Additive Functionals Of The First Hitting Time For Discrete-time Single-birth Processes

Abstract:The discrete-time single-birth processes are a class of important Markov processes. The class of processes are often used to model the reality phenomenon and are also used to study more complicated Markov processes. Therefore, investigating the single-birth processes is of theoretical and practical meanings.There are three main contents in this thesis. First, we introduce the main results of the additive functionals for the discrete-time birth-death processes (see Chapter3). Secondly, we extend the above results to more complicated single-birth processes. We study the additive functionals of the first hitting time and the central limit theorem for discrete-time single-birth processes on a countable state space by using the theories of ergodicity and the minimal nonnegative solution. Then we get a sufficient condition for the central limit theorem for the discrete-time single-birth processes (see Chapter4). Thirdly, we obtain the explicit expressions of the asymptotic variance and the sensitivity quantity by using the theory of Poisson’s equation. In the end, these results are applied to the discrete-time birth-death processes and the embedded chains of the GI/M/1queues (see Chapter5).