Stochastic integral representation and properties of the wavelet coefficients of linear fractional stable motion

Let 0 < alpha less than or equal to 2 and let T subset of or equal to R. Let {X(t),t epsilon T} be a linear fractional alpha-stable (0 < alpha less than or equal to 2) motion with scaling index H (0 < H < 1) and with symmetric alpha-stable random measure. Suppose that psi is a bounded real function with compact support [a,b] and at least one null moment. Let the sequence of the discrete wavelet coefficients of the process X be { D-j,D-k = integral(R) X(t)psi(j,k)(t) dt, j, k epsilon Z } We use a stochastic integral representation of the process X to describe the wavelet coefficients as alpha-stable integrals when H - 1/alpha > -1. This stochastic representation is used to prove that the stochastic process of wavelet coefficients {D-j,D-k, k epsilon Z}, with fixed scale index j epsilon Z, is strictly stationary. Furthermore, a property of self-similarity of the wavelet coefficients of X is proved. This property has been the motivation of several wavelet-based estimators for the scaling index H. (C) 2000 Elsevier Science B.V. All rights reserved.