Persistence exponents via perturbation theory : autoregressive and moving average processes

In this thesis, the persistence problem in the context of Markov chains is studied. We are mainly concerned with processes where the persistence probability converges to zero at exponential speed and we are interested in the rate of decay, the so-called persistence exponent. For the main results, we use methods from perturbation theory. This approach is completely new in the field of persistence. For this reason, we provide a mostly self-contained presentation of the used theorems of perturbation theory. We show that the persistence exponent of an autoregressive process of order one can be expressed as a power series in the parameter of the autoregressive process. Additionally, we derive an iterative formula for the coefficients of this power series representation. For moving average processes of order one similar results as in the autoregressive case are derived