Generalized Gamma Convolutions (GGC)

• Generalized Gamma Convolutions (GGC) are infinitely divisible distributions defined via Thorin measure representations with key closure properties.
• They support algebraic operations like scaling, addition, multiplication, and power transformations, essential for stochastic modeling and financial applications.
• Recent advances extend GGCs to multivariate and matrix settings, enabling robust statistical inference in high-dimensional data contexts.

A generalized gamma convolution (GGC) is a class of infinitely divisible distributions defined either via Laplace–Thorin measure representations, or equivalently as weak limits of finite sums of independent gamma random variables. The univariate GGCs appear as marginal distributions of Thorin subordinators and possess closure under convolution, scaling, weak limits, and more recently, under arbitrary $q$-th powers for $q>1$, as established in recent work (Sjödin, 7 Jan 2026). Multivariate and cone-valued extensions admit rich dependence structures and play a crucial role in stochastic modeling, especially for variance gamma and general Lévy processes, matrix-valued distributions, and emerging applications in high-dimensional statistics and mathematical finance.

1. Characterizing Generalized Gamma Convolutions

A random variable $X\ge0$ is said to be a generalized gamma convolution if its Laplace transform admits the canonical Thorin form: $$\mathbb{E}[e^{-sX}] = \exp\left(- a s + \int_{0}^{\infty} \log\frac{t}{t+s}\; U(dt)\right),\quad s \ge 0,$$ where $a \ge 0$ and $U$ is a nonnegative Radon measure (the Thorin measure) satisfying

$$\int_0^1 |\log t|\,U(dt)<\infty,\qquad \int_1^\infty t^{-1}\,U(dt)<\infty.$$

Equivalently, the class consists of all weak limits of finite sums of independent gamma random variables with arbitrary positive shape and scale parameters. A defining analytic property is that the Laplace transform $\phi_X(s)$ is hyperbolically completely monotone (HCM)—for each $u>0$, the function $H(w) = \phi_X(u v)\,\phi_X(u/v)$, $q>1$0, is completely monotone in $q>1$1 (Sjödin, 7 Jan 2026, Behme et al., 2015).

GGCs form a proper subclass of the nonnegative infinitely divisible laws, containing all gamma, lognormal, positive $q>1$2-stable ($q>1$3), Pareto, and generalized inverse Gaussian (GIG) distributions (Sayit, 2024, Jedidi et al., 2013).

A mixture and series representation is available in terms of an infinite sum of independent exponentially distributed random variables whose rate parameters are sampled according to the Thorin measure, corresponding to a Poisson random measure with intensity $q>1$4 (Polson, 2018, Laverny, 2022).

2. Closure, Structural, and Power Properties

The GGC class is closed under the following operations (Sjödin, 7 Jan 2026):

• Scaling: $q>1$5 for $q>1$6.
• Addition: If $q>1$7 are independent, so is $q>1$8.
• Multiplication: $q>1$9 is GGC when $X\ge0$0 independent GGCs.
• Weak limits: Any weak limit of GGCs is again GGC.
• Division: $X\ge0$1 is also GGC (Behme et al., 2015).

A major advance is the proof that $X\ge0$2 for any $X\ge0$3 if $X\ge0$4 is GGC. The proof uses the HCM characterization and explicit formulas for finite gamma sums, together with an induction on the number of summands, leveraging complete monotonicity under the hyperbolic substitution (Sjödin, 7 Jan 2026).

Table: Fundamental Closure Properties

Operation	Preserved for GGC?	Details
Scaling	Yes	$X\ge0$5 for $X\ge0$6
Addition	Yes	Sum of independent GGCs
Multiplication	Yes	Product of independent GGCs
Power $X\ge0$7 ($X\ge0$8)	Yes	(Sjödin, 7 Jan 2026)
Division	Yes	$X\ge0$9, $$\mathbb{E}[e^{-sX}] = \exp\left(- a s + \int_{0}^{\infty} \log\frac{t}{t+s}\; U(dt)\right),\quad s \ge 0,$$0 and $$\mathbb{E}[e^{-sX}] = \exp\left(- a s + \int_{0}^{\infty} \log\frac{t}{t+s}\; U(dt)\right),\quad s \ge 0,$$1 independent GGC
Weak limit	Yes	Convergence in law

This robust algebraic structure underpins much of the analytic tractability of GGCs, permitting functional transformations and constructions critical for applications.

3. Multivariate, Cone-Valued, and Matrix GGCs

Multivariate GGCs are defined via Thorin measures on $$\mathbb{E}[e^{-sX}] = \exp\left(- a s + \int_{0}^{\infty} \log\frac{t}{t+s}\; U(dt)\right),\quad s \ge 0,$$2 and admit Laplace transforms generalizing the univariate case: $$\mathbb{E}[e^{-sX}] = \exp\left(- a s + \int_{0}^{\infty} \log\frac{t}{t+s}\; U(dt)\right),\quad s \ge 0,$$3 for a proper Thorin measure $$\mathbb{E}[e^{-sX}] = \exp\left(- a s + \int_{0}^{\infty} \log\frac{t}{t+s}\; U(dt)\right),\quad s \ge 0,$$4 and $$\mathbb{E}[e^{-sX}] = \exp\left(- a s + \int_{0}^{\infty} \log\frac{t}{t+s}\; U(dt)\right),\quad s \ge 0,$$5 (Buchmann et al., 2015, Buchmann et al., 2017).

Cone-valued generalizations, especially on the cone $$\mathbb{E}[e^{-sX}] = \exp\left(- a s + \int_{0}^{\infty} \log\frac{t}{t+s}\; U(dt)\right),\quad s \ge 0,$$6 of positive semidefinite matrices, yield models whose Lévy measure admits a spherical-radial decomposition:

$$\mathbb{E}[e^{-sX}] = \exp\left(- a s + \int_{0}^{\infty} \log\frac{t}{t+s}\; U(dt)\right),\quad s \ge 0,$$7

with radial kernel $$\mathbb{E}[e^{-sX}] = \exp\left(- a s + \int_{0}^{\infty} \log\frac{t}{t+s}\; U(dt)\right),\quad s \ge 0,$$8 completely monotone and angular measure $$\mathbb{E}[e^{-sX}] = \exp\left(- a s + \int_{0}^{\infty} \log\frac{t}{t+s}\; U(dt)\right),\quad s \ge 0,$$9 on $a \ge 0$0 (Pérez-Abreu et al., 2012). The matrix $a \ge 0$1 distribution provides a canonical example, with explicit Laplace transforms and tractable moment formulas tied to the Wishart law and the Marchenko–Pastur asymptotic regime.

4. Estimation and Inference in High Dimensions

Statistical inference for GGCs, especially in high dimensions, leverages the semi-parametric convolutional structure of their densities (Laverny, 2022, Laverny et al., 2021). The estimation paradigm involves reduction to a continuum of univariate problems via random projections, with the density in each direction expanded in a Laguerre basis. The key ingredients are:

• Laguerre-integrated square error loss: Matching theoretical and empirical Laguerre coefficients of projected marginals.
• Shifted cumulants optimization: Reduces estimation to fitting moments via low-dimensional, polynomial-in-cumulant representations.
• Random projections and Grassmannian cubatures: Control the approximation error and circumvent the curse of dimensionality. Error scales $a \ge 0$2 for $a \ge 0$3 random directions and $a \ge 0$4 using $a \ge 0$5-cubatures.
• Sparse Thorin measure parametrization: The Thorin measure is discretized as a sum of weighted atoms. Sparsity is enforced by $a \ge 0$6 regularization.
• Stochastic gradient descent in measure space: Wasserstein space interpretations yield convergence rates $a \ge 0$7 for the loss and $a \ge 0$8 for projected cumulants.

This pipeline permits fitting GGC laws to high-dimensional data efficiently and stably, encompassing dependence via the support of the Thorin measure in angular components (Laverny, 2022).

5. Applications and Analytical Structure

GGCs arise naturally as distributions of:

• Mixtures and scale mixtures involving gamma variables, particularly when the mixing density is hyperbolically monotone (HM) of order $a \ge 0$9 (Behme et al., 2015, Sjödin, 2018).
• Exponential functionals of Lévy processes and perpetuity solutions (Behme et al., 2012), leveraging closure properties and explicit Thorin measure constructions.
• Time-fractional and subordinated diffusions: Mellin convolution of generalized gamma densities leads to explicit solutions of time-fractional diffusion equations in terms of nested Bessel integrals or Fox H-functions (D'Ovidio, 2010).
• Variance-Gamma and multidimensional financial models: Multivariate subordination of general Lévy processes by Thorin subordinators generates processes (weak variance–GGC) with flexible jump and dependence structures, supporting option pricing and robust risk aggregation (Buchmann et al., 2015, Buchmann et al., 2016, Buchmann et al., 2017).

6. Structural Properties: Self-Decomposability and Weak Limits

GGCs are self-decomposable, making them admissible as marginal laws of stationary Ornstein–Uhlenbeck processes (Polson, 2018, Buchmann et al., 2017), with precise extensions to weakly subordinated variance–GGC processes, under driftless and mild moment conditions. Moreover, recent results establish that weak convergence in the family of GGCs implies convergence in mean—a property that underlies the robustness of inferred functionals (such as expected utility optimal portfolios) with respect to model misspecification within the GGC class (Sayit, 2024).

7. Open Problems and Future Directions

• Boundary cases for stable laws: The question of whether all negative powers of positive $U$0-stable laws are GGCs remains open, with strong evidence for the range $U$1 and positive powers $U$2 (Jedidi et al., 2013).
• Functional transformations: The closure of GGCs under more general convex or Bernstein functionals is an emerging area (Sjödin, 7 Jan 2026).
• Extensions to operator-valued and infinite-dimensional settings: Cone-valued GGCs suggest extensions to operator self-decomposability and further connections to random matrix theory (Pérez-Abreu et al., 2012).

The GGC framework thus provides a unifying paradigm for infinitely divisible distributions on the positive axis and cones, with deep connections to classical analysis (HCM, Bernstein functions), stochastic process theory, and multivariate statistical modeling.

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Key References:

• "The Power Problem for Generalized Gamma Convolutions (GGC) and Related Questions" (Sjödin, 7 Jan 2026)
• "Estimation of high dimensional Gamma convolutions through random projections" (Laverny, 2022)
• "A class of scale mixtures of gamma(k)-distributions that are generalized gamma convolutions" (Behme et al., 2015)
• "Weak convergence implies convergence in mean within GGC" (Sayit, 2024)
• "On Mixtures of Gamma Distributions, Distributions with Hyperbolically Monotone Densities and Generalized Gamma Convolutions (GGC)" (Sjödin, 2018)
• "Riemann Hypothesis: a GGC factorisation" (Polson, 2018)
• "A Class of Infinitely Divisible Multivariate and Matrix Gamma Distributions and Cone-valued Generalised Gamma Convolutions" (Pérez-Abreu et al., 2012)