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

Let 0<α⩽2 and let T⊆R. Let {X(t),t∈T} be a linear fractional α-stable (0<α⩽2) motion with scaling index H (0<H<1) and with symmetric α-stable random measure. Suppose that ψ 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 beDj,k=∫RX(t)ψj,k(t) dt, j,k∈Z.We use a stochastic integral representation of the process X to describe the wavelet coefficients as α-stable integrals when H−1/α>−1. This stochastic representation is used to prove that the stochastic process of wavelet coefficients {Dj,k, k∈Z}, with fixed scale index j∈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.