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 $1\leq i\leq n$, \[ V_i=\frac{X_i}{|X|_1}=\frac{X_i}{X_1+\cdots+X_n} \sim\mathrm{Beta}_{[0,1]}\Bigr(a_i,\sum_{j\neq i}a_j\Bigr). \] Indeed, if $A\sim\mathrm{Gamma}(a,\lambda)$ and $B\sim\mathrm{Gamma}(b,\lambda)$ are independent then $A+B\sim\mathrm{Gamma}(a+b,\lambda)$ and $A/(A+B)\sim\mathrm{Beta}_{[0,1]}(a,b)$. In particular \[ X_1+\cdots+X_n\sim\mathrm{Gamma}(a_1+\cdots+a_n,\lambda) \quad\text{and}\quad V_i=\frac{X_i}{X_i+Y_i} \] where $Y_i:=\sum_{j\neq i}X_j\sim\mathrm{Gamma}(\sum_{j\neq i}a_j,\lambda)$ is independent of $X_i\sim\mathrm{Gamma}(a_i,\lambda)$.
$\ell^2$ 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.
Funk-Hecke and Barenblatt. In particular, for all $1\leq i\leq n$, $U_i^2\sim\mathrm{Beta}_{[0,1]}(\frac{1}{2},\frac{n-1}{2})$, hence \[ U_i=\varepsilon|U_i|\sim\mathrm{Beta}_{[-1,1]}\Bigr(\frac{n-1}{2},\frac{n-1}{2}\Bigr). \] This is nothing else but the Funk-Hecke formula in harmonic analysis. More generally, we get, for all $1\leq k\leq n-1$, that $\mathrm{proj}_{\mathbb{R}^k}(U)$ follows a Barenblatt distribution with density proportional to \[ x\mapsto(1-|x|_2^2)^{\frac{n-k-2}{2}}\mathbf{1}_{|x|_2\leq1}. \] This can be seen as a multivariate Beta distribution.
Beta, Gamma, Normal. The Euler formula $\lim_{n\to\infty}(1+\frac{x}{n})^n=\mathrm{e}^x$ gives \begin{align*} x^{a-1}(1-\tfrac{x}{n})^{n-1}&\to x^{a-1}\mathrm{e}^{-x}\\ (1-\tfrac{x^2}{2n})^{n-1}&\to\mathrm{e}^{-\frac{x^2}{2}} \end{align*} hence, for the narrow convergence, \begin{align*} \lim_{n\to\infty} \mathrm{Beta}_{[0,n]}(a,n) &=\mathrm{Gamma}(a,1)\\ \lim_{n\to\infty} \mathrm{Beta}_{[-\sqrt{2n},\sqrt{2n}]}(n,n) &=\mathcal{N}(0,1). \end{align*} In terms of random variables and using Beta distributions on $[0,1]$ we have, in distribution,
- if $X_n\sim\mathrm{Beta}_{[0,1]}(a,n)$ then $nX_n\to\mathrm{Gamma}(a,1)$
- if $X_n\sim\mathrm{Beta}_{[0,1]}(n,n)$ then $\sqrt{2n}(2X_n-1)\to\mathcal{N}(0,1)$
while using Beta distributions on $[-1,1]$ this reads, in distribution,
- if $X_n\sim\mathrm{Beta}_{[-1,1]}(a,n)$ then $n(\frac{X_n+1}{2})\to\mathrm{Gamma}(a,1)$
- if $X_n\sim\mathrm{Beta}_{[-1,1]}(n,n)$ then $\sqrt{2n}X_n\to\mathcal{N}(0,1)$.
This makes the Beta distribution a proxy for the Gamma and the Normal distributions. In other words, Gamma and Normal are translation-dilation deformations of Beta !
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 : $(1-x^2)f''+(a-b-(a+b)x)f'$ 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 orthogonal polynomials. They are obtained by using the Gram-Schmidt algorithm with the canonical algebraic basis $1,X,X^2,\ldots$ of $\mathbb{R}[X]$ and 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. Jacobi polynomials are proxies for 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 is one aspect of the general way of organizing orthogonal polynomials using hypergeometric series, known as the Askey scheme, due to Richard Askey (1933 - 2019). \begin{align*} J^{a,b}_n(x) &=\frac{(b)_n}{n!}\ {}_2F_1(-n;n+a+b-1;b;\tfrac{1-x}{2})\\ &=\frac{\Gamma(b+n)}{n!\Gamma(a+b+n-1)}\sum_{m=0}^n\binom{n}{m}\frac{\Gamma(a+b+n+m-1)}{\Gamma(b+m)}\Bigr(\frac{x-1}{2}\Bigr)^m\\ L_n^a(x)&=(-1)^n\lim_{b\to\infty}J^{a,b}_n(\tfrac{2x}{b}-1)\\ H_n(x)&=n!2^n\lim_{a\to\infty}a^{-\frac{n}{2}}J^{a,a}_n(\tfrac{x}{\sqrt{2a}}) \end{align*}
Random matrices. Let $N$ be an $n\times m$ random matrix with iid entries of law $\mathcal{N}(0,1)$, a matrix analogue of the Normal distribution. The law of \[ A=NN^\top \] is known as the Wishart ensemble, a matrix analogue of $\chi^2$ hence Gamma distribution. If $B$ is an independent copy of $A$, then the law of \[ C=(A+B)^{-1/2}A(A+B)^{-1/2} \] is a matrix analogue of the Beta distribution. The eigenvalues of $A$ and $C$ turn out to be Coulomb gases with weight Gamma and Beta, known as Laguerre and Jacobi ensembles. These are Coulomb gases studied notably by Freeman Dyson (1923 - 2020) and for which the normalizing constant was computed by Atle Selberg (1917 - 2007). We refer to Edelman and Rao for more.
Ervin Feldheim (1912 - 1944). A Hungarian mathematician. He studied in Paris from 1931 to 1934 and obtained a doctorate in 1937 from Université de Paris with a committee formed by Émile Borel (1871 - 1956), Georges Darmois (1888 - 1960), and Maurice Fréchet (1878 - 1973). He knew Wolfgang Döblin or Vincent Doblin (1915 - 1940), and 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. This was common at that time in Europe. He wrote notably on characteristic functions as well as on special functions. According to László Lovász and Vera T. Sós (2015), he was, together with Paul Erdős (1913 - 1996), Tibor Gallai (1912 - 1992), Pál Turán (1910 - 1976), and some few others, a member of the legendary Anonymus group, meeting regularly during their university years at the Statue of Anonymus in City Park in Budapest. Lifelong friendships were formed between them, with deep impact on their professional lives. According to Bernard Bru (1992), Ervin Feldheim was arrested in summer 1942, deported until late 1943, returned to Hungary in poor health, then deported again in 1944, and probably killed during the evacuation massacres.
Further reading.
- Richard Askey and James Wilson
Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials
Memoirs of the American Mathematical Society 54 (1985) - 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) - Alan Edelman and N. Raj Rao
Random matrix theory
Acta Numerica 14 233-297 (2005) - 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)
Includes an editorial note by Gábor Szegő (1895 - 1985) explaining that it is essentially a letter dated March 12, 1944, found among Lipót Fejér (1880 - 1959) papers by Pál Turán (1910 - 1976). - 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) - László Lovász and Vera T. Sós
Erdős Centennial
Notices of the American Mathematical Society 62(2) (February 2015)
Further reading on this blog.
- Back to basics : Student and Barenblatt (2024)
- Archimedes theorem on sphere and cylinder (2024)
- The Funk-Hecke formula (2021)
- Back to basics - Hypergeometric functions (2020)
- Beta laws: arcsine, uniform, semicircle (2011)
