Poisson point process limits in size-biased Galton-Watson trees
Let \(Z_t\), \(t\geq 0,\) be a continuous-time critical binary Galton-Watson branching process starting with a single founding ancestor, and let \(\widehat{T}_t\) be the tree of the process size-biased according to the number of particles at time \(t.\) By keeping track of 1) the times when different families grow out of the distinguished line of descent of a particle picked at random among existing at time \(t,\) 2) the sizes of the respective families at time \(t,\) the author constructs a point process in a time-size plane and studies its properties for the cases of single-type and some multi-type continuous-time Galton-Watson processes. As a corollary, he shows in particular, that the number of edges of height \(t\) in \(\widehat{T}_t\) has a gamma limit law with shape parameter 2.
scientific article; zbMATH DE number 1549920
scientific article