This lecture deals with branching process, which is a simple stochastic model describing the time-evolution of a population in which each individual can produce an i.i.d. random number of off-springs.

We begin with the basic extinction-explosion properties of the discrete-time simple branching process on non-negative integers, also called Galton-Watson process (GWP). We introduce the classical tool – generating function and calculate the extinction probabilities. Then we study the asymptotic long-time behaviors of GWPs: Heyde-Seneta Theorem and Kesten-Stigum Theorem for super-critical case, Kolmogorov and Yaglom limits for sub-critical and critical cases. We present short analogs of extinction, generating function, limit behaviors for the continuous-time GWPs.

We also introduce the ideas of Galton-Watson tree and size-biased trees that give another proof of Kesten-Stigum Theorem. We close this course with some classical results, such as law of large numbers and central limit theory, on a related topic – branching random walks.