Stochastic integral representation and properties of the wavelet coefficients of linear fractional stable motion
For a linear fractional \(\alpha \)-stable motion \(\{X(t), t\in T\},\;T\subset R\), with scaling index \(H\) (\(0<\alpha \leq 2,\;0<H<1\)) and symmetric \(\alpha \)-stable random measure the discrete wavelet coefficients are defined as \(D_{j,k}= \int _R X(t)\psi _{j,k}(t) dt\), \(j,k\in Z\), where \(\psi _{j,k}(t)=2^{-j/2}\psi (2^{-j}t-k)\) and \(\psi \) is a bounded real function with compact support and at least one null moment. A stochastic integral representation of \(X\) is used to describe the wavelet coefficients as \(\alpha \)-stable integrals when \(H-1/\alpha >-1\). The process \(k\to D_{j,k}\) is proved to be strictly stationary, and a self-similarity property of the wavelet coefficients of \(X\) is shown.
scientific article; zbMATH DE number 2000167