Chapter 1: Branching Processes 1.1 Introduction: Background and Definitions In 1874, Sir Francis Galton and H.W. Watson developed a mathematical model to study the extinction of family names in England. This model, known as the Galton-Watson Branching Process, was the first significant application of probability theory to study random fluctuations in family or population development. The model describes the reproduction of objects (such as humans, bacteria, or neutrons) across generations. Starting with an initial set of objects (zeroth generation), each member produces offspring according to a fixed probability distribution. The total number of objects in each generation forms a sequence of random variables, representing the evolution of the population. The process is modeled as a stochastic process where each generation branches into the next. The number of offspring produced by each individual follows a fixed distribution, and the offspring numbers are independent across individuals. The process is characterized by the sequence $ (X_n)_{n \geq 0} $, where $ X_n $ is the number of objects in the n-th generation. The process is of interest for both finite-time and asymptotic behavior, including the expected size and variance of each generation, and the probability of eventual extinction. The initial population is assumed to be fixed, with $ X_0 = k_0 $, a positive integer. Each individual produces offspring independently according to the distribution $ \widetilde{p} $. The total number of offspring in the first generation is $ X_1 = Y_1^0 + \cdots + Y_{k_0}^0 $, where $ Y_i^0 $ are i.i.d. random variables with distribution $ \widetilde{p} $. If $ X_1 = 0 $, the family becomes extinct. Otherwise, the process continues with the next generation. The key question is the probability of eventual extinction of the family.Chapter 1: Branching Processes 1.1 Introduction: Background and Definitions In 1874, Sir Francis Galton and H.W. Watson developed a mathematical model to study the extinction of family names in England. This model, known as the Galton-Watson Branching Process, was the first significant application of probability theory to study random fluctuations in family or population development. The model describes the reproduction of objects (such as humans, bacteria, or neutrons) across generations. Starting with an initial set of objects (zeroth generation), each member produces offspring according to a fixed probability distribution. The total number of objects in each generation forms a sequence of random variables, representing the evolution of the population. The process is modeled as a stochastic process where each generation branches into the next. The number of offspring produced by each individual follows a fixed distribution, and the offspring numbers are independent across individuals. The process is characterized by the sequence $ (X_n)_{n \geq 0} $, where $ X_n $ is the number of objects in the n-th generation. The process is of interest for both finite-time and asymptotic behavior, including the expected size and variance of each generation, and the probability of eventual extinction. The initial population is assumed to be fixed, with $ X_0 = k_0 $, a positive integer. Each individual produces offspring independently according to the distribution $ \widetilde{p} $. The total number of offspring in the first generation is $ X_1 = Y_1^0 + \cdots + Y_{k_0}^0 $, where $ Y_i^0 $ are i.i.d. random variables with distribution $ \widetilde{p} $. If $ X_1 = 0 $, the family becomes extinct. Otherwise, the process continues with the next generation. The key question is the probability of eventual extinction of the family.