Maximal Displacement of Critical Branching Symmetric Stable Processes

Abstract: We consider a critical continuous-time branching process (a Yule process) in which the individuals independently execute symmetric $\alpha-$stable random motions on the real line starting at their birth points. Because the branching process is critical, it will eventually die out, and so there is a well-defined maximal location $M$ ever visited by an individual particle of the process. We prove that the distribution of $M$ satisfies the asymptotic relation $P{M\geq x }\sim (2/\alpha)^{1/2}x^{-\alpha /2}$ as $x \rightarrow \infty$.