| In this thesis, we discuss tail asymptotics of the stationary distribution, including rough decay called decay in a weak sense and exact decay, polynomial ergodicity and factorization of generating functions of the transition probability for the GI/G/1-type Markov chain with finitely many background states. In chapter 3, we derive the decay rate in a weak sense for the stationary distribution which is the largest one of { 1fA+, 1fB+, 1a }. The results show that the smallest singularities of A +(z), B+( z) and [I - A(z)] -1 play an important role in determining decay rate for the stationary distribution, and both boundary components of the transition probability matrix and non-boundary components of the transition probability matrix affect the tail behavior. In chapter 4, we further analyze sufficient conditions on an exact geometric decay, and provide sufficient conditions for other exact decay when the stationary distributions is light-tailed but not exact asymptotics. Especially, we give more specific description about the effect of the boundary components of the transition probability matrix on exact decay. In chapter 5, we provide two necessary and sufficient conditions for polynomial ergodicity, and also obtain a lower bound for the first hitting time, which is useful to discuss ergodicity, especially for necessary conditions. In chapter 6, we provide factorization results for the generating functions of the transition probability matrix. In chapter 7, we provide some concluding remarks and future work based on the work in this thesis. |
