Random Sums and Branching Stochastic Processes

I. Sums of a Random Number of Random Variables.- §1.1. Sampling sums of dependent variables and mixtures of infinitely divisible distributions.- §1a. Sums of a random number of random variables.- §1b. Multiple sums of dependent random variables.- §1c. Sampling sums from a finite population.- §1.2. Limit theorems for a sum of randomly indexed sequences.- §2a. Sufficient conditions.- §2b. Necessary and sufficient conditions.- §2c. An application.- §1.3. Necessary and sufficient conditions and limit theorems for sampling sums.- §3a. Convergence theorems.- §3b. The rate of convergence.- II. Branching Processes with Generalized Immigration.- §2.1.Classical models of branching processes.- §1a. Bellman-Harris processes.- §1b. Moments and extinction probabilities.- §1c. Asymptotics of non-extinct Ion probability and exponential unit distribution.- §1d. Branching processes with stationary immigration.- §1e. Continuous tine branching processes with immigration.- §2.2 General branching processes with reproduction dependent immigration.- §2a. The model.- §2b. The main theorem.- §2c. The proof of the twin theorem.- §2d. Applications of the main theorem.- §2.3.Discrete time processes.- §3a. The model.- §3b. Limit theorems for discrete time processes.- §3c. Some examples.- §3d.Randomly stopped immigration.- §2.4.Convergence to Jirina processes and transfer theorems for branching processes.- §4a. The model.- §4b. The main theorem and corollaries.- §4c. The proof of the main theorem.- III. Branching Processes with Time-Dependent Immigration.- §3. 1.Decreasing immigration.- §1a. The main theorem.- §1b. The proof of the main theorem.- §1c. State-dependent immigration.- §3.2.Increasing immigration.- §2a. The process with Infinite variance.- §2b. The process with finite variance.- §3.3.Local limit theorems.- §3a. Occupation of an increasing state.- §3b. Occupation of a fixed state.- IV. The Asymptotic Behavior of Families of Particles in Branching Processes.- §4.1. Sums of dependent indicators.- §1a. Sums of functions of independent random variables.- §1b. Sampling sums of dependent indicators.- §4.2.Family of particles in critical processes.- §2a. The model.- §2b. Limit theorems.- §4.3.Families of particles in supercritical and subcritical processes.- §3a. Supercritical processes.- §3b. Subcritical processes.- References.