Let be a positive integer, and let
be a random vector which has a multivariate Gaussian distribution with probability density function,
with a nonsingular covariance matrix . We refer to the random vector as having a centered Gaussian distribution because , and we write .
The Gaussian product inequality (GPI) conjecture states that, for any centered Gaussian random vector and any , there holds the inequality,
for arbitrary . Wei  derived hypotheses under which the inequality
Russell and Sun  recently related the GPI to a class of combinatorial inequalities, and thereby established numerous cases of the GPI for . One of the results obtained by Russell and Sun  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  to lead to new inequalities for the bivariate Gaussian distributions.
The weak form of the GPI was established by Frenkel  for and arbitrary , and by Lan, et al.  for and integer exponents , , and with equality between at least two exponents. Genest and Ouimet  developed recently a novel and far-reaching approach to the GPI, proving (1.2) for arbitrary with nonnegative even integer exponents when the covariance matrix is completely positive, i.e., where is a matrix with for all . Nevertheless, it is still unknown whether the weak form of the GPI (1.2) is valid for general .
or by using moment-generating functions, we obtain
If we set , , and with then with , the matrix is positive definite and yet
concrete examples are and .
Wei  showed, however, that the strong form of the GPI holds for the case in which all exponents are negative. It is also obvious that the strong form (1.3) holds if , the vector of absolute values, is associated, i.e., if for all component-wise non-decreasing functions . Thus, for a centered Gaussian random vector , if is associated then it follows immediately that the strong form of the GPI holds. In particular, if the vector is multivariate totally positive of order , denoted (cf., ) then, as the property implies associatedness, it follows that the strong form of the GPI holds.
Moreover, for Gaussian vectors , the property of its absolute values can be characterized explicitly in terms of the covariance matrix . For this purpose (and in the sequel) we call a diagonal matrix a sign matrix if for all . It was proved by Karlin and Rinott  that for the vector of absolute values, , is if and only if there exists a sign matrix such that all off-diagonal entries of 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  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 with for all . 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 ) to obtain in Section 4 an alternative and succinct proof of the strong form of the GPI for negative exponents, derived originally by Wei ; further, we show that this result extends to the multivariate gamma distributions.
2 The strong form of the GPI for nonnegative correlations
) for the multivariate normal distributionand with even integers and completely positive covariance matrix , i.e., , where is a matrix with nonnegative entries . In Theorem 2.1, we extend this result in three directions. First and most importantly, we extend the result in  to the case of nonnegative correlations, where is such that for all . For , the assumption of nonnegative correlations is known to be less restrictive than complete positivity .
On the other hand, a famous counterexample of Šidák  established the existence of Gaussian random vectors with completely positive covariance matrices and for which the vector 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.
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 -dimensional gamma distribution (in the sense of Krishnamoorthy and Parthasarathy ) may be defined by means of its moment-generating function. Denote by
the identity matrix of orderand, for sufficiently small , define . Then we say that
has a multivariate gamma distribution with a not necessarily integer “degree-of-freedom parameter”and positive semidefinite matrix parameter , written , if the moment-generating function of is
This -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
-Wishart distributed random matrix with. In this regard, it is remarkable that the distribution also exists for all values , where denotes the integer part of ; see .
The distribution is infinitely divisible (i.e., the distribution exists for all ) if and only if the distribution is multivariate totally positive of order (); see Bapat . For example, one can show that is infinitely divisible if is of “structure ” , i.e., with for all , or if is of “tree-type”; see  for details.
If then distribution. Consequently, in the case of even exponents, as considered in , the weak form (1.2) and the strong form (1.3) of the GPI intrinsically are inequalities on the distribution. Therefore it is natural to extend these inequalities to the more general multivariate gamma distributions having moment-generating function (2.1).
Let , where is positive semidefinite. Suppose there exists a sign matrix such that all the elements in are nonnegative, i.e., for all . Then for all subsets , and for all nonnegative integers , there holds the strong form of the GPI,
Proof. Since the moment-generating function (2.1) is invariant under the transformation we can, without loss of generality, assume that for all . Moreover, by permuting the coordinates of , we may also assume that where .
With sufficiently small and , the moment-generating function of the distribution is
the eigenvalues of the matrix. For sufficiently small , we have . Then we have
Inserting into this sum the series expansions,
, and interchanging the order of summation, we obtain
Next, we partition and into block matrices,
where and are , is , and and are . Then,
be a symmetric matrix which has been partitioned similarly to . By induction on , we find that
where each matrix is a homogeneous polynomial in with nonnegative coefficients; for instance,
where each is a polynomial in the entries of and . It is evident from (2.6) that the coefficients of each are nonnegative; therefore, since the entries of are nonnegative, we obtain for all .
Next, the Maclaurin expansion of the exponential function leads to
Since and for all then and for all .
Collecting terms in on the right-hand side of the above expression, we obtain
Next, we decompose the inner summation into terms corresponding to the cases in which and . Noting that , we obtain
with . Comparing the coefficients of the monomials , we obtain
which yields (2.2), the strong form of the GPI.
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 , , and , 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 for all and then we obtain the strong form of the GPI.
We note that Theorem 2.1 can be extended further to distributions more general than the multivariate gamma distributions. Consider mutually independent random vectors such that, for all , where all entries of the matrix are nonnegative. Denote by the -th component of , and define a random vector by , . Then it is straightforward to show that the moment-generating function of is
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 .
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 . As already pointed out in the introduction, (1.3) is valid if , the vector of absolute values, is associated. It is also clear that (1.3) holds when is weakly associated .
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 . Moreover the weak form (1.2) follows from the notion of positive upper orthant dependence (PUOD).
Let us recall  that a random vector is said to be positively upper orthant dependent (PUOD) if
for all . We will also say that the vector is strongly positively upper orthant dependent (SPUOD) if
for all .
We begin with a result which, in the one-dimensional case, is classical in the literature on the statistical analysis of survival data.
Let be a random vector with nonnegative components and such that . Then
Proof. For completeness, we provide a direct proof; cf., . For , let be the indicator function of the interval ; i.e., if , and if . Then
and it follows by an application of Fubini’s theorem that
It is trivial that
Let be a random vector such that for fixed exponents .
If , the vector of absolute values of , is PUOD then there holds the weak form of the GPI,
If is SPUOD then the strong form of the GPI holds, i.e., for any ,
Proof. Suppose that is PUOD. Replacing each by and simplifying the various inequalities on the , , we obtain
The strong form of the GPI (3.4) can be derived analogously starting from the assumption that 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 . We now derive this result succinctly by an application of the Gaussian correlation inequality  and the method of integrating the multivariate survival function, as applied earlier in Section 3.
Suppose that and that . Then
Proof. Without loss of generality, we can assume that . We note that the conditions on are necessary to ensure that the moments in (4.1) are finite.
For , we apply the Gaussian correlation inequality  to obtain
Integrating the first and last terms of this inequality with respect to , we obtain (4.1).
The argument used to prove Proposition 4.1 also establishes the novel finding that if any random vector with (almost surely) positive components satisfies the Gaussian-type correlation inequality,
for all , then
for all such that the expectations exist, and for all . In particular, the strong form of the GPI with negative exponents holds for the multivariate gamma distributions treated in .
Acknowledgments. We are grateful to Frédéric Ouimet for drawing our attention to the article  which motivated us to take another look at the GPI.
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