Self similar stochastic processes

Abstract

The thesis is concerned with the development of a representation and the behaviour of self similar stochastic processes, and for the testing of a Gaussian sequence with finite variance for self similarity. By considering limits of sums of nonlinear functions of Gaussian sequences, which display a long term dependence structure, a particular class of self similar process is realised together with part of its domain of attraction. Fractional Gaussian noise has self similar increments and is part of the above class. After reviewing results of R/S analysis, often used to determine long range dependence, a β-optimal test is derived to test the hypothesis of white noise. If a process is self similar, but has stationary independent increments, then its increments belong to a stable law. A review of a canonical representation for these processes is given and their sample path properties briefly discussed.