A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector of arbitrary length can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. This condition on is shown to be strictly weaker than the assumption that the density of the random vector is multivariate totally positive of order 2, abbreviated , for which the GPI is already known to hold. Under this condition, the paper highlights a new link between the GPI and the monotonicity of a certain ratio of gamma functions.
The Gaussian product inequality (GPI) is a long-standing conjecture, which states that for any centered Gaussian random vector of dimension and every integer , one has
This inequality is known to imply the real polarization problem conjecture in functional analysis , and it is related to the so-called -conjecture to the effect that if and are two nonconstant polynomials on such that the random variables and are independent, then there exist an orthogonal transformation on and an integer such that is a function of and is a function of ; see, e.g., [7,14] and the references therein.
Inequality (1) is well known to be true when ; see, e.g., Frenkel . Karlin and Rinott  also showed that it holds when the random vector has a multivariate totally positive density of order 2, denoted . As stated in Remark 1.4 of their paper, the latter condition is verified, among others, in dimension for all nonsingular Gaussian random pairs.
Interest in the problem has recently gained traction when Lan et al.  established the inequality in dimension . Hope that the result might be true in general is also fueled by the fact, established by Wei , that for any reals , one has
Li and Wei  have further conjectured that the latter inequality holds for all reals and any centered Gaussian random vector .
The purpose of this paper is to report a combinatorial proof of inequality (2) in the special case where the reals are nonnegative integers and when each of the components of the centered Gaussian random vector can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. A precise statement of this assumption is given as Condition (III) in Section 2, and the proof of the main result, Proposition 2, appears in Section 3. It is then shown in Section 4, see Proposition 3, that this condition is strictly weaker than the assumption that the random vector is .
Coincidentally, shortly after the first version of the present paper was posted on arXiv, inequality (2) for all nonnegative integers was established under an even weaker assumption, stated as Condition (IV) in Section 2. The latter condition states that up to a change of sign, the components of the Gaussian random vector are all nonnegatively correlated; see Lemma 2.3 of Russell and Sun . Therefore, the present paper’s main contribution resides in the method of proof using a combinatorial argument closely related to the complete monotonicity of multinomial probabilities previously shown by Ouimet  and Qi et al. .
All background materials required to understand the contribution and put it in perspective are provided in Section 2. The statements and proofs of the paper’s results are then presented in Sections 3 and 4. This article concludes with a brief discussion in Section 5. For completeness, a technical lemma due to Ouimet , which is used in the proof of Proposition 2, is included in the Appendix.
First, recall the definition of multivariate total positivity of order 2 ( ) on a set .
A density supported on is said to be multivariate totally positive of order 2, denoted , if and only if, for all vectors , one has
where and .
Densities in this class have many interesting properties, including the following result, which corresponds to equation (1.7) of Karlin and Rinott .
Let be an random vector on , and let be a collection of nonnegative and (component-wise) nondecreasing functions on . Then
In particular, let be a -variate Gaussian random vector with zero mean and nonsingular covariance matrix . Suppose that the following condition holds.
The random vector belongs to the class on .
When is a centered Gaussian random vector with covariance matrix , Theorem 3.1 of Karlin and Rinott  finds an equivalence between Condition (I) and the requirement that the off-diagonal elements of the inverse of are all nonpositive up to a change of sign for some of the components of . The latter condition can be stated more precisely as follows using the notion of signature matrix, which refers to a diagonal matrix whose diagonal elements are .
There exists a signature matrix such that the covariance matrix only has nonpositive off-diagonal elements.
Two other conditions of interest on the structure of the random vector are as follows.
There exist a signature matrix and a matrix with entries in such that the random vector has the same distribution as the random vector , where is a Gaussian random vector with zero mean vector and identity covariance matrix .
There exists a signature matrix such that the covariance matrix has only nonnegative elements.
Recently, Russell and Sun  used Condition (IV) to show that, for all integers , and , and up to a change of sign for some of the components of , one has
This result was further extended by Edelmann et al.  to the case where the random vector has a multivariate gamma distribution in the sense of Krishnamoorthy and Parthasarathy . See also  for a use of Condition (IV) in the context of the Gaussian correlation inequality (GCI) conjecture.
In the following section, it will be shown how Condition (III) can be exploited to give a combinatorial proof of a weak form of inequality (3). It will then be seen in Section 4 that Condition (II) implies Condition (III), thereby proving the implications illustrated in Figure 1 between Conditions (I)–(IV). That the implications Condition (II) Condition (III) and Condition (III) Condition (IV) are strict can be checked using, respectively, the covariance matrices
for some appropriate .
In the first example, the matrix is completely positive (meaning that it can be written as for some matrix with nonnegative entries) and positive definite by construction. Furthermore, the matrix has at least one positive off-diagonal element for any of the eight possible choices of signature matrix . Another way to see this is to observe that if , then the cyclic product , which is invariant to , is strictly positive in the aforementioned example, so that the off-diagonal elements of cannot all be nonpositive. This shows that (III) (II). This example was adapted from ideas communicated to the authors by Thomas Royen.
For the second example, when , it is mentioned by Maxfield and Minc , using a result from Hall , that the matrix is positive semidefinite and has only nonnegative elements but that it is not completely positive. Using the fact that the set of completely positive matrices is closed, there exists small enough that the matrix is positive definite and has only nonnegative elements but is not completely positive. More generally, given that the elements of are all nonnegative, the matrix is not completely positive for any of the 32 possible choices of signature matrix , which shows that (IV) (III). This idea was adapted from comments by Stein .
3 A combinatorial proof of the GPI conjecture
The following result, which is this paper’s main result, shows that the extended GPI conjecture of Li and Wei  given in (2) holds true under Condition (III) when the reals are nonnegative integers. This result also follows from inequality (3), due to Russell and Sun , but the argument below is completely different from the latter authors’ derivation based on Condition (IV).
Let be a d-variate centered Gaussian random vector. Assume that there exist a signature matrix and a matrix C with entries in such that the random vector has the same distribution as the random vector , where is a -dimensional standard Gaussian random vector. Then, for all integers ,
In terms of , the claimed inequality is equivalent to
By the multinomial formula, the left-hand side of inequality (4) can be expanded as follows:
Calling on the linearity of expectations and the mutual independence of the components of the random vector , one can then rewrite this expression as follows:
Given that the coefficients are all nonnegative by assumption, and exploiting the fact that, for every integer and ,
one can bound the left-hand side of inequality (4) from below by
The right-hand side of (4) can be expanded in a similar way. Upon using the fact that for every integer when , one finds
Next, compare the coefficients of the corresponding powers in expressions (5) and (6). To prove inequality (4), it suffices to show that, for all integer-valued vectors satisfying for every integer , one has
Taking into account the fact that , and after cancelling some factorials, one finds that the aforementioned inequality reduces to
Therefore, the proof is complete if one can establish inequality (7). To this end, one can assume without loss of generality that the integers are all nonzero; otherwise, inequality (7) reduces to a lower dimensional case. For any given integers and every integer , define the function
on the interval , where denotes Euler’s gamma function.
To prove inequality (7), it thus suffices to show that, for every integer , the map is nondecreasing on the interval . Direct computations yield, for every real ,
where denotes the digamma function. Now call on the integral representation [1, p. 260]
valid for every real , to write
Given that by construction, the quantity within braces in equation (8) is always nonnegative by Lemma 1.4 of Ouimet ; this can be checked upon setting and for every integer in that paper’s notation. Alternatively, see p. 516 in the study by Qi et al. . Therefore,
In fact, the map is even completely monotonic. Moreover, given that
one can deduce from inequality (9) that
Hence, the map is nondecreasing on . This concludes the argument.□
4 Condition (II) implies Condition (III)
This paper’s second result, stated below, shows that Condition (II) implies Condition (III). In view of Figure 1, one may then conclude that Condition (I) also implies Conditions (III) and (IV), and hence, also Condition (II) implies Condition (IV). The implication Condition (II) Condition (IV) was already established in Theorem 2 (i) of Karlin and Rinott , and its strictness was mentioned on top of p. 427 of the same paper.
Let be a symmetric positive definite matrix with Cholesky decomposition . If the off-diagonal entries of are all nonpositive, then the elements of C are all nonnegative.
The proof is by induction on the dimension of . The claim trivially holds when . Assume that it is verified for some integer , and fix . Given the assumptions on , one can write
in terms of a real , an vector with nonpositive components, and an matrix with nonpositive off-diagonal entries.
Given that is symmetric positive definite by assumption, so is , and hence so are and . Moreover, the off-diagonal entries of are nonpositive, and hence by induction, the factor in the Cholesky decomposition has nonnegative entries. Letting denote the Schur complement, which is strictly positive, one has
where is an vector of zeros. Accordingly,
Recall that is strictly positive and all the entries of and are nonnegative. Hence, all the elements of are nonnegative, and the argument is complete.□
This paper shows that the Gaussian product inequality (2) holds under Condition (III) when the reals are nonnegative integers. This assumption is further seen to be strictly weaker than Condition (II). It thus follows from the implications in Figure 1 that when the reals are nonnegative integers, inequality (2) holds more generally than under the condition of Karlin and Rinott .
Shortly after the first draft of this article was posted on arXiv, extensions of Proposition 2 were announced by Russell and Sun  and Edelmann et al. ; see Lemma 2.3 and Theorem 2.1, respectively, in their manuscripts. Beyond priority claims, which are nugatory, the originality of the present work lies mainly in its method of proof, and in the clarification it provides of the relationship between various assumptions made in the relevant literature, as summarized by Figure 1.
Beyond its intrinsic interest, the approach to the proof of the GPI presented herein, together with its link to the complete monotonicity of multinomial probabilities previously shown by Ouimet  and Qi et al. , hints to a deep relationship between the class for the multivariate gamma distribution of Krishnamoorthy and Parthasarathy , the range of admissible parameter values for their infinite divisibility, and the complete monotonicity of their Laplace transform; see the work by Royen on the GCI conjecture [18,19, 20,21] and Theorems 1.2 and 1.3 of Scott and Sokal . These topics, and the proof or refutation of the GPI in its full generality, provide interesting avenues for future research.
The following result, used in the proof of Proposition 2, extends Lemma 1 of Alzer  from the case to an arbitrary integer . It was already reported by Ouimet , see his Lemma 4.1, but its short statement and proof are included here to make the article more self-contained.
For every integer , and real numbers and such that , one has
The proof is by induction on the integer . The case is the statement of Lemma 1 of Alzer . Fix an integer and assume that inequality (A.1) holds for every smaller integer. Fix any reals and such that . Write . Calling on Alzer’s result, one has
Therefore, the conclusion follows if one can show that
Upon setting and , one finds that the aforementioned inequality is equivalent to
which is true by the induction assumption. Therefore, the argument is complete.□
The authors are grateful to Donald Richards and Thomas Royen for their comments on earlier versions of this note. The authors also thank the referees for their comments.
Funding information: C. Genest’s research is funded in part by the Canada Research Chairs Program, the Trottier Institute for Science and Public Policy, and the Natural Sciences and Engineering Research Council of Canada. F. Ouimet received postdoctoral fellowships from the Natural Sciences and Engineering Research Council of Canada and the Fond québécois de la recherche – Nature et technologies (B3X supplement and B3XR).
Conflict of interest: The authors state no conflict of interest.
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© 2022 Christian Genest and Frédéric Ouimet, published by De Gruyter
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