Stable self-similar processes with stationary increments.

摘要

The focus of this thesis is on stochastic processes which are self-similar, have stationary increments and whose finite-dimensional distributions are stable. In the Gaussian stable case, there is only one self-similar process with stationary increments, called fractional Brownian motion. In the non-Gaussian stable case, there are infinitely many different self-similar processes with stationary increments, for example, linear fractional stable motion and harmonizable fractional stable motion.; The thesis is divided into five parts dealing with different topics related to self-similar processes with stationary increments, namely: (1) Weierstrass-Mandelbrot processes; (2) convergence of renewal reward processes; (3) fractional Brownian motion and fractional calculus; (4) structure of non-Gaussian stable self-similar processes with stationary increments; and (5) wavelet-based estimators in linear fractional stable motion.; In the first part of the thesis, which covers Chapters 1, 2 and 3, we study the convergence of modified and randomized versions of the celebrated Weierstrass function. Depending on the type of randomizations used, these versions can have fractional Brownian motion or harmonizable fractional stable motion as their limits.; In the second part of the thesis, we study renewal reward processes which are of interest in telecommunications. In Chapter 4, we identify the limit of these processes when the renewals and the rewards are heavy-tailed. The limit is a new non-Gaussian stable selfsimilar process with stationary increments. In Chapter 5, we refine convergence results for renewal reward processes by letting some of their parameters become large at the same time.; In the third part of the thesis, which covers Chapters 6, 7, 8 and 9, we explore numerous connections of fractional Brownian motion and deterministic fractional calculus. We show that these connections have important implications for integration questions and can be used to derive deconvolution and Girsanov formulas and to study the prediction and filtering problems for fractional Brownian motion.; In the fourth part of the thesis, Chapters 10, 11 and 12, we classify a subclass of non-Gaussian stable self-similar processes with stationary increments. We do so by drawing connections of such processes to deterministic flows and then by using the structure of these flows as a basis for a classification scheme.; In the fifth part of the thesis, Chapters 13 and 14, we establish the asymptotic normality of some wavelet-based self-similarity parameter estimators in linear fractional stable motion.