Linear Multifractional Stable Motion: Representation via Haar basis

The goal of this paper is to provide a wavelet series representation for Linear Multifractional Stable Motion (LMSM). Instead of using Daubechies wavelets, which are not given in closed form, we use a Haar wavelet, thus yielding a more explicit expression than that in Ayache and Hamonier (in press).The main ingredient of this construction is a Haar expansion of the integrands which define the high and low frequency components of the SαSSαS random field XX generating LMSM. Then, by using Abel summation, we show that these series are a.s. convergent in the space of continuous functions. Finally, we determine their a.s. rates of convergence in the latter space.In the end, the representations of the high and low frequency components of XX, provide a new method for simulating the high and low frequency components of LMSM. Moreover, this new way is faster than the way based on Fast Fourier Transform (Le Guevel and Levy-vehel, 2012; Stoev and Taqqu, 2004).