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Gérard Letac

Publications and source records attributed to Gérard Letac.

14 recordsLinked to original sources
arXiv2026

Probability laws associated to the independence preserving quadrirational Yang-Baxter maps -- the ultimate case

A map $F\colon\mathcal X\times\mathcal Y\to \mathcal U\times \mathcal V$ is said to be independence preserving (IP) if there exists a pair of independent random variables $(X,Y)$ valued in $\mathcal X\times\mathcal Y$ such that the two coordinates of $(U,V)=F(X,Y)$ are also independent. Recently, Sasada and Uozumi (2024) observed that a hierarchy of quadrirational Yang-Baxter maps gives rise to independence preserving transformations, and identified corresponding families of probability distributions. In view of the limiting properties of these IP maps, the newly defined generalized second kind beta ($\mathrm{GB}_{II}$) model stands at the top of the hierarchy: for independent random variables $X$ and $Y$ following a $\mathrm{GB}_{II}$ distribution, Sasada and Uozumi (2024) showed that when a special quadrirational Yang-Baxter map $F^{(α,β)}$, parameterized by $(α,β)\in(0,\infty)^2$, is applied to the pair $(X,Y)$, it produces another pair $(U,V)$ of independent $\mathrm{GB}_{II}$-distributed random variables. The aim of this paper is to show that the IP property of $F^{(α,β)}$ uniquely identifies distributions of $X,Y,U$ and $V$ as belonging to the $\mathrm{GB}_{II}$ family. To this end, we introduce specially designed Laplace-type transforms. First, we carefully explain the connection between the results from Sasada and Uozumi (2024) and Koudou and Vallois (2012). Next, we focus on the characterization of the second kind beta and the generalized second kind beta distributions through the IP map $F^{(α,\infty)}$. Finally, extending considerably the methodology developed for the case $(α,\infty)$, we prove the characterization of $\mathrm{GB}_{II}$ distributions in the case $(α,β)\in(0,\infty)^2$ with $α\neqβ$, which implies uniqueness in the ultimate missing case of the quadrirational Yang-Baxter hierarchy of IP models.

math.PR↗
arXiv2026

GIG random matrices and a Yang-Baxter extension of the Matsumoto-Yor property

Sasada and Uozumi, \cite{SasUoz2024}, identified independence preserving $[2:2]$ quadrirational parametric Yang-Baxter maps, see \eqref{YBEQ}, on $(0,\infty)$. In particular, the map denoted there by $H_{III,B}^{(α,β)}$, see \eqref{CS}, was connected to the independence preserving property of the GIG distributions on $(0,\infty)$. Remarkably, the property appears also naturally in probabilistic integrable models of discrete Korteweg de Vries type, as observed by Croydon and Sasada, \cite{CroSas2020}. In the case of $(α,β)=(1,0)$ the independence reduces to the classical Matsumoto-Yor property, \cite{MatYor2001}. In \cite{LetWes2024} we proposed an extension of $H_{III,B}^{(α,β)}$ to a map on the cone of symmetric positive definite matrices of a fixed dimension, showing that such extended map preserves independence of GIG random matrices. In the present paper we prove two results: (i) the matrix GIG distributions are characterized by the independence property governed by this map; (ii) the matrix variate extension of $H_{III,B}^{(α,β)}$ we use, is a parametric Yang-Baxter map.

math.PR↗
arXiv2025

Simple Cubic Variance Functions on $\R^n$, Part one

The classification of natural exponential families started with the paper \cite {Morri} where Carl Morris unifies six very familiar families by the fact that their variance functions are polynomials of degree less or equal to two. Extension of this classification to $\R^n$ and to degree three is the subject of this paper. Keywords: Actions of the group $GL(n+1,\R)$, classification of natural exponential families, multivariate Lagrange formula. variance functions.

math.ST↗
arXiv2022

About an extension of the Matsumoto-Yor property

If $α,β>0$ are distinct and if $A$ and $B$ are independent non-degenerate positive random variables such that $$S=\tfrac{1}{B}\,\tfrac{βA+B}{αA+B}\quad \mbox{and}\quad T=\tfrac{1}{A}\,\tfrac{βA+B}{αA+B} $$ are independent, we prove that this happens if and only if the $A$ and $B$ have generalized inverse Gaussian distributions with suitable parameters. Essentially, this has already been proved in Bao and Noack (2021) with supplementary hypothesis on existence of smooth densities. The sources of these questions are an observation about independence properties of the exponential Brownian motion due to Matsumoto and Yor (2001) and a recent work of Croydon and Sasada (2000) on random recursion models rooted in the discrete Korteweg - de Vries equation, where the above result was conjectured. We also extend the direct result to random matrices proving that a matrix variate analogue of the above independence property is satisfied by independent matrix-variate GIG variables. The question of characterization of GIG random matrices through this independence property remains open.

math.PR↗
arXiv2021

Duality for real and multivariate exponential families

Consider a measure $μ$ on $\R^n$ generating a natural exponential family $F(μ)$ with variance function $V_{F(μ)}(m)$ and Laplace transform $$ \exp(\ell_μ(s))=\int_{\R^n} \exp(-\ )μ(dx).$$ A dual measure $μ^*$ satisfies $-\ell'_{μ^*}(-\ell'_μ(s))=s.$ Such a dual measure does not always exist. One important property is $\ell"_{μ^*}(m)=(V_{F(μ)}(m))^{-1},$ leading to the notion of duality among exponential families (or rather among the extended notion of T exponential families $T\hskip-2pt F$ obtained by considering all translations of a given exponential family $F$).

math.PR↗
arXiv2019

Gaussian approximation of Gaussian scale mixture

For a given positive random variable $V>0$ and a given $Z\sim N(0,1)$ independent of $V$, we compute the scalar $t_0$ such that the distance between $Z\sqrt{V}$ and $Z\sqrt{t_0}$ in the $L^2(\R)$ sense, is minimal. We also consider the same problem in several dimensions when $V$ is a random positive definite matrix.

math.ST↗
arXiv2019

Multivariate reciprocal inverse Gaussian distributions from the Sabot -Tarrès -Zeng integral

In Sabot and Tarrès (2015), the authors have explicitly computed the integral $$STZ_n=\int \exp( -\langle x,y\rangle)(\det M_x)^{-1/2}dx$$ where $M_x$ is a symmetric matrix of order $n$ with fixed non positive off-diagonal coefficients and with diagonal $(2x_1,\ldots,2x_n)$. The domain of integration is the part of $\mathbb{R}^n$ for which $M_x$ is positive definite. We calculate more generally for $ b_1\geq 0,\ldots b_n\geq 0$ the integral $$GSTZ_n=\int \exp \left(-\langle x,y\rangle-\frac{1}{2}b^*M_x^{-1}b\right)(\det M_x)^{-1/2}dx,$$ we show that it leads to a natural family of distributions in $\mathbb{R}^n$, called the $GSTZ_n$ probability laws. This family is stable by marginalization and by conditioning, and it has number of properties which are multivariate versions of familiar properties of univariate reciprocal inverse Gaussian distribution. We also show that if the graph with the set of vertices $V=\{1,\ldots,n\}$ and the set $E$ of edges $\{i,j\}'$ s of non zero entries of $M_x$ is a tree, then the integral $$\int \exp( -\langle x,y\rangle)(\det M_x)^{q-1}dx$$ where $q>0,$ is computable in terms of the MacDonald function $K_q.$

math.PR↗
arXiv2018

Location and scale behaviour of the quantiles of a natural exponential family

Let $P_0$ be a probability on the real line generating a natural exponential family $(P_t)_{t\in \mathbb {R}}$. Fix $α$ in $ (0,1).$ We show that the property that $P_t((-\infty,t)) \leq α\leq P_t((-\infty,t])$ for all $t$ implies that there exists a number $μ_α$ such that $P_0$ is the Gaussian distribution $N(μ_α,1).$ In other terms, if for all $t$, $t$ is a quantile of $P_t$ associated to some threshold $α\in (0,1)$, then the exponential family must be Gaussian. The case $α=1/2$, \textit{i.e.} $t$ is always a median of $P_t,$ has been considered in Letac \textit{et al.} (2018). Analogously let $Q$ be a measure on $[0,\infty)$ generating a natural exponential family $(Q_{-t})_{t>0}$. We show that $Q_{-t}([0,t^{-1}))\leq α\leq Q_{-t}([0,t^{-1}])$ for all $t>0$ implies that there exists a number $p=p_α>0$ such that $Q(dx)\propto x^{p-1}dx,$ and thus $Q_{-t}$ has to be a gamma distribution with parameters $p$ and $t.$

math.ST↗
arXiv2018

Is the Sibuya distribution a progeny?

For $0<a<1$ the Sibuya distribution $s_a$ is concentrated on the set $\mathbb{N}^+$ of positive integers and is defined by the generating function $\sum_{n=1}^{\infty}s_a(n)z^n=1-(1-z)^a.$ A distribution $q$ on $\mathbb{N}^+$ is called a progeny if there exists a Galton-Watson process $(Z_n)_{n\geq 0}$ such that $Z_0=1$, such that $\mathbb{E}(Z_1)\leq 1$ and such that $q$ is the distribution of $\sum _{n=0}^{\infty}Z_n. $ The paper proves that $s_a$ is a progeny if and only if $\frac{1}{2}\leq a<1.$ The point is to find the values of $b=1/a$ such that the power series expansion of $u(1-(1-u)^b)^{-1}$ has non negative coefficients. The proof is not short, but elementary.

math.PR↗
arXiv2018

Cumulative distribution functions for the five simplest natural exponential families

Suppose that the distribution of $X_a$ belongs to a natural exponential family concentrated on the nonegative integers and is such that $\E(z^{X_a})=f(az)/f(a)$. Assume that $\Pr(X_a\leq k)$ has the form $c_k\int_a ^{\infty}u^kμ(du)$ for some number $c_k$ and some positive measure $μ,$ both independent of $a.$ We show that this asumption implies that the exponential family is either a binomial, or the Poisson, or a negative binomial family. Next, we study an analogous property for continuous distributions and we find that it is satisfied if and only the families are either Gaussian or Gamma. Ultimately, the proofs rely on the fact that only Moebius functions preserve the cross ratio, \textsc{Keywords:} Binomial, Poisson and negative binomial distributions. Gaussian and Gamma distributions. Moebius transforms. Cross ratio.

math.ST↗
arXiv2017

Existence and non-existence of the non-central Wishart distributions

The problem considered in this paper is to find when the non-central Wishart distribution, defined on the cone $\bar{\mathcal{P}_d}$ of semi positive definite matrices of order $d$ and with a real valued shape parameter, exists. We reduce this problem to the problem of existence of the measures $m(n,k,d)$ defined on $\bar{\mathcal{P}_d}$ and with Laplace transform $(\det s)^{-n/2}\exp \tr(s^{-1}w)$ where $n$ is an integer and where $w=\mathrm{diag}(0,...,0,1,...,1)$ has order $d$ and rank $k.$ We compute $m(d-1,d,d)$ and we show that neither $m(d-2,d,d)$ nor $m(d-2,d-1,d)$ exist. This proves a conjecture of E. Mayerhofer.

math.PR↗
arXiv2010

Random continued fractions with beta hypergeometric distribution

In a recent paper (Asci \textit{et al.}, 2008) it has been shown that certain random continued fractions have a density which is a product of a beta density and a hypergeometric function $_{2}F_{1}$. In the present paper we fully exploit a formula due to Thomae (1879) in order to generalize substantially the class of random continuous fractions with a density of the above form. This involves the design of seven particular graphs. Infinite paths on them lead to random continued fractions with an explicit distribution. A careful study about the set of five real parameters leading to a beta-hypergeometric distribution is required, relying on almost forgotten results mainly due to Felix Klein.

math.PR↗
arXiv2007

Wishart distributions for decomposable graphs

When considering a graphical Gaussian model ${\mathcal{N}}_G$ Markov with respect to a decomposable graph $G$, the parameter space of interest for the precision parameter is the cone $P_G$ of positive definite matrices with fixed zeros corresponding to the missing edges of $G$. The parameter space for the scale parameter of ${\mathcal{N}}_G$ is the cone $Q_G$, dual to $P_G$, of incomplete matrices with submatrices corresponding to the cliques of $G$ being positive definite. In this paper we construct on the cones $Q_G$ and $P_G$ two families of Wishart distributions, namely the Type I and Type II Wisharts. They can be viewed as generalizations of the hyper Wishart and the inverse of the hyper inverse Wishart as defined by Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272--1317]. We show that the Type I and II Wisharts have properties similar to those of the hyper and hyper inverse Wishart. Indeed, the inverse of the Type II Wishart forms a conjugate family of priors for the covariance parameter of the graphical Gaussian model and is strong directed hyper Markov for every direction given to the graph by a perfect order of its cliques, while the Type I Wishart is weak hyper Markov. Moreover, the inverse Type II Wishart as a conjugate family presents the advantage of having a multidimensional shape parameter, thus offering flexibility for the choice of a prior.

math.ST↗