Limit theorems for branching Markov processes
In this thesis we develop some limit theorems for branching processes. In the first chapter we present basic definitions, some known results in branching processes, and renewal theory both of which play an important role in later chapters. In chapter two we consider a supercritical branching Markov process in which particles move according to a process with stationary independent increments. We present conditions for convergence of the normalized empirical distribution of the positions of the particles at time t, as t goes to infinity. In chapter three we prove a central limit theorem for functionals of the empirical age distribution of supercritical Bellman-Harris processes. In other words, let f be a real valued function on the nonnegative reals that integrates to zero with respect to the stable age distribution in a supercritical Bellman-Harris process with no extinction. We present sufficient conditions for the asymptotic normality of the mean of f with respect to the empirical age distribution at time t. In the last chapter we deal with Galton-Watson process whose particles are moving according to a Markov process. The Markov process is assumed positive recurrent for the discrete state space case and Harris-recurrent for continuous case. We prove first the law of large numbers for the empirical position distribution and then discuss the large deviation aspects of these convergences.