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Random Sequences
Published in Athanasios Christou Micheas, Theory of Stochastic Objects, 2018
Example 6.23 (Branching processes) Let μ be a probability distribution on Z0+. A branching process with branching distribution μ (also known as the offspring distribution) is a Markov chain on Z0+ having transition probability measures μx = μ*x. Branching processes can be interpreted as population models (like Birth-Death processes). More precisely, if the process is at state x we think of the population as having x members each reproducing independently. During a “birth” the member dies and produces a random number of offspring with μ({k}) denoting the probability that there are k offspring from a particular individual. Thus, the convolution μ*x is the distribution of the upcoming population given that the population is currently at state x.
Risk-sensitive control of branching processes
Published in IISE Transactions, 2021
Since their introduction by Bienaymé and then by Galton and Watson in the 19th century, branching processes and their generalizations have attracted considerable attention from various fields including mathematics, management sciences, economics, finance, biology, and epidemiology. A branching process is a stochastic process that represents the evolution of a population of individuals (or entities) whose reproduction, survival, and death dynamics are governed by probabilistic laws. In a discrete-time branching process, the population consists of a non-negative number of individuals at any discrete time period, and each of these individuals can survive or fail to survive to the next period, and can generate a random number of individuals that join the population in the next period. It is often assumed that individuals reproduce independently, according to the same probability distribution. This assumption allows the formulation of any branching process as a discrete-time Markov process by letting the state variable be the population size.