A theory tour of the GH family

Choosing a positive mixing law produces several familiar relatives of the generalized hyperbolic distribution. This note follows their mathematical nesting; the upstream tutorial retains executable constructions and plots.

One mixing mechanism¶

Let $Y>0$ and $Z\sim\mathcal N_d(0,I_d)$ be independent, and put

X=\mu+\gamma Y+\sqrt Y\,LZ,\qquad LL^\top=\Sigma\succ0.

(1)

Conditional on $Y=y$ , the mean is $\mu+\gamma y$ and the covariance is $y\Sigma$ . The conditioning introduction uses $W=Y$ , $\beta=\gamma$ , and $\sigma^2=\Sigma$ in one dimension. Drawing $Y$ then the conditional normal gives the joint law; keeping only $X$ gives its marginal.

The source calls $Y$ a subordinator. Here it is just a positive random variable. A stochastic-process subordinator is an increasing Lévy process; using a one-time mixing law does not by itself specify a process, time scale, or independent increments.

Four mixing laws¶

The general GIG kernel is

g(y)\propto y^{p-1}\exp\!\left[-\tfrac12(ay+b/y)\right],\qquad y>0.

(2)

Its interior has $p\in\mathbb R$ and $a,b>0$ . Gamma and inverse gamma arise as normalized boundary limits, while inverse Gaussian is an exact subfamily:

Mixing law	GIG relation	Parameters used here
Gamma	$b\downarrow0$ , $p>0$	Shape $\alpha=p$ , rate $\rho=a/2$
Inverse gamma	$a\downarrow0$ , $p<0$	Shape $\alpha=-p$ , scale $\rho=b/2$
Inverse Gaussian	$p=-1/2$	Mean $m=\sqrt{b/a}$ , shape $\lambda=b$
Generalized inverse Gaussian	Interior $a,b>0$	$(p,a,b)$

For clarity, the two boundary densities are

\begin{aligned} g_{\mathrm{Gamma}}(y)&=\frac{\rho^\alpha}{\Gamma(\alpha)} y^{\alpha-1}e^{-\rho y},\\ g_{\mathrm{InvGamma}}(y)&=\frac{\rho^\alpha}{\Gamma(\alpha)} y^{-\alpha-1}e^{-\rho/y},\qquad \alpha,\rho>0. \end{aligned}

(3)

The inverse-Gaussian density is

g_{\mathrm{IG}}(y)=\sqrt{\frac{\lambda}{2\pi y^3}} \exp\!\left[-\frac{\lambda(y-m)^2}{2m^2y}\right].

(4)

Expanding its exponent shows the GIG embedding $(-1/2,\lambda/m^2,\lambda)$ . Unlike the gamma limits, this equality holds at positive finite parameters without a limiting operation.

The induced marginal families¶

Positive mixing law	Distribution of $X$
Gamma	Variance gamma (VG)
Inverse gamma	Normal-inverse gamma
Inverse Gaussian	Normal-inverse Gaussian (NIG)
Generalized inverse Gaussian	Generalized hyperbolic (GH)

The historical references retained from the source are Barndorff-Nielsen (1977), Exponentially decreasing distributions for the logarithm of particle size; Madan and Seneta (1990), The Variance Gamma (V.G.) model for share market returns; and Barndorff-Nielsen (1997), Normal inverse Gaussian distributions and stochastic volatility modelling. These identify the distribution families; no finance-model derivation is imported here.

The phrase GH family often includes its gamma and inverse-gamma boundary limits. An arbitrary positive mixing law gives a normal variance–mean mixture, but need not be GH. The GH density note is specifically the result of integrating a GIG-normal joint law.

A familiar boundary example is Student $t$ . If $Y\sim\operatorname{InvGamma}(\nu/2,\nu/2)$ and $\gamma=0$ , integration gives

f_X(x)=\frac{\Gamma((\nu+d)/2)} {\Gamma(\nu/2)(\nu\pi)^{d/2}|\Sigma|^{1/2}} \left(1+\frac{(x-\mu)^\top\Sigma^{-1}(x-\mu)}{\nu}\right)^{-(\nu+d)/2}.

(5)

Here $\nu>0$ , and $\Sigma$ is the scale matrix; the covariance is $\nu\Sigma/(\nu-2)$ only when $\nu>2$ .

Symmetry, tails, and comparable scales¶

When $\gamma=0$ , the marginal is symmetric about $\mu$ . With finite second mixing moment, every coordinate has excess kurtosis

\gamma_2=3\frac{\operatorname{Var}(Y)}{\mathbb E[Y]^2}.

(6)

This follows from the conditional fourth normal moment and is strictly positive for nonconstant $Y$ . It becomes zero for constant mixing, which gives a Gaussian. When $\gamma\ne0$ , the conditional mean also varies with $Y$ ; the GH third and fourth moments quantify the resulting asymmetry whenever those moments exist.

Interior GIG mixing and its inverse-Gaussian subfamily have finite power moments of every order. Inverse-gamma mixing has positive moments only below its shape parameter. Consequently a symmetric normal-inverse-gamma marginal can have a finite fourth moment even when the mixing variable has no fourth moment: symmetry requires $\mathbb E[Y^2]$ , whereas a nonzero mean coupling can require $\mathbb E[Y^4]$ . A plot alone cannot establish these boundaries.

For scalar comparisons with $\mathbb E[Y^2]<\infty$ , centering and scaling use

\frac{X-\mu-\gamma\mathbb E[Y]} {\sqrt{\Sigma\mathbb E[Y]+\gamma^2\operatorname{Var}(Y)}}.

(7)

Comparing distributions at equal conditional $\Sigma$ does not generally put them at equal marginal variance. The original tutorial’s plotting cells and notebook outputs are omitted here; no numerical plot is offered as a proof of tail behavior.

Continue with normal-mixture moments, joint and marginal structure, or EM for GH. The upstream tutorial and package API own construction and sampling examples.

Source and adaptation¶

Rewritten as a mathematical note from xshi19/normix, docs/tutorials/core/02_gh_family_tour.md, at revision 763bb3608920661a012cf089888d349fbf680aad (2026-09-13 import). Copyright (c) 2020 xshi19. Licensed under MIT. The pinned source records the original version. Notation, mathematical qualifications, and links were adapted for this site; API recipes, executable package cells, and notebook plots are omitted. No upstream benchmark execution or formal-proof verification is claimed for this adaptation.