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A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector $\boldsymbol{X} = (X_1, \ldots, X_d)$ 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 $\boldsymbol{X}$ is shown to be strictly weaker than the assumption that the density of the random vector $(|X_1|, \ldots, |X_d|)$ is multivariate totally positive of order $2$, abbreviated MTP${}_2$, 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., 9 pages, 1 figure

Stimulated by the work of Rzqdkowski et al. (2015, J. Nonlinear Math. Phys., 22 (2015), 374–380), the authors derive representations for certain classes of univariate and bivariate lifetime distributions in terms of sequences of Bell, Bernoulli, and Stirling numbers of the second kind, their generalizations, and associated polynomials. Gould–Hopper polynomials are used in the bivariate case, leading to representations for large classes of distributions satisfying a law of uniform seniority for dependent lives formulated by Genest and Kolev (Scan. Act. J., 2021-8 (2021), 726–743).

Bayes spaces were initially designed to provide a geometric framework for the modeling and analysis of distributional data. It has recently come to light that this methodology can be exploited to provide an orthogonal decomposition of bivariate probability distributions into an independent and an interaction part. In this paper, new insights into these results are provided by reformulating them using Hilbert space theory and a multivariate extension is developed using a distributional analog of the Hoeffding-Sobol identity. A connection between the resulting decomposition of a multivariate density and its copula-based representation is also provided.

Deheuvels [J. Multivariate Anal. 11 (1981) 102--113] and Genest and R��millard [Test 13 (2004) 335--369] have shown that powerful rank tests of multivariate independence can be based on combinations of asymptotically independent Cram��r--von Mises statistics derived from a M��bius decomposition of the empirical copula process. A result on the large-sample behavior of this process under contiguous sequences of alternatives is used here to give a representation of the limiting distribution of such test statistics and to compute their relative local asymptotic efficiency. Local power curves and asymptotic relative efficiencies are compared under familiar classes of copula alternatives., Published at http://dx.doi.org/10.1214/009053606000000984 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Vincentization is a convenient method of aggregating a set of $n \geq 2$ probability distributions $F_1, \ldots, F_n$ in such a way that their synthesis, $F = T(F_1, \ldots, F_n)$, be of the same functional form as the $F_i$'s when the latter are identical up to a location-scale transformation. A characterization of this combination rule is proposed and some of its consequences are outlined.