Power closure of generalized gamma convolutions: a proposed proof

Abstract

We present a proposed proof that generalized gamma convolutions are closed under every real power greater than one. The argument varies the exponent in the logarithmic Laplace transform. At a zero of an analytic Thorin density, its exponent derivative is expressed as the expectation of a positive functional of a gamma random measure. Regularized contour integrals and a gamma covariance identity yield this formula; the covariance identity is also derived by finite partitions and ordinary integration by parts. Uniform endpoint estimates and zero-freeness give persistence for compact gamma rate mixtures. A smoothing construction, successive weak limits, and iteration reduce arbitrary generalized gamma convolutions to this setting. The argument also gives strictly positive real-analytic Thorin densities for powers greater than one of finite gamma sums, with total mass equal to the input shape divided by the exponent. Generative-AI disclosure and author responsibility. OpenAI's ChatGPT was used for mathematical development, internal proof checking, calculations, and drafting; GPT-6 Astra Pro was used for the final editorial and submission preparation. The author directed this work and is responsible for the final manuscript. This is a proposed proof for independent mathematical review, not a claim of independent verification. 2020 Mathematics Subject Classification: Primary 60E07; Secondary 60E10, 60G57. Files: the compiled manuscript GGC_Power_Closure_arxiv.pdf (36 pages) and the source archive GGC_Power_Closure_arxiv_source.zip, which contains the self-contained LaTeX source main.tex (bibliography included; compiles with pdfLaTeX).