Mellin变换在分支过程中的应用
Some applications of the Mellin transform to branching processes
摘要：
我们引入了一个梅林变换的功能，生活在所有的$\bR$和讨论其应用的Bellman-Harris过程的极限理论，特别是Luria-Delbrück过程。更精确地说，我们计算粒子的寿命分布在Bellman-Harris过程从他们的第一代后代和极限分布，并证明了一个公式的拉普拉斯变换的类型分布在Luria-Delbrück过程中的Mittag-Leffler极限。作为一个副产品，我们展示了如何容易地获得（经典）梅林变换的某些稳定的概率分布从他们的傅里叶变换。
英文摘要：
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.
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