The aim of this monograph is to show how random sums (that is, the summation of a random number of dependent random variables) may be used to analyse the behaviour of branching stochastic processes. The author shows how these techniques may yield insight and new results when applied to a wide range of branching processes. In particular, processes with reproduction-dependent and non-stationary immigration may be analysed quite simply from this perspective. On the other hand some new characterizations of the branching process without immigration dealing with its genealogical tree can be studied. Readers are assumed to have a firm grounding in probability and stochastic processes, but otherwise this account is self-contained. As a result, researchers and graduate students tackling problems in this area will find this makes a useful contribution to their work.

Inhalt

I. Sums of a Random Number of Random Variables.- 1. Sampling sums of dependent variables and mixtures of infinitely divisible distributions.- . Sums of a random number of random variables.- . Multiple sums of dependent random variables.- . Sampling sums from a finite population.- 2. Limit theorems for a sum of randomly indexed sequences.- . Sufficient conditions.- . Necessary and sufficient conditions.- . An application.- 3. Necessary and sufficient conditions and limit theorems for sampling sums.- . Convergence theorems.- . The rate of convergence.- II. Branching Processes with Generalized Immigration.- 1.Classical models of branching processes.- . Bellman-Harris processes.- . Moments and extinction probabilities.- . Asymptotics of non-extinct Ion probability and exponential unit distribution.- . Branching processes with stationary immigration.- . Continuous tine branching processes with immigration.- 2 General branching processes with reproduction dependent immigration.- . The model.- . The main theorem.- . The proof of the twin theorem.- . Applications of the main theorem.- 3.Discrete time processes.- . The model.- . Limit theorems for discrete time processes.- . Some examples.- .Randomly stopped immigration.- 4.Convergence to Jirina processes and transfer theorems for branching processes.- . The model.- . The main theorem and corollaries.- . The proof of the main theorem.- III. Branching Processes with Time-Dependent Immigration.- 1.Decreasing immigration.- . The main theorem.- . The proof of the main theorem.- . State-dependent immigration.- 2.Increasing immigration.- . The process with Infinite variance.- . The process with finite variance.- 3.Local limit theorems.- . Occupation of an increasing state.- . Occupation of a fixed state.- IV. The Asymptotic Behavior of Families of Particles in Branching Processes.- 1. Sums of dependent indicators.- . Sums of functions of independent random variables.- . Sampling sums of dependent indicators.- 2.Family of particles in critical processes.- . The model.- . Limit theorems.- 3.Families of particles in supercritical and subcritical processes.- . Supercritical processes.- . Subcritical processes.- References.