LIMIT THEOREMS FOR ONE CLASS OF ERGODIC MARKOV CHAINS
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Abstract
The goal of this dissertation is to develop some classical limit theorems for the additive functionals of the homogeneous Markov chains in the special class of the so-called, Loop Markov Chains. The additive functionals of the Markov chains have the numerous applications; especially in Mathematical Finance, Optimal control, and Random game theory. The first limit theorems for the finite Markov chains were proven by the founder of the theory Andrei Markov, later on by A. Kolmogorov, W. Döblin, J. Doob, W. Feller and many other experts on the topic. However, the situation with infinite Markov chains is more complicated including the case of Loop Markov chains. Therefore, we present a detailed work to prove the limit theorems for the Loop Markov chains in this dissertation.This dissertation consists of five chapters. Our most significant contribution to the theory is presented in chapters four and five after we familiarize the reader on the essentials of the theory by reviewing the preliminary work given by others in the first three chapters. The structure of this dissertation is organized as follows. In the first chapter, we provide an intuitive background for the theory and introduce the case of Loop Markov chains. We address the difficulties we face during the construction of the limit theorems, especially in the case of the Gaussian limiting law for the Loop Markov chains. Since we construct the theory for both discrete and continuous-time Loop Markov chains, we review the essentials on both cases. Later on in the third chapter we give the specifics on the Döblin method and the Martingale approach to prove the CLT. In the fourth chapter, we introduce three models of Loop Markov chains, namely, Discrete-time Loop Markov chain with countable phase space, Continuous-time Loop Markov chain with countable phase space and Continuous-time Loop Markov chain with continuous phase space, which are the main objectives of this dissertation. We present the CLT for the first two models and calculate the corresponding limiting variance by using both the Döblin method and the Martingale approach. Moreover, we talk about the Random Number Generators (RNG’s) which are appropriate applications of the models constructed on the countable phase. Lastly, in chapter five we analyze and present a complete work on the convergence to the Stable limiting laws on both countable and continuous phase space, in which the latter case enlightens the complexity of the third model.