Month: December 2025

This post is about the Beta/Gamma/Normal or Jacobi/Laguerre/Hermite trilogy.

Beta. The Beta distribution $\mathrm{Beta}_{[0,1]}(a,b)$ on $[0,1]$ has density proportional to \[ x\mapsto x^{a-1}(1-x)^{b-1}\mathbf{1}_{x\in [0,1]}. \] Its image by $x\mapsto 2x-1$ is $\mathrm{Beta}_{[-1,1]}(a,b)$ on $[-1,1]$, with density proportional to \[ x\mapsto (1+x)^{a-1}(1-x)^{b-1}\mathbf{1}_{x\in[-1,1]}. \] In particular $\mathrm{Beta}_{[-1,1]}(a,a)$ on $[-1,1]$ has density proportional to \[ x\mapsto (1-x^2)^{a-1}\mathbf{1}_{x\in[-1,1]}, \] also known as a Barenblatt profile. More generally, the $\mathrm{Beta}_{[\alpha,\beta]}(a,b)$ distribution is \[ x\mapsto(x-\alpha)^{a-1}(\beta-x)^{b-1}\mathbf{1}_{x\in[\alpha,\beta]} \] up to the normalizing constant. In particular

• $\mathrm{Beta}_{[\alpha,\beta]}(1,1)$ is the uniform distribution
• $\mathrm{Beta}_{[\alpha,\beta]}(\frac{3}{2},\frac{3}{2})$ is the semi-circle distribution
• $\mathrm{Beta}_{[\alpha,\beta]}(\frac{1}{2},\frac{1}{2})$ is the arcsine distribution.

Beta distributions are strongly related to many other distributions.

$\ell^1$ geometry. If $X_1,\ldots,X_n$ are independent with $X_i\sim\mathrm{Gamma}(a_i,\lambda)$ then \[ V=\frac{X}{|X|_1}=\frac{(X_1,\ldots,X_n)}{X_1+\cdots+X_n} \sim\mathrm{Dirichlet}(a_1,\ldots,a_n) \] which is a probability distribution on the simplex $\{(v_1,\ldots,v_n)\in\mathbb{R}_+:|v|_1=1\}$, uniform when $a_1=\cdots=a_n=1$ and in this case $X_i\sim\mathrm{Exp}(\lambda)$. For all $1\leq i\leq n$, \[ V_i=\frac{X_i}{|X|_1}=\frac{X_i}{X_1+\cdots+X_n} \sim\mathrm{Beta}_{[0,1]}\Bigr(a_i,\sum_{j\neq i}a_j\Bigr). \]

$\ell^2$ geometry. If $X_1,\ldots,X_n$ are iid with $X_i\sim\mathcal{N}(0,1)$ then \[ U=\frac{X}{|X|_2}=\frac{(X_1,\ldots,X_n)}{\sqrt{X_1^2+\cdots+X_n^2}} \] is uniformly distributed on the sphere $\mathbb{S}^{n-1}=\{u\in\mathbb{R}^n:|u|_2^2=1\}$. For all $1\leq k\leq n$, \[ |\mathrm{proj}_{\mathbb{R}^k}(U)|_2^2 =U_1^2+\cdots+U_k^2\sim\mathrm{Beta}_{[0,1]}\Bigr(\frac{k}{2},\frac{n-k}{2}\Bigr). \] Indeed, we have \[ U_1^2+\cdots+U_k^2=\frac{X_1^2+\cdots+X_k^2}{(X_1^2+\cdots+X_k^2)+(X_{k+1}^2+\cdots+X_n^2)} =\frac{A}{A+B} \] and $\begin{cases} A\sim\chi^2(k)=\mathrm{Gamma}(\frac{k}{2},\frac{1}{2})\\ B\sim\chi^2(n-k)=\mathrm{Gamma}(\frac{n-k}{2},\frac{1}{2}) \end{cases}$ are independent, hence the result.

Beta, Gamma, Normal. The Euler formula $\lim_{n\to\infty}(1+\frac{x}{n})^n=\mathrm{e}^x$ gives \begin{align*} (1-\tfrac{x^2}{2n})^{n-1}&\to\mathrm{e}^{-\frac{x^2}{2}}\\ x^{a-1}(1-\tfrac{x}{n})^{n-1}&\to x^{a-1}\mathrm{e}^{-x} \end{align*} hence \begin{align*} \lim_{n\to\infty} \mathrm{Beta}_{[-\sqrt{2n},\sqrt{2n}]}(n,n) &=\mathcal{N}(0,1)\\ \lim_{n\to\infty} \mathrm{Beta}_{[0,n]}(a,n) &=\mathrm{Gamma}(a,1). \end{align*} In terms of random variables we have

• if $X_n\sim\mathrm{Beta}_{[-1,1]}(n,n)$ then $\sqrt{2n}X_n\to\mathcal{N}(0,1)$ in law
• if $X_n\sim\mathrm{Beta}_{[0,1]}(a,n)$ then $nX_n\to\mathrm{Gamma}(a,1)$ in law.

This makes the Beta distribution a proxy for the Gamma and the Normal distributions. In other words, Gamma and Normal distributions are translation-dilation deformations of Beta distribution !

This was used for instance by Atle Selberg (1917 - 2007) to get his explicit formula for the normalizing constant of Coulomb gases coming from random matrix theory. This was also used by Dominique Bakry (1954 - ) for the study of univariate Markov diffusion operators

• Jacobi operator : $(1-x^2)f''+(b-a-(a+b)x)f'$ on $[-1,1]$
  in other words $\frac{1}{w}((1-x^2)wf')'$ where $w(x)=(1-x)^{a-1}(1+x)^{b-1}$
• Laguerre operator : $xf''+(a-x)f'$ on $\mathbb{R}_+$
  in other words $\frac{1}{w}(xwf')'$ where $w(x)=x^{a-1}\mathrm{e}^{-x}$
• Hermite (or Ornstein-Uhlenbeck) operator : $f''-xf'$ on $\mathbb{R}$
  in other words $\frac{1}{w}(wf')'$ where $w(x)=\mathrm{e}^{-\frac{x^2}{2}}$

for which these orthogonal polynomials are the eigenfunctions of the operator. The triple of distributions Beta/Gamma/Normal corresponds to the triple of operators Jacobi/Laguerre/Hermite, via reversible invariant measures and spectral decomposition.

Jacobi, Laguerre, Hermite. The Jacobi, Laguerre, and Hermite orthogonal polynomials are obtained by using the Gram-Schmidt algorithm with the canonical algebraic basis $1,X,X^2,\ldots$ of $\mathbb{R}[X]$ with respect to the scalar product of $L^2(\mu)$ where $\mu$ is the Beta, Gamma, and Normal distribution respectively. The passage from Beta to Gamma and Normal allows to pass from Jacobi to Laguerre and Hermite polynomials. This makes the Jacobi polynomials a proxy for the Laguerre and Hermite polynomials.

Askey scheme. It seems that the observation goes back at least to Ervin Feldheim (1912 - 1944) in the context of orthogonal polynomials, and to Ernst Eduard Kummer (1810 - 1893) in the context of confluent hypergeometric series, something like $\lim_{b\to\infty}{}_2F_1(a,b;c;\frac{z}{b})={}_1F_1(a;c;z)$. Nowadays, this can be seen as a simple aspect of the general way of organizing orthogonal polynomials using hypergeometric series, known as the Askey scheme, due to Richard (Dick) Allen Askey (1933 - 2019).

Ervin Feldheim (1912 - 1944). A Hungarian mathematician. He studied in Paris from 1931 to 1934 and obtained a doctorate in 1937 on characteristic functions under the guidance of Georges Darmois (1888 - 1960). Ervin Feldheim and Vincent Döblin (1915 - 1940) were students in Paris and knew each other. They helped Paul Lévy (1886 - 1971) proofreading his famous manuscript Théorie de l'addition de variables aléatoires. After returning to Hungary in 1934, Ervin Feldheim worked in insurance while continuing research in mathematics. He wrote notably on the stability of probability laws, as well as on special functions. His later paper is posthumous, and includes an editorial note by Gábor Szegő explaining it is essentially a letter dated March 12, 1944, found among Lipót Fejér’s papers by Pál Turán (1910 - 1976). According to Bernard Bru (1942 - ) , Feldheim was arrested in summer 1942, deported until late 1943, returned to Hungary in poor health, then deported again in 1944 and was probably killed during the evacuation massacres.

Further reading.

• Dominique Bakry
  Remarques sur les semigroupes de Jacobi
  Astérisque 236 23-39 (1996)
• Bernard Bru
  La vie et l’œuvre de W. Doeblin (1915-1940) d’après les archives parisiennes
  Mathématiques et sciences humaines 119 5-51 (1992)
• Ervin Feldheim
  Relations entre les polynômes de Jacobi, Laguerre et Hermite
  Acta Mathematica 75 117-138 (1942)
• Ervin Feldheim
  On the positivity of certain sums of ultraspherical polynomials
  Journal d’Analyse Mathématique 11 275-284 (1963) (postumous publication)
• Peter J. Forrester and S. Ole Warnaar
  The importance of the Selberg integral
  Bulletin of the American Mathematical Society 45 489-534 (2008)
• Tom H. Koornwinder
  Group theoretic interpretations of Askey's scheme of hypergeometric orthogonal polynomials
  Orthogonal polynomials and their applications (Segovia, 1986) 46-72
  Lecture Notes in Mathematics 1329 Springer (1988)