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Maximal Displacement of Critical Branching Symmetric Stable Processes

We consider a critical continuous-time branching process (a Yule process) in which the individuals independently execute symmetric $α-$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 $P\{M\geq x \}\sim (2/α)^{1/2}x^{-α/2}$ as $x \rightarrow \infty$.