribution, multivariate Gaussian
distribution, multivariate survival function, Wishart distribution.

###### Abstract

The Gaussian product inequality is an important conjecture concerning the
moments of Gaussian random vectors. While all attempts to prove the Gaussian
product inequality in full generality have been unsuccessful to date, numerous
partial results have been derived in recent decades and we provide here
further results on the problem. Most importantly, we establish a strong
version of the Gaussian product inequality for multivariate gamma
distributions in the case of nonnegative correlations, thereby extending a
result recently derived by Genest and Ouimet [5]. Further, we show that the
Gaussian product inequality holds with nonnegative exponents for all random
vectors with positive components whenever the underlying vector is positively
upper orthant dependent. Finally, we show that the Gaussian product inequality
with negative exponents follows directly from the Gaussian correlation
inequality.

##  1 Introduction

Let $d$ be a positive integer, and let ${\boldsymbol{X}}=(X_{1},\ldots,X_{d})$
be a random vector which has a multivariate Gaussian distribution with
probability density function,

$(2\pi)^{-n/2}|\Sigma|^{-1/2}\exp(-\tfrac{1}{2}x^{\prime}\Sigma^{-1}x),\qquad
x\in\mathbb{R}^{d},$

with a nonsingular covariance matrix $\Sigma$. We refer to the random vector
${\boldsymbol{X}}$ as having a centered Gaussian distribution because
$\mathbb{E}\hskip 1.0pt({\boldsymbol{X}})=0$, and we write
${\boldsymbol{X}}\sim\mathcal{N}_{d}(0,\Sigma)$.

The Gaussian product inequality (GPI) conjecture states that, for any centered
Gaussian random vector ${\boldsymbol{X}}=(X_{1},\ldots,X_{d})$ and any
$n\in\mathbb{N}$, there holds the inequality,

$\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j=1}^{d}X_{j}^{2n}\Big{)}\geq\prod_{j=1}^{d}\mathbb{E}\hskip
1.0pt(X_{j}^{2n}).$ (1.1)

We refer readers to Frenkel [4] and Genest and Ouimet [5] for details of the
history, motivation, and literature on this inequality.

Several generalizations of (1.1) have been studied recently. Li and Wei [13]
considered, as an extension of (1.1), conditions such that

$\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j=1}^{d}|X_{j}|^{n_{j}}\Big{)}\geq\prod_{j=1}^{d}\mathbb{E}\hskip
1.0pt\big{(}|X_{j}|^{n_{j}}\big{)},$ (1.2)

for arbitrary $n_{1},\ldots,n_{d}>0$. Wei [20] derived hypotheses under which
the inequality

$\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j=1}^{d}|X_{j}|^{n_{j}}\Big{)}\geq\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j\in I}|X_{j}|^{n_{j}}\Big{)}\cdot\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j\in I^{c}}|X_{j}|^{n_{j}}\Big{)},$ (1.3)

holds for all index sets $I\subset\\{1,\ldots,d\\}$, where
$I^{c}=\\{1,\ldots,d\\}\setminus I$. To distinguish between these
inequalities, we call (1.2) the weak form and (1.3) the strong form of the
GPI.

Russell and Sun [18] recently related the GPI to a class of combinatorial
inequalities, and thereby established numerous cases of the GPI for $d=3$. One
of the results obtained by Russell and Sun [18] is derived here by different
methods, and we present that result in Corollary 2.2. The approach by way of
combinatorial inequalities is noteworthy because it is also shown in [18] to
lead to new inequalities for the bivariate Gaussian distributions.

The weak form of the GPI was established by Frenkel [4] for
$n_{1}=\cdots=n_{d}=2$ and arbitrary $d$, and by Lan, et al. [11] for $d=3$
and integer exponents $n_{1}$, $n_{2}$, and $n_{3}$ with equality between at
least two exponents. Genest and Ouimet [5] developed recently a novel and far-
reaching approach to the GPI, proving (1.2) for arbitrary $d$ with nonnegative
even integer exponents $n_{j}$ when the covariance matrix $\Sigma$ is
completely positive, i.e., $\Sigma=CC^{\prime}$ where $C=(c_{ij})$ is a
$d\times d$ matrix with $c_{ij}\geq 0$ for all $i,j=1,\ldots,d$. Nevertheless,
it is still unknown whether the weak form of the GPI (1.2) is valid for
general $\Sigma$.

On the other hand, the strong form of the GPI (1.3) fails even for $d=3$ with
$n_{1}=n_{2}=n_{3}=2$. Consider a Gaussian random vector,
${\boldsymbol{X}}\sim\mathcal{N}_{3}(0,\Sigma)$, with $\Sigma=(\sigma_{ij})$.
By the Isserlis-Wick formula [7] or by using moment-generating functions, we
obtain

$\mathbb{E}\hskip 1.0pt(X_{1}^{2}X_{2}^{2}X_{3}^{2})-\mathbb{E}\hskip
1.0pt(X_{1}^{2}X_{2}^{2})\mathbb{E}\hskip
1.0pt(X_{3}^{2})=2\big{(}\sigma_{11}\sigma_{23}^{2}+4\sigma_{12}\sigma_{13}\sigma_{23}+\sigma_{13}^{2}\sigma_{22}\big{)}.$

If we set $\sigma_{ii}=1$, $i=1,2,3$, and $\sigma_{13}=\sigma_{23}=\rho$ with
$0<|\rho|<1/2$ then with $\sigma_{12}\in(2\rho^{2}-1,-1/2)$, the matrix
$\Sigma$ is positive definite and yet

$\mathbb{E}\hskip 1.0pt(X_{1}^{2}X_{2}^{2}X_{3}^{2})-\mathbb{E}\hskip
1.0pt(X_{1}^{2}X_{2}^{2})\mathbb{E}\hskip
1.0pt(X_{3}^{2})=4\rho^{2}(1+2\sigma_{12})<0;$

concrete examples are $(\sigma_{12},\rho)=(-0.6,0.4)$ and
$(\sigma_{12},\rho)=(-0.75,0.3)$.

Wei [20] showed, however, that the strong form of the GPI holds for the case
in which all exponents $n_{1},\ldots,n_{d}$ are negative. It is also obvious
that the strong form (1.3) holds if
$|{\boldsymbol{X}}|:=(|X_{1}|,\ldots,|X_{d}|)$, the vector of absolute values,
is associated, i.e., if
$\mbox{Cov}\big{(}f(|{\boldsymbol{X}}|),g(|{\boldsymbol{X}}|)\big{)}\geq 0$
for all component-wise non-decreasing functions
$f,g:\mathbb{R}_{+}^{n}\to\mathbb{R}$ [3]. Thus, for a centered Gaussian
random vector ${\boldsymbol{X}}$, if $|{\boldsymbol{X}}|$ is associated then
it follows immediately that the strong form of the GPI holds. In particular,
if the vector $|{\boldsymbol{X}}|$ is multivariate totally positive of order
$2$, denoted $\mbox{MTP}_{2}$ (cf., [8]) then, as the $\mbox{MTP}_{2}$
property implies associatedness, it follows that the strong form of the GPI
holds.

Moreover, for Gaussian vectors ${\boldsymbol{X}}$, the $\mbox{MTP}_{2}$
property of its absolute values $|{\boldsymbol{X}}|$ can be characterized
explicitly in terms of the covariance matrix $\Sigma$. For this purpose (and
in the sequel) we call a diagonal matrix $S=\mbox{diag}(s_{1},\ldots,s_{d})$ a
sign matrix if $s_{j}=\pm 1$ for all $j=1,\ldots,d$. It was proved by Karlin
and Rinott [9] that for $X\sim\mathcal{N}_{d}(0,\Sigma)$ the vector of
absolute values, $|{\boldsymbol{X}}|$, is $\mbox{MTP}_{2}$ if and only if
there exists a sign matrix $S$ such that all off-diagonal entries of
$-S\Sigma^{-1}S$ are nonnegative; hence the strong form of the GPI holds for
that class of covariance matrices.

In this article, we derive new and more general hypotheses under which the
weak form of the GPI (1.2) and the strong form of the GPI (1.3) hold. We
extend the results of Genest and Ouimet [5] in several directions, one of
which is a proof of the strong form of the GPI (1.3) for nonnegative
correlations, i.e., for any covariance matrix $\Sigma=(\sigma_{ij})$ with
$\sigma_{ij}\geq 0$ for all $1\leq i<j\leq d$. Additionally, we show that the
weak form of the GPI and the strong form of the GPI follow from the properties
of positive upper orthant dependence (PUOD) and strongly positive upper
orthant dependence (SPUOD), respectively. Finally, we apply the Gaussian
correlation inequality (Royen [14]) to obtain in Section 4 an alternative and
succinct proof of the strong form of the GPI for negative exponents, derived
originally by Wei [20]; further, we show that this result extends to the
multivariate gamma distributions.

##  2 The strong form of the GPI for nonnegative correlations

Genest and Ouimet [5] established the weak form of the GPI (1.2) for the
multivariate normal distribution $\mathcal{N}_{d}(0,\Sigma)$ and with even
integers $n_{1},\ldots,n_{d}$ and completely positive covariance matrix
$\Sigma$, i.e., $\Sigma=CC^{\prime}$, where $C=(c_{ij})$ is a matrix with
nonnegative entries $c_{ij}\geq 0$. In Theorem 2.1, we extend this result in
three directions. First and most importantly, we extend the result in [5] to
the case of nonnegative correlations, where $\Sigma=(\sigma_{ij})$ is such
that $\sigma_{ij}\geq 0$ for all $1\leq i<j\leq d$. For $d\geq 5$, the
assumption of nonnegative correlations is known to be less restrictive than
complete positivity [6].

On the other hand, a famous counterexample of Šidák [19] established the
existence of Gaussian random vectors ${\boldsymbol{X}}=(X_{1},X_{2},X_{3})$
with completely positive covariance matrices and for which the vector
$|{\boldsymbol{X}}|=(|X_{1}|,|X_{2}|,|X_{3}|)$ of absolute values is not
positively upper orthant dependent (PUOD) and hence not associated. Hence the
result of Genest and Ouimet and the more general result presented here both
extend the strong form of the GPI (1.3) beyond the straightforward case in
which the vector of absolute values is associated.

The second direction in which Theorem 2.1 extends results known hitherto is
that the we obtain the strong form (1.3), hence also the weak form (1.2).

Third, in considering the case of even exponents, the weak form of the GPI
(1.2) and the strong form (1.3) each correspond to inequalities for special
cases of the multivariate gamma distributions. Precisely, the $d$-dimensional
gamma distribution (in the sense of Krishnamoorthy and Parthasarathy [10]) may
be defined by means of its moment-generating function. Denote by $I_{d}$ the
identity matrix of order $d$ and, for sufficiently small
$t_{1},\ldots,t_{d}\in\mathbb{R}$, define $T={\rm{diag}}(t_{1},\ldots,t_{d})$.
Then we say that ${\boldsymbol{X}}=(X_{1},\ldots,X_{d})$ has a multivariate
gamma distribution with a not necessarily integer “degree-of-freedom
parameter” $2\alpha$ and positive semidefinite matrix parameter $\Sigma$,
written ${\boldsymbol{X}}\sim{\hbox{Gamma}\hskip 1.0pt}_{d}(\alpha,\Sigma)$,
if the moment-generating function of ${\boldsymbol{X}}$ is

$\mathbb{E}\hskip
1.0pt\exp\Big{(}\sum_{j=1}^{d}t_{j}X_{j}\Big{)}=\det(I_{d}-\Sigma
T)^{-\alpha}.$ (2.1)

This $d$-dimensional gamma distribution is also known as the Wishart-Gamma
distribution since it was derived originally as the distribution of one-half
of the diagonal entries of a $W_{d}(2\alpha,\Sigma)$-Wishart distributed
random matrix with $2\alpha\in\mathbb{N}\cup(d-1,\infty)$. In this regard, it
is remarkable that the ${\hbox{Gamma}\hskip 1.0pt}_{d}(\alpha,\Sigma)$
distribution also exists for all values $2\alpha>[(d-1)/2]$, where $[u]$
denotes the integer part of $u\in\mathbb{R}$; see [15].

The ${\hbox{Gamma}\hskip 1.0pt}_{d}(\frac{1}{2},\Sigma)$ distribution is
infinitely divisible (i.e., the ${\hbox{Gamma}\hskip
1.0pt}_{d}(\alpha,\Sigma)$ distribution exists for all $\alpha>0$) if and only
if the ${\hbox{Gamma}\hskip 1.0pt}_{d}(\frac{1}{2},\Sigma)$ distribution is
multivariate totally positive of order $2$ ($\mbox{MTP}_{2}$); see Bapat [1].
For example, one can show that ${\hbox{Gamma}\hskip
1.0pt}_{d}(\frac{1}{2},\Sigma)$ is infinitely divisible if
$\Sigma=(\sigma_{ij})$ is of “structure $\ell$” [17], i.e.,
$\sigma_{ij}=a_{i}a_{j}(\sigma_{ii}\sigma_{jj})^{1/2}$ with $|a_{j}|<1$ for
all $j=1,\ldots,d$, or if $\Sigma^{-1}$ is of “tree-type”; see [16] for
details.

If $(X_{1},\ldots,X_{d})\sim N_{d}(0,\Sigma)$ then
$\tfrac{1}{2}(X_{1}^{2},\ldots,X_{d}^{2})\sim{\hbox{Gamma}\hskip
1.0pt}_{d}(\frac{1}{2},\Sigma)$ distribution. Consequently, in the case of
even exponents, as considered in [5], the weak form (1.2) and the strong form
(1.3) of the GPI intrinsically are inequalities on the ${\hbox{Gamma}\hskip
1.0pt}_{d}(\frac{1}{2},\Sigma)$ distribution. Therefore it is natural to
extend these inequalities to the more general multivariate gamma distributions
having moment-generating function (2.1).

######  Theorem 2.1.

Let $(X_{1},\ldots,X_{d})\sim{\hbox{Gamma}\hskip 1.0pt}_{d}(\alpha,\Sigma)$,
where $\Sigma=(\sigma_{ij})$ is positive semidefinite. Suppose there exists a
sign matrix $S$ such that all the elements in $S\Sigma S$ are nonnegative,
i.e., $\sigma_{ij}s_{i}s_{j}\geq 0$ for all $1\leq i<j\leq d$. Then for all
subsets $I\subset\\{1,\ldots,d\\}$, and for all nonnegative integers
$n_{1},\ldots,n_{d}$, there holds the strong form of the GPI,

$\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j=1}^{d}X_{j}^{n_{j}}\Big{)}\geq\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j\in I}X_{j}^{n_{j}}\Big{)}\cdot\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j\in I^{c}}X_{j}^{n_{j}}\Big{)}.$ (2.2)

Proof. Since the moment-generating function (2.1) is invariant under the
transformation $\Sigma\to S\Sigma S$ we can, without loss of generality,
assume that $\sigma_{ij}\geq 0$ for all $1\leq i<j\leq d$. Moreover, by
permuting the coordinates of ${\boldsymbol{X}}$, we may also assume that
$I=\\{1,\ldots,p\\}$ where $1\leq p\leq d-1$.

With sufficiently small $t_{1},\ldots,t_{d}\in[0,1)$ and
$T={\rm{diag}}(t_{1},\ldots,t_{d})$, the moment-generating function of the
${\hbox{Gamma}\hskip 1.0pt}_{d}(\alpha,\Sigma)$ distribution is

$\mathbb{E}\hskip
1.0pt\exp\Big{(}\sum_{j=1}^{d}t_{j}X_{j}\Big{)}=\det(I_{d}-\Sigma
T)^{-\alpha}=\det(I_{d}-T^{1/2}\Sigma T^{1/2})^{-\alpha}.$ (2.3)

Denote by $\varepsilon_{1},\ldots,\varepsilon_{d}$ the eigenvalues of the
matrix $T^{1/2}\Sigma T^{1/2}$. For sufficiently small
$t_{1},\ldots,t_{d}\in[0,1)$, we have
$\varepsilon_{1},\ldots,\varepsilon_{d}\in[0,1)$. Then we have

$\displaystyle\det(I_{d}-T^{1/2}\Sigma T^{1/2})^{-\alpha}$
$\displaystyle=\exp\big{(}-\alpha\log\det(I_{d}-T^{1/2}\Sigma T^{1/2}\big{)}$
$\displaystyle=\exp\Big{(}-\alpha\sum_{j=1}^{d}\log(1-\varepsilon_{j})\Big{)}.$

Inserting into this sum the series expansions,

$-\log(1-\varepsilon_{j})=\sum_{n=1}^{\infty}\frac{\varepsilon_{j}^{n}}{n},$

$j=1,\ldots,n$, and interchanging the order of summation, we obtain

$\displaystyle\det(I_{d}-\Sigma T)^{-\alpha}$
$\displaystyle=\exp\Big{(}\alpha\sum_{j=1}^{d}\sum_{n=1}^{\infty}\frac{\varepsilon_{j}^{n}}{n}\Big{)}$
$\displaystyle=\exp\Big{(}\alpha\sum_{n=1}^{\infty}\frac{1}{n}\sum_{j=1}^{d}\varepsilon_{j}^{n}\Big{)}=\exp\Big{(}\alpha\sum_{n=1}^{\infty}\frac{1}{n}{\rm{tr}}\hskip
1.0pt[(\Sigma T)^{n}]\Big{)}.$ (2.4)

Next, we partition $\Sigma$ and $T$ into block matrices,

$\Sigma=\begin{pmatrix}\Sigma_{11}&\Sigma_{12}\\\
\Sigma_{21}&\Sigma_{22}\end{pmatrix},\qquad T=\begin{pmatrix}T_{1}&0\\\
0&T_{2}\end{pmatrix},$

where $\Sigma_{11}$ and $T_{1}={\rm{diag}}(t_{1},\ldots,t_{p})$ are $p\times
p$, $\Sigma_{12}=\Sigma_{21}^{\prime}$ is $p\times(d-p)$, and $\Sigma_{22}$
and $T_{2}={\rm{diag}}(t_{p+1},\ldots,t_{d})$ are $(d-p)\times(d-p)$. Then,

$T^{1/2}\Sigma
T^{1/2}=\begin{pmatrix}T_{1}^{1/2}\Sigma_{11}T_{1}^{1/2}&T_{1}^{1/2}\Sigma_{12}T_{2}^{1/2}\\\
T_{2}^{1/2}\Sigma_{21}T_{1}^{1/2}&T_{2}^{1/2}\Sigma_{22}T_{2}^{1/2}\end{pmatrix}$
(2.5)

Let

$A=\begin{pmatrix}A_{11}&A_{12}\\\ A_{21}&A_{22}\end{pmatrix}$

be a symmetric matrix which has been partitioned similarly to $\Sigma$. By
induction on $n$, we find that

$A^{n}=\begin{pmatrix}A_{11}^{n}+Q_{11,n}(A)&Q_{12,n}(A)\\\
Q_{21,n}(A)&A_{22}^{n}+Q_{22,n}(A)\end{pmatrix}$ (2.6)

where each matrix $Q_{ij,n}(A)$ is a homogeneous polynomial in $A$ with
nonnegative coefficients; for instance,

$\displaystyle Q_{11,2}(A)=A_{12}A_{21},\
Q_{12,2}(A)=[Q_{21,2}(A)]^{\prime}=A_{11}A_{12}+A_{12}A_{22},\
Q_{22,2}=A_{21}A_{12}.$

Denote by $\mathbb{N}_{0}$ the set of nonnegative integers. Applying (2.6) to
(2.5), and taking traces, we obtain

$\displaystyle\frac{1}{n}{\rm{tr}}\hskip 1.0pt[(\Sigma T)^{n}]$
$\displaystyle=\frac{1}{n}{\rm{tr}}\hskip 1.0pt\big{[}(T^{1/2}\Sigma
T^{1/2})^{n}\big{]}$ $\displaystyle=\frac{1}{n}{\rm{tr}}\hskip
1.0pt\big{[}(T_{1}^{1/2}\Sigma_{11}T_{1}^{1/2})^{n}\big{]}+\frac{1}{n}{\rm{tr}}\hskip
1.0pt\big{[}(T_{2}^{1/2}\Sigma_{22}T_{2}^{1/2})^{n}\big{]}+\frac{1}{n}\sum_{i=1}^{2}Q_{ii,n}(T^{1/2}\Sigma
T^{1/2})$ $\displaystyle=\frac{1}{n}{\rm{tr}}\hskip
1.0pt[(\Sigma_{11}T_{1})^{n}]+\frac{1}{n}{\rm{tr}}\hskip
1.0pt[(\Sigma_{22}T_{2})^{n}]+\sum_{\begin{subarray}{c}{\boldsymbol{n}}\in\mathbb{N}_{0}^{d},\\\
n_{1}+\cdots+n_{d}=n\end{subarray}}c_{\boldsymbol{n}}t_{1}^{n_{1}}\cdots
t_{d}^{n_{d}},$ (2.7)

where each $c_{\boldsymbol{n}}$ is a polynomial in the entries of $\Sigma$ and
$c_{(0,\ldots,0)}=0$. It is evident from (2.6) that the coefficients of each
$c_{\boldsymbol{n}}$ are nonnegative; therefore, since the entries of $\Sigma$
are nonnegative, we obtain $c_{\boldsymbol{n}}\geq 0$ for all
${\boldsymbol{n}}$.

Substituting (2) into (2), we obtain

$\displaystyle\det(I_{d}-\Sigma T)^{-\alpha}$
$\displaystyle=\exp\Big{(}\alpha\sum_{n=1}^{\infty}\frac{1}{n}{\rm{tr}}\hskip
1.0pt[(\Sigma_{11}T_{1})^{n}]\Big{)}\exp\Big{(}\alpha\sum_{n=1}^{\infty}\frac{1}{n}{\rm{tr}}\hskip
1.0pt[(\Sigma_{22}T_{2})^{n}]\Big{)}\exp\Big{(}\sum_{{\boldsymbol{n}}\in\mathbb{N}_{0}^{d}}\alpha
c_{\boldsymbol{n}}t_{1}^{n_{1}}\cdots t_{d}^{n_{d}}\Big{)}$
$\displaystyle=\det(I_{p}-\Sigma_{11}T_{1})^{-\alpha}\det(I_{d-p}-\Sigma_{22}T_{2})^{-\alpha}\cdot\exp\Big{(}\sum_{{\boldsymbol{n}}\in\mathbb{N}_{0}^{d}}\alpha
c_{\boldsymbol{n}}t_{1}^{n_{1}}\cdots t_{d}^{n_{d}}\Big{)}.$ (2.8)

Next, the Maclaurin expansion of the exponential function leads to

$\exp\Big{(}\sum_{{\boldsymbol{n}}\in\mathbb{N}_{0}^{d}}\alpha
c_{\boldsymbol{n}}t_{1}^{n_{1}}\cdots
t_{d}^{n_{d}}\Big{)}=\sum_{{\boldsymbol{m}}\in\mathbb{N}_{0}^{d}}b_{\boldsymbol{m}}t_{1}^{m_{1}}\cdots
t_{d}^{m_{d}},$ (2.9)

Since $c_{(0,\ldots,0)}=0$ and $c_{\boldsymbol{n}}\geq 0$ for all
${\boldsymbol{n}}$ then $b_{(0,\ldots,0)}=1$ and $b_{\boldsymbol{m}}\geq 0$
for all ${\boldsymbol{m}}$.

Applying (2.3) and (2.9) to (2.8), we obtain

$\displaystyle\sum_{{\boldsymbol{n}}\in\mathbb{N}_{0}^{d}}$
$\displaystyle\Big{(}\mathbb{E}\hskip
1.0pt\prod_{j=1}^{d}\frac{X_{j}^{n_{j}}}{n_{j}!}\Big{)}t_{1}^{n_{1}}\cdots
t_{d}^{n_{d}}$
$\displaystyle=\Big{(}\sum_{{\boldsymbol{k}}\in\mathbb{N}_{0}^{p}}\mathbb{E}\hskip
1.0pt\prod_{j=1}^{p}\frac{(t_{j}X_{j})^{k_{j}}}{k_{j}!}\Big{)}\cdot\Big{(}\sum_{{\boldsymbol{l}}\in\mathbb{N}_{0}^{d-p}}\mathbb{E}\hskip
1.0pt\prod_{j=p+1}^{d}\frac{(t_{j}X_{j})^{l_{j}}}{l_{j}!}\Big{)}\cdot\Big{(}\sum_{{\boldsymbol{m}}\in\mathbb{N}_{0}^{d}}b_{\boldsymbol{m}}t_{1}^{m_{1}}\cdots
t_{d}^{m_{d}}\Big{)}.$

Collecting terms in ${\boldsymbol{n}}$ on the right-hand side of the above
expression, we obtain

$\sum_{{\boldsymbol{n}}\in\mathbb{N}_{0}^{d}}\Big{(}\mathbb{E}\hskip
1.0pt\prod_{j=1}^{d}\frac{X_{j}^{n_{j}}}{n_{j}!}\Big{)}t_{1}^{n_{1}}\cdots
t_{d}^{n_{d}}=\sum_{{\boldsymbol{n}}\in\mathbb{N}_{0}^{d}}\bigg{[}\sum_{({\boldsymbol{k}},{\boldsymbol{l}})+{\boldsymbol{m}}={\boldsymbol{n}}}b_{\boldsymbol{m}}\Big{(}\mathbb{E}\hskip
1.0pt\prod_{j=1}^{p}\frac{X_{j}^{k_{j}}}{k_{j}!}\Big{)}\cdot\Big{(}\mathbb{E}\hskip
1.0pt\prod_{j=p+1}^{d}\frac{X_{j}^{l_{j}}}{l_{j}!}\Big{)}\bigg{]}\prod_{j=1}^{d}t_{j}^{n_{j}}.$

Next, we decompose the inner summation into terms corresponding to the cases
in which ${\boldsymbol{m}}=(0,\ldots,0)$ and
${\boldsymbol{m}}\neq(0,\ldots,0)$. Noting that $b_{(0,\ldots,0)}=1$, we
obtain

$\sum_{{\boldsymbol{n}}\in\mathbb{N}_{0}^{d}}\Big{(}\mathbb{E}\hskip
1.0pt\prod_{j=1}^{d}\frac{X_{j}^{n_{j}}}{n_{j}!}\Big{)}t_{1}^{n_{1}}\cdots
t_{d}^{n_{d}}=\sum_{{\boldsymbol{n}}\in\mathbb{N}_{0}^{d}}\bigg{[}\Big{(}\mathbb{E}\hskip
1.0pt\prod_{j=1}^{p}\frac{X_{j}^{n_{j}}}{n_{j}!}\Big{)}\cdot\Big{(}\mathbb{E}\hskip
1.0pt\prod_{j=p+1}^{d}\frac{X_{j}^{n_{j}}}{n_{j}!}\Big{)}+\delta_{\boldsymbol{n}}\bigg{]}\prod_{j=1}^{d}t_{j}^{n_{j}},$

with $\delta_{\boldsymbol{n}}\geq 0$. Comparing the coefficients of the
monomials $t_{1}^{n_{1}}\cdots t_{d}^{n_{d}}$, we obtain

$\mathbb{E}\hskip
1.0pt\prod_{j=1}^{d}\Big{(}\frac{X_{j}^{n_{j}}}{n_{j}!}\Big{)}=\Big{(}\mathbb{E}\hskip
1.0pt\prod_{j=1}^{p}\frac{X_{j}^{n_{j}}}{n_{j}!}\Big{)}\cdot\Big{(}\mathbb{E}\hskip
1.0pt\prod_{j=p+1}^{d}\frac{X_{j}^{n_{j}}}{n_{j}!}\Big{)}+\delta_{\boldsymbol{n}}\geq\Big{(}\mathbb{E}\hskip
1.0pt\prod_{j=1}^{p}\frac{X_{j}^{n_{j}}}{n_{j}!}\Big{)}\cdot\Big{(}\mathbb{E}\hskip
1.0pt\prod_{j=p+1}^{d}\frac{X_{j}^{n_{j}}}{n_{j}!}\Big{)},$

which yields (2.2), the strong form of the GPI. ∎

The following result was obtained by Russell and Sun [18] by different
methods. In the context of Theorem 2.1, the corollary follows from the well-
known result that if $(X_{1},\ldots,X_{d})\sim N_{d}(0,\Sigma)$ then
$\tfrac{1}{2}(X_{1}^{2},\ldots,X_{d}^{2})\sim{\hbox{Gamma}\hskip
1.0pt}_{d}(\frac{1}{2},\Sigma)$.

######  Corollary 2.2.

(Russell and Sun [18]) Let $(X_{1},\ldots,X_{d})\sim N_{d}(0,\Sigma)$, and
suppose that there exists a sign matrix $S$ such that all off-diagonal
elements of the matrix $S\Sigma S$ are nonnegative. Then the strong form of
the GPI (1.3) holds for all even integers $n_{1},\ldots,n_{d}$.

######  Remark 2.3.

An alternative approach to establishing Corollary 2.2 is by means of the
classical Isserlis-Wick formula [7]. To see this, we write

$\displaystyle\mathbb{E}\hskip 1.0pt\prod_{j=1}^{p}X_{j}^{2m_{j}}$
$\displaystyle=\mathbb{E}\hskip 1.0pt\Big{(}\underbrace{X_{1}\cdots
X_{1}}_{2m_{1}\,\rm{terms}}\cdot\underbrace{X_{2}\cdots
X_{2}}_{2m_{2}\,\rm{terms}}\cdots\underbrace{X_{p}\cdots
X_{p}}_{2m_{p}\,\rm{terms}}\Big{)},$ (2.10) $\displaystyle\mathbb{E}\hskip
1.0pt\prod_{j=p+1}^{d}X_{j}^{2m_{j}}$ $\displaystyle=\mathbb{E}\hskip
1.0pt\Big{(}\underbrace{X_{p+1}\cdots
X_{p+1}}_{2m_{p+1}\,\rm{terms}}\cdot\underbrace{X_{p+2}\cdots
X_{p+2}}_{2m_{p+2}\,\rm{terms}}\cdots\underbrace{X_{d}\cdots
X_{d}}_{2m_{d}\,\rm{terms}}\Big{)},$ (2.11) and $\displaystyle\mathbb{E}\hskip
1.0pt\prod_{j=1}^{d}X_{j}^{2m_{j}}$ $\displaystyle=\mathbb{E}\hskip
1.0pt\Big{(}\underbrace{X_{1}\cdots
X_{1}}_{2m_{1}\,\rm{terms}}\cdot\underbrace{X_{2}\cdots
X_{2}}_{2m_{2}\,\rm{terms}}\cdots\underbrace{X_{d}\cdots
X_{d}}_{2m_{d}\,\rm{terms}}\Big{)}.$ (2.12)

By the Isserlis-Wick formula, the expectations (2.10), (2.11), and (2.12) can
be written as a sum of products of the elements of $\Sigma_{11}$,
$\Sigma_{22}$, and $\Sigma$, respectively. Moreover, a simple inspection of
the terms arising in the evaluation of (2.10) and (2.11) show that the product
of those two expectations yields a collection of terms that are a subset of
the terms arising from evaluation of (2.12). Since we assume that
$\sigma_{ij}\geq 0$ for all $i$ and $j$ then we obtain the strong form of the
GPI.

######  Remark 2.4.

We note that Theorem 2.1 can be extended further to distributions more general
than the multivariate gamma distributions. Consider mutually independent
random vectors $Y_{1},\ldots,Y_{p}\in\mathbb{R}^{d}$ such that, for all
$i=1,\ldots,p$, $Y_{i}\sim{\hbox{Gamma}\hskip
1.0pt}_{d}(\alpha_{i},\Sigma_{i})$ where all entries of the matrix
$\Sigma_{i}$ are nonnegative. Denote by $Y_{ij}$ the $j$-th component of
$Y_{i}$, and define a random vector $(Z_{1},\ldots,Z_{d})$ by
$Z_{j}=\sum_{i=1}^{p}Y_{ij}$, $j=1,\ldots,d$. Then it is straightforward to
show that the moment-generating function of $(Z_{1},\ldots,Z_{d})$ is

$\prod_{i=1}^{p}\det(I_{d}-\Sigma_{i}T)^{-\alpha_{i}}=\exp\Big{(}\sum_{i=1}^{p}\alpha_{i}\sum_{n=1}^{\infty}\frac{1}{n}{\rm{tr}}\hskip
1.0pt[(\Sigma_{i}T)^{n}]\Big{)}.$

Exploiting the additivity of the traces and using similar arguments as in the
proof of Theorem 2.1 yields an analogous theorem for the vector
$(Z_{1},\ldots,Z_{d})$.

##  3 Positive upper orthant dependence and the GPI

In this section, we investigate the validity of the inequalities (1.2) and
(1.3) without making specific assumptions on the distribution of the marginals
of ${\boldsymbol{X}}$. As already pointed out in the introduction, (1.3) is
valid if $|{\boldsymbol{X}}|$, the vector of absolute values, is associated.
It is also clear that (1.3) holds when $|{\boldsymbol{X}}|$ is weakly
associated [21].

In this section, we show that (1.3) follows from the notion of strong positive
upper orthant dependence (SPUOD), which has been shown to be strictly weaker
than weak association [21]. Moreover the weak form (1.2) follows from the
notion of positive upper orthant dependence (PUOD).

Let us recall [2] that a random vector
$V=(V_{1},\ldots,V_{d})\in\mathbb{R}^{d}$ is said to be positively upper
orthant dependent (PUOD) if

$\mathbb{P}(V_{1}\geq t_{1},\ldots,V_{d}\geq
t_{d})\geq\prod_{j=1}^{d}\mathbb{P}(V_{j}\geq t_{j})$

for all $t_{1},\ldots,t_{d}\in\mathbb{R}$. We will also say that the vector
$V$ is strongly positively upper orthant dependent (SPUOD) if

$\mathbb{P}(V_{1}\geq t_{1},\ldots,V_{d}\geq t_{d})\geq\prod_{j\in
I}\mathbb{P}(V_{j}\geq t_{j})\cdot\prod_{j\in I^{c}}\mathbb{P}(V_{j}\geq
t_{j})$

for all $I\subset 1,\ldots,d$.

We begin with a result which, in the one-dimensional case, is classical in the
literature on the statistical analysis of survival data.

######  Lemma 3.1.

Let $Y=(Y_{1},\ldots,Y_{d})$ be a random vector with nonnegative components
$Y_{1},\ldots,Y_{d}$ and such that $\mathbb{E}\hskip 1.0pt(Y_{1}\cdots
Y_{d})<\infty$. Then

$\mathbb{E}\hskip 1.0pt(Y_{1}\cdots
Y_{d})=\int_{0}^{\infty}\cdots\int_{0}^{\infty}\mathbb{P}(Y_{1}\geq
t_{1},\ldots,Y_{d}\geq t_{d})\,\textrm{d}t_{1}\cdots\,\textrm{d}t_{d}.$ (3.1)

Proof. For completeness, we provide a direct proof; cf., [12]. For $y\geq 0$,
let $\chi_{y}$ be the indicator function of the interval $[0,y]$; i.e.,
$\chi_{y}(t)=1$ if $0\leq t\leq y$, and $\chi_{y}(t)=0$ if $t>y$. Then

$\int_{0}^{\infty}\chi_{y}(t)\,\textrm{d}t=y,$

and it follows by an application of Fubini’s theorem that

$\displaystyle\mathbb{E}\hskip 1.0pt(Y_{1}\cdots Y_{d})$
$\displaystyle=\mathbb{E}\hskip
1.0pt\prod_{j=1}^{d}\int_{0}^{\infty}\chi_{Y_{j}}(t_{j})\,\textrm{d}t_{j}$
$\displaystyle=\int_{0}^{\infty}\cdots\int_{0}^{\infty}\mathbb{E}\hskip
1.0pt\prod_{j=1}^{d}\chi_{Y_{j}}(t_{j})\,\textrm{d}t_{j}.$ (3.2)

It is trivial that

$\prod_{j=1}^{d}\chi_{Y_{j}}(t_{j})=\begin{cases}1,&\hbox{ if }Y_{1}\geq
t_{1},\ldots,Y_{d}\geq t_{d}\\\ 0,&\hbox{ otherwise}\end{cases};$

therefore

$\mathbb{E}\hskip 1.0pt\prod_{j=1}^{d}\chi_{Y_{j}}(t_{j})=\mathbb{P}(Y_{1}\geq
t_{1},\ldots,Y_{d}\geq t_{d}).$

Substituting the latter result into (3), we obtain (3.1). ∎

######  Theorem 3.2.

Let $(X_{1},\ldots,X_{d})$ be a random vector such that $\mathbb{E}\hskip
1.0pt(|X_{1}|^{n_{1}}\cdots|X_{d}|^{n_{d}})<\infty$ for fixed exponents
$n_{1},\ldots,n_{d}\geq 0$.

If $|{\boldsymbol{X}}|$, the vector of absolute values of
$(X_{1},\ldots,X_{d})$, is PUOD then there holds the weak form of the GPI,

$\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j=1}^{d}|X_{j}|^{n_{j}}\Big{)}\geq\prod_{j=1}^{d}\mathbb{E}\hskip
1.0pt\big{(}|X_{j}|^{n_{j}}\big{)}.$ (3.3)

If $|{\boldsymbol{X}}|$ is SPUOD then the strong form of the GPI holds, i.e.,
for any $I\subset\\{1,\ldots,d\\}$,

$\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j=1}^{d}|X_{j}|^{n_{j}}\Big{)}\geq\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j\in I}|X_{j}|^{n_{j}}\Big{)}\cdot\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j\in I^{c}}|X_{j}|^{n_{j}}\Big{)}.$ (3.4)

Proof. Suppose that $|{\boldsymbol{X}}|$ is PUOD. Replacing each $t_{j}$ by
$t_{j}^{1/n_{j}}$ and simplifying the various inequalities on the $X_{j}$,
$j=1,\ldots,d$, we obtain

$\mathbb{P}(|X_{1}|^{n_{1}}\geq t_{1},\ldots,|X_{d}|^{n_{d}}\geq
t_{d})\geq\prod_{j=1}^{d}\mathbb{P}(|X_{j}|^{n_{j}}\geq t_{j})$

for all $t_{1},\ldots,t_{d}\geq 0$. Integrating both sides of this inequality
with respect to $t_{1},\ldots,t_{d}$ over $\mathbb{R}^{d}_{\geq 0}$ then, by
applying Lemma 3.1, we obtain (3.3).

The strong form of the GPI (3.4) can be derived analogously starting from the
assumption that $|{\boldsymbol{X}}|$ is SPUOD. ∎

##  4 The strong form of the GPI for negative exponents

The strong form of the GPI (1.3) for the case in which all exponents are
negative was proved by Wei [20]. We now derive this result succinctly by an
application of the Gaussian correlation inequality [14] and the method of
integrating the multivariate survival function, as applied earlier in Section
3.

######  Proposition 4.1.

Suppose that $(X_{1},\ldots,X_{d})\sim N_{d}(0,\Sigma)$ and that
$n_{1},\ldots,n_{d}\in(0,1)$. Then

$\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j=1}^{d}|X_{j}|^{-n_{j}}\Big{)}\geq\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j\in I}|X_{j}|^{-n_{j}}\Big{)}\cdot\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j\in I^{c}}|X_{j}|^{-n_{j}}\Big{)},$ (4.1)

for all $I\subset\\{1,\ldots,n\\}$

Proof. Without loss of generality, we can assume that $I=\\{1,\ldots,p\\}$. We
note that the conditions on $n_{1},\ldots,n_{d}$ are necessary to ensure that
the moments in (4.1) are finite.

For $t_{1},\ldots,t_{d}>0$, we apply the Gaussian correlation inequality [14]
to obtain

$\displaystyle\mathbb{P}(|$ $\displaystyle X_{1}|^{-n_{1}}\geq
t_{1},\ldots,|X_{d}|^{-n_{d}}\geq t_{d})$
$\displaystyle=\mathbb{P}(|X_{1}|\leq t_{1}^{-1/n_{1}},\ldots,|X_{d}|\leq
t_{d}^{-1/n_{d}})$ $\displaystyle\geq\mathbb{P}(|X_{1}|\leq
t_{1}^{-1/n_{1}},\ldots,|X_{p}|\leq
t_{p}^{-1/n_{p}})\,\mathbb{P}(|X_{p+1}|\leq
t_{p+1}^{-1/n_{p+1}},\ldots,|X_{d}|\leq t_{d}^{-1/n_{d}})$
$\displaystyle=\mathbb{P}(|X_{1}|^{-n_{1}}\geq
t_{1},\ldots,|X_{p}|^{-n_{p}}\geq t_{p})\,\mathbb{P}(|X_{p+1}|^{-n_{p+1}}\geq
t_{p+1},\ldots,|X_{d}|^{-n_{d}}\geq t_{d}).$

Integrating the first and last terms of this inequality with respect to
$t_{1},\ldots,t_{d}\in(0,\infty)$, we obtain (4.1). ∎

The argument used to prove Proposition 4.1 also establishes the novel finding
that if any random vector $(Y_{1},\ldots,Y_{d})$ with (almost surely) positive
components satisfies the Gaussian-type correlation inequality,

$\mathbb{P}(Y_{1}\leq t_{1},\ldots,Y_{d}\leq t_{d})\geq\mathbb{P}(Y_{1}\leq
t_{1},\ldots,Y_{p}\leq t_{p})\mathbb{P}(Y_{p+1}\leq t_{p+1},\ldots,Y_{d}\leq
t_{d})$

for all $t_{1},\ldots,t_{d}>0$, then

$\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j=1}^{d}Y_{j}^{-n_{j}}\Big{)}\geq\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j=1}^{p}Y_{j}^{-n_{j}}\Big{)}\cdot\mathbb{E}\hskip
1.0pt\Big{(}\prod_{j=p+1}^{d}Y_{j}^{-n_{j}}\Big{)},$

for all $n_{1},\ldots,n_{d}<0$ such that the expectations exist, and for all
$1\leq p\leq d-1$. In particular, the strong form of the GPI with negative
exponents holds for the multivariate gamma distributions treated in [14].

Acknowledgments. We are grateful to Frédéric Ouimet for drawing our attention
to the article [5] which motivated us to take another look at the GPI.

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