Extended Gaussian Family

Updated 21 August 2025

Extended Gaussian family is defined as a set of probability distributions that generalize the classical Gaussian by adding parameters to control tails, shape, and asymmetry.
Key constructions such as the exp-G, scale mixtures, and q-Gaussian models enable precise modulation of distribution behavior for robust applications.
The framework supports advanced stochastic processes and geometric methods, like the symmetric PSD bicone, to analyze multivariate, degenerate, and complex dependency structures.

The extended Gaussian family comprises a set of probability distributions, stochastic processes, and associated parameter spaces that systematically generalize the classical Gaussian (normal) distribution and Gaussian processes. These extensions include distributions with heavy or light tails, additional shape or asymmetry parameters, degenerate covariance or precision matrices, non-Gaussian stochastic process models, and transformations governing the geometry of parameter spaces. The family also contains limit distributions arising in generalized central limit theorems, generalized integrals and convolution structures, and algebraic-geometric frameworks subsuming and extending classical Gaussianity.

1. Constructions and Parameterizations

The central construction techniques for the extended Gaussian family involve adding parameters or deformations to the Gaussian law to create richer model classes.

The exp-G construction augments any baseline distribution $G(x;\theta)$ by applying a truncated exponential transformation to its cumulative distribution function (cdf). Specifically, for real parameter $\lambda$ , the exp-G cdf is

$F_\lambda^G(x) = \frac{1 - \exp(-\lambda G(x;\theta))}{1 - \exp(-\lambda)}$

For continuous $G$ , the density is

$f(x) = \frac{\lambda}{1 - \exp(-\lambda)}\,g(x;\theta)\exp(-\lambda G(x;\theta))$

For the Gaussian base, this yields an "exp-Gaussian" model with a warping parameter $\lambda$ that modulates tail and body behavior, providing control over departures from normality while including the classical Gaussian as $\lambda\to 0$ (Barreto-Souza et al., 2010).

Scale mixtures of Gaussians such as elliptical distributions represent another fundamental extension. The marginal of an elliptical process is

$p_\theta(u) = |\Sigma|^{-1/2}\int_0^\infty \left(\frac{\xi}{2\pi}\right)^{n/2}\exp\left(-\frac{u\xi}{2}\right)p_\theta(\xi)d\xi$

with $u=(y-\mu)^\top\Sigma^{-1}(y-\mu)$ and flexible mixing density $p_\theta(\xi)$ . This encompasses Student- $t$ , Cauchy, and more general fat-tailed models, and is operationalized in elliptical processes for robust regression and anomaly detection tasks (Bånkestad et al., 2020).

Multi-Gaussian distributions introduce an additional shape parameter $M>0$ , generating pdfs as alternating finite or infinite series of Gaussians with varying widths. For $M=1$ , one recovers the classical Gaussian; as $M$ varies, the profile transitions from flat-topped ( $M>1$ ) to cusped ($0Korotkova, 2020).
q-Gaussian and their asymmetric generalizations, which arise in non-extensive statistical mechanics and generalized central limit theorems, are obtained via nonlinear transformation of pairs of independent Gamma random variables. The "complexity parameter" $q$ and an asymmetry parameter are determined by the Gamma shape indices. The resulting densities are

$P(x)\propto(1+\sqrt{\beta}x)^{\alpha-1}(1-\sqrt{\beta}x)^{\alpha'-1}$

upon normalization, with the classical Gaussian recovered as $q\to 1$ . Such constructions are crucial for modeling heavy-tailed empirical phenomena (Budini, 2015).

Hierarchical or group-theoretic perspectives, e.g., representation-theoretic methods, allow construction of canonical exponential families over homogeneous spaces by pairing group representations $(V, v_0)$ and continuous characters of the group. In the $(\mathbb{R}_{>0},\{1\})$ case with a two-dimensional representation, the result is precisely the generalized inverse Gaussian (GIG) family—a central component of the reciprocal inverse Gaussian and related families (Tojo et al., 2019).
The parameter space geometry of extended Gaussians is captured by the symmetric positive semi-definite (PSD) bicone, the union of all covariance and precision matrices—degenerate and non-degenerate—forming a "bicone" manifold with Hilbert geometry and explicit metric invariance structure (Karwowski et al., 20 Aug 2025).

2. Limiting Distributions and Deformation Principles

Several extended Gaussian families emerge as limiting distributions in generalized central limit settings or as attractors under model deformations.

For correlated or scale-invariant N-body probabilistic models, $q$ -Gaussians appear as the limiting distributions under deformations that retain asymptotic scale invariance (so-called $\alpha$ - and $\beta$ -deformations). In contrast, $\gamma$ -deformations based on Q-numbers, which lose scale invariance, do not yield $q$ -Gaussians but either standard Gaussians or degenerate distributions. This highlights the centrality of scale-invariance for the robustness of the $q$ -Gaussian attractor (Sicuro et al., 2015).
The transformation groupoid structure of the $q$ -Gaussian family provides a systematic framework for mapping one $q$ -Gaussian to another via probability-preserving scaling. All normalizable $q$ -Gaussians with $q\in(-\infty,3)$ are interrelated through these groupoid transformations, cementing the $q$ -Gaussian's role as a structurally robust extension (Tateishi et al., 2013).

3. Extensions in Stochastic Processes and Covariance Functions

Extended Gaussian families include broad classes of stochastic processes and covariance structures not encompassed by the standard Gaussian process paradigm.

Elliptical processes generalize Gaussian processes by allowing the finite-dimensional marginals to be any elliptical distribution (i.e., scale-mixtures of Gaussians with mixing distribution over precision); such processes retain tractable marginalization and conditioning, and exhibit improved robustness and tail-accuracy for modeling heavy-tailed noise in regression/extrapolation (Bånkestad et al., 2020).
Weighted Gaussian processes characterized by a covariance function

$K_f(s,t) = 2\int_0^{s\wedge t}f(u)\{[s+t-2u]\ln(s+t-2u)-(s-u)\ln(s-u)-(t-u)\ln(t-u)\}du$

arise as scaling limits of occupation time fluctuations in branching particle systems. By appropriate choice of the weight $f$ , one obtains processes exhibiting long-range dependence and logarithmic memory, often with non-semimartingale paths. Such processes enable modeling of empirical phenomena with non-standard memory properties, as in animal movement telemetry data (Gonzalez et al., 30 May 2024).

Stationary Gaussian processes with parametrically-rich covariance functions—including those derived as general solutions of damped/underdamped second-order stochastic ODEs—form the "2Dsys" family. Explicit parametrization allows modeling both oscillatory and overdamped regimes, with direct interpretability in terms of physical and stochastic system properties (MacKay et al., 2018).

4. Geometric and Analytical Structures

The extended Gaussian family can be situated within explicit geometric frameworks:

The parameter space for the family, including degenerate covariance and/or precision matrices, is the symmetric PSD bicone. The Hilbert metric on this set, with explicit formula

$d_H(A,B) = \log\left(\frac{\max\{\lambda_{\max}(B^{-1}A),\lambda_{\max}((I-B)^{-1}(I-A))\}}{\min\{\lambda_{\min}(B^{-1}A),\lambda_{\min}((I-B)^{-1}(I-A))\}}\right)$

is invariant under orthogonal congruence and symmetry under $X \mapsto I-X$ . This provides both theoretical underpinning and computational advantage over traditional affine-invariant Riemannian metrics when dealing with degenerate cases. Geodesics are affine lines, enabling efficient computation of geometric primitives such as balls or Voronoi diagrams, with direct applications in statistics, imaging, and clustering (Karwowski et al., 20 Aug 2025).

Within the natural exponential family framework, the Gaussian law is uniquely characterized (up to translation) by the property that its cumulative distribution function can be expressed as an exponential tilt integral over a base Gaussian measure. This is underpinned by a rigidity result: only Moebius transformations preserve the associated cross ratio, isolating the Gaussian and Gamma families as possessing this exact property among continuous exponential families (Letac, 2018).

5. Generalized Integrals, Kernel Structures, and Functional Analytic Extensions

The extended Gaussian family underpins a wide array of integral and analytic calculations, both in probability and in applied analysis.

Generalized Gaussian integrals of the form

$\int_0^\infty e^{-x^n}dx = \frac{1}{n}\Gamma\left(\frac{1}{n}\right)$

and Gaussian-like integrals incorporating arbitrary continuous functions—either in the exponent or as weights—span a spectrum of functionals involving error functions, Bessel functions, or the Gamma function. These integrals are crucial in various domains from statistical mechanics to quantum theory and model real-world heavy-tail behavior (Pant et al., 11 Aug 2025).

The set of functions for which convolution with an isotropic Gaussian kernel yields closed-form expressions (the "Extended Gaussian Kernel Family," Editor's term) includes not only polynomials (via Hermite polynomial structure) and Gaussian radial basis functions, but also certain trigonometric and linearly-parameterized functions. This class enables exact or efficient computation of diffusion processes, scale-space representations, and smoothing in numerous applications (Mobahi, 2016).

6. Multivariate and Hierarchical Generalization

Several works extend the univariate Gaussian family to multivariate and structurally complex settings:

The construction of multivariate reciprocal inverse Gaussian (MRIG) distributions, via integration over positive-definite constraints involving symmetric matrices with non-positive off-diagonal components, retains key stability properties (under marginalization and conditioning), generalizes the 1D RIG, and is intimately related to tree-structured random graphs and Laplacian-like interactions. In this context, explicit connections to Bessel (MacDonald) functions provide computable formulas for tree graphs (Letac et al., 2017).
Unified GP models whose likelihoods are in a multivariate exponential family allow the development of processes such as Von Mises GP for angular data and Dirichlet GP for simplex-valued outputs, each extending classical Gaussianity and expanding application to angular, compositional, or otherwise constrained domains (Chan, 2013, Klami et al., 2012).

7. Transformations, Algebraic Structure, and Invariance

The extended Gaussian family is structurally robust under a variety of algebraic and analytic transformations:

The beta-gamma algebra unifies fixed-point, invariance, and Stein-type characterizations, showing that the Gaussian law arises as a unique fixed point under specific transformations (e.g., zero-bias) and as an attractor in algebraic frameworks. This perspective strengthens the ties between probabilistic symmetries, geometric invariance, and analytical characterizations, and extends to other exponential family members (Pitman et al., 2012).
The comparison of output distributions from Gaussian and Poisson (or other smoothing) channels indicates that small total variation distances are reflected quantitatively across channels, with exponents capturing the relative strength of smoothing—even when the underlying output spaces (continuous vs. discrete) differ. These results facilitate optimal transport analyses and improved estimation in mixture models (Teh et al., 2023).

In summary, the extended Gaussian family comprises a unified, richly structured assemblage of distributions, processes, and analytic tools that generalize the classical Gaussian framework along parametric, geometric, algebraic, and functional axes. This generalization affords robust modeling of tail behavior, dependence structure, degeneracy, and multivariate complexity, all while retaining—when possible—the analytical tractability, invariance, and interpretability characteristic of the Gaussian paradigm. The theoretical developments, explicit formulas, and structural invariances highlighted above provide effective methodologies for statistical modeling, stochastic process theory, geometric inference, and applied analysis across a wide array of disciplines.