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.- §1.1. Sampling sums of dependent variables and mixtures of infinitely divisible distributions.- §1.2. Limit theorems for a sum of randomly indexed sequences.- §1.3. Necessary and sufficient conditions and limit theorems for sampling sums.- II. Branching Processes with Generalized Immigration.- §2.1.Classical models of branching processes.- §2.2 General branching processes with reproduction dependent immigration.- §2.3.Discrete time processes.- §2.4.Convergence to Jirina processes and transfer theorems for branching processes.- III. Branching Processes with Time-Dependent Immigration.- §3. 1.Decreasing immigration.- §3.2.Increasing immigration.- §3.3.Local limit theorems.- IV. The Asymptotic Behavior of Families of Particles in Branching Processes.- §4.1. Sums of dependent indicators.- §4.2.Family of particles in critical processes.- §4.3.Families of particles in supercritical and subcritical processes.- References.