Linear Multifractional Stable Motion: representation via Haar basis

The aim of this paper is to give a wavelet series representation of Linear Multifractional Stable Motion (LMSM in brief), which is more explicit than that introduced in (Ayache & Hamonier 2012). Instead of using Daubechies wavelet, which is not given by a closed form, we use the Haar wavelet. In order to obtain this new representation, we introduce a Haar expansion of the high and low frequency parts of the $\mathcal{S}α\mathcal{S}$ random field $X$ generating LMSM. Then, by using Abel transforms, we show that these series are convergent, almost surely, in the space of continuous functions. Finally, we determine their almost sure rates of convergence in the latter space. Note that these representations of the high and low frequency parts of $X$, provide a new method for simulating the high and low frequency parts of LMSM.

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