Discrete-time models for statistically self-similar signals

Wide-sense statistical self-similarity in continuous-time random processes is defined through invariance of its first-order and second-order statistics to scaling in time. Since scaling has an unambiguous definition in continuous-time but not in discrete-time, researchers have provided various definitions of discrete-time self-similarity without reference to scaling. This paper proposes a discrete-time continuous-dilation scaling operator and develops a framework based on it for formulating statistical self-similarity from first principles in a manner analogous to the continuous-time development. Relationship between the resulting model and fractional order transfer function systems is presented. The potential for using this model in applications involving long-range dependent phenomena is explored.