Branching Processes which Grow Faster than Binary Splitting

摘要

With a branching process (Zn) one may associate a (random) family tree: there is a distinguished node called the root associated to the ancestor, which is connected to Z1 first level nodes. In general, each nth level node is connected to the n+1th level nodes associated with the offspring of the corresponding particle at time n. The extinction problem can then be interpreted as the question 'with what probability does the family tree contain the unary tree as a subtree,' where the unary tree is the family tree associated to the (deterministic) branching process where each particle gives rise to exactly one particle (the offspring distribution is determined by p1 = 1). Of concern is the question 'with what probability does the family tree contain the binary tree as a subtree,' where the binary tree is the family tree associated to the branching process with offspring distribution given by p2 = 1. If this probability is positive, then it can be said that the branching process grows faster than binary splitting, or has the binary splitting property. (Copyright (c) 1988 by Faculty of Technical Mathematics and Informatics, Delft, The Netherlands.)