Stable self-similar and locally self-similar random processes: Stochastic properties, parameter estimation, and simulation

The contributions of this thesis are in the area of stochastic processes with stable distributions. The focus is on statistically self-similar, asymptotically and locally self-similar processes. It consists of two parts. In part one, wavelet-based estimators for the self-similarity parameter of the Linear Fractional Stable Motion (LFSM) are studied. Their consistency and asymptotic normality are established. Similar statistical results are obtained in the case of Hurst parameter estimation for Fractional Autoregressive Integrated Moving Average (FARIMA) time series with stable innovations. The performance of these estimators is evaluated over simulated data. This part of the dissertation concludes with the presentation of efficient methods for computer simulation of the LFSM and FARIMA processes, based on the Fast Fourier Transform.; The second part of the thesis is devoted to the study of a class of locally self-similar processes called Linear Multifractional Stable Motions. These processes extend the Gaussian Multifractional Brownian Motions and provide versatile models where the index of self-similarity can vary with time and where the distributions can have heavy tails and be non-symmetric. Results on the stochastic properties, boundedness and Holder regularity of the paths of these processes are established. New results for the class of Gaussian Multifractional Brownian Motions are obtained.