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The Power Problem for Generalized Gamma Convolutions (GGC) and Related Questions

Tord Sjödin

TL;DR

This paper addresses whether the generalized gamma convolution class ($GGC$) is closed under taking $q$-th powers for $q>1$, resolving the long-standing Power Problem. The authors prove that if $X\sim GGC$, then $X^q\sim GGC$ by decomposing $X$ into finite sums of gamma variables, employing Bondesson's $GGC$ characterization via hyperbolically completely monotone (HCM) Laplace transforms, and establishing a key CM structure through a detailed induction (with Lemma 2 supporting the step). The result is leveraged to show closure under sums and products of powered $GGC$ variables, extend the framework to symmetric extended GGCs (symEGGC), and provide new proofs and representations for related transformations such as $e^X-1$ and Bondesson’s theorems. Overall, the work broadens the structural understanding of $GGC$s, enabling new probabilistic constructions and potentially impacting applications in infinite divisibility and related stochastic models.

Abstract

The class of generalized gamma convolutions (GGC) is closed with respect to (wrt) change of scales, weak limits and addition and multiplication of independent random variables. Our main result adds the new property that GGC is also closed wrt q-th powers, q>1. The proof uses explicit formulas for the densities of finite sums of independent gamma variables, hyperbolically completely monotone functions (HCM) and the Laplace transform. The result is applied to sums and products of independent gamma variables and to symmetric extended GGC (symEGGC).

The Power Problem for Generalized Gamma Convolutions (GGC) and Related Questions

TL;DR

This paper addresses whether the generalized gamma convolution class () is closed under taking -th powers for , resolving the long-standing Power Problem. The authors prove that if , then by decomposing into finite sums of gamma variables, employing Bondesson's characterization via hyperbolically completely monotone (HCM) Laplace transforms, and establishing a key CM structure through a detailed induction (with Lemma 2 supporting the step). The result is leveraged to show closure under sums and products of powered variables, extend the framework to symmetric extended GGCs (symEGGC), and provide new proofs and representations for related transformations such as and Bondesson’s theorems. Overall, the work broadens the structural understanding of s, enabling new probabilistic constructions and potentially impacting applications in infinite divisibility and related stochastic models.

Abstract

The class of generalized gamma convolutions (GGC) is closed with respect to (wrt) change of scales, weak limits and addition and multiplication of independent random variables. Our main result adds the new property that GGC is also closed wrt q-th powers, q>1. The proof uses explicit formulas for the densities of finite sums of independent gamma variables, hyperbolically completely monotone functions (HCM) and the Laplace transform. The result is applied to sums and products of independent gamma variables and to symmetric extended GGC (symEGGC).
Paper Structure (7 sections, 8 theorems, 37 equations)

This paper contains 7 sections, 8 theorems, 37 equations.

Table of Contents

  1. AMS 2010 Subject Classsification:
  2. Key words and phrases:
  3. Introduction
  4. Background
  5. Powers
  6. Applications
  7. Acknowledgement

Key Result

Proposition 1

A function $\phi (s)$ defined on $(0,\infty )$ such that $\phi (0+)=1$ is the Laplace transform of a GGC if and only if $\phi$ is HCM.

Theorems & Definitions (8)

  • Proposition 1
  • Lemma 1: Feller F, Criterium 2, p. 441
  • Lemma 2
  • Lemma 3: Feller F, Chap. XIII.I,Theorem 1, p. 408
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4