Locally Self-Similar Processes and Their Wavelet Analysis

Introduction A stochastic process Y (t) is defined as self-similar with self-similarity parameter H if for any positive stretching factor c, the distribution of the rescaled and reindexed process c Y (c t) is equivalent to that of the original process Y (t). This means for any sequence of time points t 1 ; : : : ; t n and any positive constant c, the collections fc Y (c t 1 ); : : : ; c Y (ct n )g and fY (t 1 ); : : : ; Y (t n )g are governed by the same probability law. As a consequence, the qualitative features of a sample path of a self-similar process are invariant to magnification or shrinkage, so that the path will retain the same general appearance regardless of the distance from which it is observed. Although self-similar processes were first introduced in a theoretical context by Kolmogorov (1941), statisticians were made aware of the practical applicability of such processes through the work of B.B. Mandelbrot (Mandelbrot and van Ness, 1968; Mandelbrot and Wallis