Some applications of the Mellin transform to branching processes

2007/06/18 by Wolfgang P. Angerer, Angerer, Wolfgang P.
Mathematics · #44A05 #60J80 #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:44A05 #msc:60J80

paper · pdf · doi:10.48550/arxiv.0706.2638

Proof of Theorem 6 revised; minor errors corrected

openalex publication_date 2007/06/18 · arxiv created 2008/04/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a Mellin transform of functions which live on all of \bR and discuss its applications to the limiting theory of Bellman-Harris processes, and specifically Luria-Delbrück processes. More precisely, we calculate the life-time distribution of particles in a Bellman-Harris process from their first-generation offspring and limiting distributions, and prove a formula for the Laplace transform of the distribution of types in a Luria-Delbrück process in the Mittag-Leffler limit. As a by-product, we show how to easily derive the (classical) Mellin transforms of certain stable probability distributions from their Fourier transform.

Related

• Refined properties for the Mellin transform, with applications to convergence of families obtained by biasing or by the stationary excess operator
• Further studies on square-root boundaries for Bessel processes
• Convergence of a Critical Multitype Bellman--Harris Process with Finite-Mean Lifetimes
• Asymptotic Results for Rough Continuous-state Branching Processes
• A two-type Bellman--Harris process initiated by a large number of particles