Basic Estimation of Markovian Pseudo Long-Range Dependent Processes

The pseudo self similar processes are quite attractive due to their simplicity but the question we are interested in this paper concerns the basic estimation of such models. How do the standard estimators (sample mean and variance) converge with time? This will give us an indication about the time we have to collect data in order to accurately model them. With no surprise we notice that this is dependant of the Hurst parameter of course and on the number of states the model has (which defines the domain in which the behavior is self-similar). One has to collect more data with higher Hurst parameters and with more states in the Markov chain to accurately estimate the mean and variance of the process. Outside the domain where the process is self similar, standard statistics methods apply.