| It is well known that random walks have many important applications in insurance, finance, queues, storage processes and branch processes. An important subject in the study of random walks is asymptotics of random walks with heavy-tailed increments. Through people's contributions in the past decades, there are rich results in the subject. Recently, the local asymptotics of random walks has caused people's research interests due to the need of theory and application. In this thesis, we study the local asympototics for maxima of Markov modulated random walks with heavy-tailed increments. Furthermore, we apply the results to insurance and queueing theory, and extend Markov renewal theorems using the similar method.In Chapter 1, we introduce the background and present situation of problems, state the main results and the structure of the thesis.In Chapter 2, we introduce two classes of heavy-tailed distributions: (?) class and△—subexponential class, and extend some basic properties.In Chapter 3, we investigate the local asympototics for maxima of Markov modulated random walks with heavy-tailed increments and application. We provide the conditions sufficient and necessary for maxima of Markov modulated random walks with long-tailed increments to have the certain asymptotic behavior, which is an extension and improvement of the existing results of Jelenkovic and Lazar (1998) and Asmussen et al. (2002). Finally, we give their applications in insurance and queues.In Chapter 4, we consider Markov renewal processes with heavy-tailed sojourn time. Based on the result of Asmussen (2003), we provide the local asympototic expression of Markov renewal measure for the processes whose sojourn time belongs to two classes of heavy-tailed distributions, and extend the key renewal theorem. |
