One-Dimensional Branching Processes

Abstract

A one-dimensional CB-process is a Markov process with branching property taking values from the positive half line. A more general model is the CB-process with immigration which deals with the situation where immigrants may come from outer sources. In this chapter, we first give some characterizations of the extinction probabilities and evolution rates of CB-processes. Then we prove some conditional limit theorems for the processes that extinguish with strictly positive probability. In particular, we shall see that a class of CB-processes with immigration can be obtained from those without immigration by conditioning on non-extinction. The proofs of those theorems are based on the asymptotic analysis of the cumulant semigroup and are easier than their discrete-state counterparts given as in Athreya and Ney (1972). The greater tractability of the CB-processes arises because both their time and state spaces are smooth, and the distributions which appear are infinitely divisible. In this sense, the continuous-state models provide a more economical way to establish the nicest conditional limit theorems for branching processes. We also show that the CB-process with immigration arises naturally as the scaling limit of a sequence of discrete Galton–Watson branching processes with immigration. The contents of this chapter are helpful for the reader in developing the intuitions of Dawson–Watanabe superprocesses.